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The principle of indifference (also called principle of insufficient reason) is a rule for assigning epistemic probabilities. Suppose that there are n > 1 mutually exclusive and collectively exhaustive possibilities. The principle of indifference states that if the n possibilities are indistinguishable except for their names, then each possibility should be assigned a probability equal to 1/n.

In Bayesian probability, this is the simplest non-informative prior. The principle of indifference is meaningless under the frequency interpretation of probability,Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. in which probabilities are relative frequencies rather than degrees of belief in uncertain propositions, conditional upon state information.

Examples

The textbook examples for the application of the principle of indifference are coins, dice, and cards.

In a macroscopic system, at least, it must be assumed that the physical laws which govern the system are not known well enough to predict the outcome. As observed some centuries ago by John Arbuthnot (in the preface of Of the Laws of Chance, 1692),

It is impossible for a Die, with such determin'd force and direction, not to fall on such determin'd side, only I don't know the force and direction which makes it fall on such determin'd side, and therefore I call it Chance, which is nothing but the want of art....

Given enough time and resources, there is no fundamental reason to suppose that suitably precise measurements could not be made, which would enable the prediction of the outcome of coins, dice, and cards with high accuracy: Persi Diaconis's work with coin-flipping machines is a practical example of this.

Coins

A symmetric coin has two sides, arbitrarily labeled heads and tails. Assuming that the coin must land on one side or the other, the outcomes of a coin toss are mutually exclusive, exhaustive, and interchangeable. According to the principle of indifference, we assign each of the possible outcomes a probability of 1/2.

It is implicit in this analysis that the forces acting on the coin are not known with any precision. If the momentum imparted to the coin as it is launched were known with sufficient accuracy, the flight of the coin could be predicted according to the laws of mechanics. Thus the uncertainty in the outcome of a coin toss is derived (for the most part) from the uncertainty with respect to initial conditions. This point is discussed at greater length in the article on coin flipping.

There is also a third possible outcome: the coin could land on its edge. However, the principle of indifference doesn't say anything about this outcome, as the labels head, tail, and edge are not interchangeable. One could argue, though, that head and tail remain interchangeable, and therefore Pr(head) and Pr(tail) are equal, and both are equal to 1/2 (1 - Pr(edge)).

Dice

A symmetric dice has n faces, arbitrarily labeled from 1 to n. Ordinary cubical dice have n = 6 faces, although symmetric dice with different numbers of faces can be constructed; see dice. We assume that the die must land on one face or another, and there are no other possible outcomes. Applying the principle of indifference, we assign each of the possible outcomes a probability of 1/n.

As with coins, it is assumed that the initial conditions of throwing the dice are not known with enough precision to predict the outcome according to the laws of mechanics. Dice are typically thrown so as to bounce on a table or other surface. This interaction makes prediction of the outcome much more difficult.

Cards

A standard deck contains 52 cards, each given a unique label in an arbitrary fashion, i.e. arbitrarily ordered. We draw a card from the deck; applying the principle of indifference, we assign each of the possible outcomes a probability of 1/52.

This example, more than the others, shows the difficulty of actually applying the principle of indifference in real situations. What we really mean by the phrase "arbitrarily ordered" is simply that we don't have any information that would lead us to favor a particular card. In actual practice, this is rarely the case: a new deck of cards is certainly not in arbitrary order, and neither is a deck immediately after a hand of cards. In practice, we therefore shuffle the cards; this does not destroy the information we have, but instead (hopefully) renders our information practically unusable, although it is still usable in principle. In fact, some expert blackjack players can track aces through the deck; for them, the condition for applying the principle of indifference is not satisfied.

Application to continuous variables

Applying the principle of indifference incorrectly can easily lead to nonsensical results, especially in the case of multivariate, continuous variables. A typical case of misuse is the following example.

Suppose there is a cube hidden in a box. A label on the box says the cube has a side length between 3 and 5 cm.
We don't know the actual side length, but we might assume that all values are equally likely and simply pick the mid-value of 4 cm.
The information on the label allows us to calculate that the surface area of the cube is between 54 and 150 cm². We don't know the actual surface area, but we might assume that all values are equally likely and simply pick the mid-value of 102 cm².
The information on the label allows us to calculate that the volume of the cube is between 27 and 125 cm³. We don't know the actual volume, but we might assume that all values are equally likely and simply pick the mid-value of 76 cm³.
However, we have now reached the impossible conclusion that the cube has a side length of 4 cm, a surface area of 102 cm², and a volume of 76 cm³!

In this example, mutually contradictory estimates of the length, surface area, and volume of the cube arise because we have assumed three mutually contradictory distributions for these parameters: a uniform distribution for any one of the variables implies a non-uniform distribution for the other two. (The same paradox arises if we make it discrete: the side is either exactly 3 cm, 4 cm, or 5 cm, mutatis mutandis.) In general, the principle of indifference does not indicate which variable (e.g. in this case, length, surface area, or volume) is to have a uniform epistemic probability distribution.

Another classic example of this kind of misuse is Bertrand's paradox. Edwin T. Jaynes introduced the principle of transformation groups, which can yield an epistemic probability distribution for this problem. This generalises the principle of indifference, by saying that one is indifferent between equivalent problems rather than indifference between propositions. This still reduces to the ordinary principle of indifference when one considers a permutation of the labels as generating equivalent problems (i.e. using the permutation transformation group). To apply this to the above box example, we have three problems, with no reason to think one problem is "our problem" more than any other - we are indifferent between each. If we have no reason to favour one over the other, then our prior probabilities must be related by the rule for changing variables in continuous distributions. Let L be the length, and V be the volume. Then we must have

f(L)=|{\partial V \over \partial L}|f(V)=3L^{2}f(L^{3})

Which has a general solution: $f(L)={K \over L}$ Where K is an arbitrary constant, determined by the range of L, in this case equal to:

K^{-1}=\int _{3}^{5}{dL \over L}=\log({5 \over 3})

To put this "to the test", we ask for the probability that the length is less than 4. This has probability of:

Pr(L<4)=\int _{3}^{4}{dL \over L\log({5 \over 3})}={\log({4 \over 3}) \over \log({5 \over 3})}\approx 0.56

.

For the volume, this should be equal to the probability that the volume is less than 4³ = 64. The pdf of the volume is

f(V^{1 \over 3}){1 \over 3}V^{-{2 \over 3}}={1 \over 3V\log({5 \over 3})}

.

And then probability of volume less than 64 is

Pr(V<64)=\int _{27}^{64}{dV \over 3V\log({5 \over 3})}={\log({64 \over 27}) \over 3\log({5 \over 3})}={3\log({4 \over 3}) \over 3\log({5 \over 3})}={\log({4 \over 3}) \over \log({5 \over 3})}\approx 0.56

.

Thus we have achieved invariance with respect to volume and length. You can also show the same invariance with respect to surface area being less than 6(4²) = 96. However, note that this probability assignment is not necessarily a "correct" one. For the exact distribution of lengths, volume, or surface area will depend on how the "experiment" is conducted. This probability assignment is very similar to the maximum entropy one, in that the frequency distribution corresponding to the above probability distribution is the most likely to be seen. So, if one was to go to N people individually and simply say "make me a box somewhere between 3 and 5 cm, or a volume between 27 and 125 cm, or a surface area between 54 and 150 cm", then unless there is a systematic influence on how they make the boxes (e.g. they form a group, and choose one particular method of making boxes), about 56% of the boxes will be less than 4 cm - and it will get very close to this amount very quickly. So, for large N, any deviation from this basically indicates the makers of the boxes were "systematic" in how the boxes were made.

The fundamental hypothesis of statistical physics, that any two microstates of a system with the same total energy are equally probable at equilibrium, is in a sense an example of the principle of indifference. However, when the microstates are described by continuous variables (such as positions and momenta), an additional physical basis is needed in order to explain under which parameterization the probability density will be uniform. Liouville's theorem justifies the use of canonically conjugate variables, such as positions and their conjugate momenta.

History of the principle of indifference

The original writers on probability, primarily Jacob Bernoulli and Pierre Simon Laplace, considered the principle of indifference to be intuitively obvious and did not even bother to give it a name. Laplace wrote:

The theory of chance consists in reducing all the events of the same kind to a certain number of cases equally possible, that is to say, to such as we may be equally undecided about in regard to their existence, and in determining the number of cases favorable to the event whose probability is sought. The ratio of this number to that of all the cases possible is the measure of this probability, which is thus simply a fraction whose numerator is the number of favorable cases and whose denominator is the number of all the cases possible.

These earlier writers, Laplace in particular, naively generalized the principle of indifference to the case of continuous parameters, giving the so-called "uniform prior probability distribution", a function which is constant over all real numbers. He used this function to express a complete lack of knowledge as to the value of a parameter. According to Stigler (page 135), Laplace's assumption of uniform prior probabilities was not a meta-physical assumption. It was an implicit assumption made for the ease of analysis.

The principle of insufficient reason was its first name, given to it by later writers, possibly as a play on Leibniz's principle of sufficient reason. These later writers (George Boole, John Venn, and others) objected to the use of the uniform prior for two reasons. The first reason is that the constant function is not normalizable, and thus is not a proper probability distribution. The second reason is its inapplicability to continuous variables, as described above. (However, these paradoxical issues can be resolved. In the first case, a constant, or any more general finite polynomial, is normalizable within any finite range: the range [0,1] is all that matters here. Alternatively, the function may be modified to be zero outside that range, as with a continuous uniform distribution. In the second case, there is no ambiguity provided the problem is "well-posed", so that no unwarranted assumptions can be made, or have to be made, thereby fixing the appropriate prior probability density function or prior moment generating function (with variables fixed appropriately) to be used for the probability itself. See the Bertrand paradox (probability) for an analogous case.)

The "Principle of insufficient reason" was renamed the "Principle of Indifference" by the economist Template:Harvs, who was careful to note that it applies only when there is no knowledge indicating unequal probabilities.

Attempts to put the notion on firmer philosophical ground have generally begun with the concept of equipossibility and progressed from it to equiprobability.

The principle of indifference can be given a deeper logical justification by noting that equivalent states of knowledge should be assigned equivalent epistemic probabilities. This argument was propounded by E.T. Jaynes: it leads to two generalizations, namely the principle of transformation groups as in the Jeffreys prior, and the principle of maximum entropy.

More generally, one speaks of non-informative priors.

Template:No footnotes

References

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Edwin Thompson Jaynes. Probability Theory: The Logic of Science. Cambridge University Press, 2003. ISBN 0-521-59271-2.
Persi Diaconis and Joseph B. Keller. "Fair Dice". The American Mathematical Monthly, 96(4):337-339, 1989. (Discussion of dice that are fair "by symmetry" and "by continuity".)
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Main Page: Difference between revisions

Revision as of 06:02, 11 August 2014

Contents

Examples

Coins

Dice

Cards

Application to continuous variables

History of the principle of indifference

References

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-'''Mass spectrometry''' (MS) is an analytical technique that measures the [[mass-to-charge ratio]] of charged particles.<ref name="isbn0-9660813-2-3">{{cite book |author=Sparkman, O. David |title=Mass spectrometry desk reference |publisher=Global View Pub |location=Pittsburgh |year=2000 |isbn=0-9660813-2-3}}</ref>
+{{No footnotes|date=April 2010}}
-It is used for determining masses of particles, for determining the elemental composition of a sample or [[molecule]], and for elucidating the chemical structures of molecules, such as [[peptide]]s and other [[chemical compound]]s.  MS works by ionizing chemical compounds to generate charged molecules or molecule fragments and measuring their [[mass-to-charge ratio]]s.<ref name="isbn0-9660813-2-3" /> In a typical MS procedure:
-# A sample is loaded onto the mass spectrometer, and undergoes [[vaporization]]
-# The components of the sample are ionized by one of a variety of methods (e.g., by impacting them with an [[electron beam]]), which results in the formation of charged particles ([[ion]]s)
-# The ions are separated according to their [[mass-to-charge ratio]] in an analyzer by [[Electromagnetism|electromagnetic]] fields
-# The ions are detected, usually by a quantitative method
-# The ion signal is processed into mass spectra
-MS instruments consist of three modules:
+{{Bayesian statistics}}
-* An ''[[ion source]]'', which can convert gas phase sample molecules into ions (or, in the case of electrospray ionization, move ions that exist in solution into the gas phase)
-* A ''mass analyzer'', which sorts the ions by their masses by applying electromagnetic fields
-* A ''detector'', which measures the value of an indicator quantity and thus provides data for calculating the abundances of each ion present
-The technique has both [[Qualitative research|qualitative]] and [[Quantitative analysis (chemistry)|quantitative]] uses. These include identifying unknown compounds, determining the [[isotope|isotopic]] composition of elements in a molecule, and determining the [[structure]] of a compound by observing its fragmentation. Other uses include quantifying the amount of a compound in a sample or studying the fundamentals of [[gas phase ion chemistry]] (the chemistry of ions and neutrals in a vacuum). MS is now in very common use in analytical laboratories that study physical, chemical, or biological properties of a great variety of compounds.
-==Etymology==
+The '''principle of indifference''' (also called '''principle of insufficient reason''') is a rule for assigning [[epistemic probability|epistemic probabilities]].
-The word ''spectrograph'' had become part of the [[international scientific vocabulary]] by 1884.<ref>"[http://dev.m-w.com/dictionary/spectrograph Definition of spectrograph]." Merriam Webster. Accessed 13 June 2008.</ref><ref>{{cite journal|title = William Aston – the man behind the mass spectrograph|author = KM Downard|journal = European Journal of Mass Spectrometry|volume = 13|issue = 3|pages = 177–190|year = 2007|doi = 10.1255/ejms.878|pmid = 17881785}}</ref>
+Suppose that there are ''n'' &gt; 1 [[mutually exclusive]] and [[collectively exhaustive]] possibilities.
-The [[Root (linguistics)|linguistic roots]] are a combination and removal of [[bound morpheme]]s and [[free morpheme]]s which relate to the terms '''''spectr'''-um'' and ''phot-'''ograph'''-ic plate''.<ref>Harper, Douglas. "[http://www.etymonline.com/index.php?search=spectrum&searchmode=none Spectrum]." [[Online Etymology Dictionary]]. Nov. 2001. Accessed 07-12-2007. '''Note''': This part of the article only makes descriptive claims about the information found in the primary source, the accuracy and applicability of which is easily verifiable by any reasonable, educated person without specialist knowledge. (See [[WP:PSTS]])</ref> Early ''spectrometry'' devices that measured the mass-to-charge ratio of ions were called ''[[spectrograph|mass spectrographs]]'' which consisted of instruments that recorded a [[spectrum]] of mass values on a [[photographic plate]].<ref name='a804629h'>{{cite journal|title=Francis Aston and the mass spectrograph|journal=[[Dalton Transactions]]|year=1998|first=Gordon|last=Squires|pages=3893–3900|doi= 10.1039/a804629h|issue=23}}</ref><ref>{{cite journal|title = Francis William Aston – the man behind the mass spectrograph|author = KM Downard|journal = European Journal of Mass Spectrometry|volume = 13|issue = 3|pages = 177–190|year = 2007|doi = 10.1255/ejms.878|pmid = 17881785}}</ref> A ''mass spectroscope'' is similar to a ''mass spectrograph'' except that the beam of ions is directed onto a [[phosphor]] screen.<ref>{{cite book|last = Thomson|first = J.J.|authorlink = J. J. Thomson|title = Rays Of Positive Electricity and Their Application to Chemical Analysis|publisher = Longman's Green and Company|year = 1913|location = London|url = http://www.archive.org/details/RaysOfPositiveElectricity}}</ref> A mass spectroscope configuration was  used in early instruments when it was desired that the effects of adjustments be quickly observed. Once the instrument was properly adjusted, a photographic plate was inserted and exposed. The term mass spectroscope continued to be used even though the direct illumination of a phosphor screen was replaced by indirect measurements with an [[oscilloscope]].<ref name='Siri_1947'>{{cite journal|title=Mass spectroscope for analysis in the low-mass range|journal=Review of Scientific Instruments|year= 1947|first=William|last=Siri|volume=18|issue=8|pages=540–545|doi= 10.1063/1.1740998|bibcode = 1947RScI...18..540S }}</ref> The use of the term ''mass spectroscopy'' is now discouraged due to the possibility of confusion with light [[spectroscopy]].<ref name="isbn0-9660813-2-3"/><ref name='Price 1991'>{{cite journal|title=Standard definitions of terms relating to mass spectrometry. A report from the Committee on Measurements and Standards of the American Society for Mass Spectrometry|journal=Journal of the American Society for Mass Spectrometry|year=1991|first=Phil|last=Price|volume=2|issue=4|pages=336–348|doi=10.1016/1044-0305(91)80025-3}}</ref> Mass spectrometry is often abbreviated as ''mass-spec'' or simply as ''MS''.<ref name="isbn0-9660813-2-3"/>
+The principle of indifference states that if the ''n'' possibilities are indistinguishable except for their names,
+then each possibility should be assigned a probability equal to 1/''n''.
-==History==
+In [[Bayesian probability]], this is the simplest [[Prior probability#Uninformative priors|non-informative prior]].
-{{Details|History of mass spectrometry}}
+The principle of indifference is meaningless under the [[Frequency probability|frequency interpretation of probability]],{{citation needed|date=July 2013}} in which probabilities are relative frequencies rather than degrees of belief in uncertain propositions, conditional upon state information.
-[[File:Early Mass Spectrometer (replica).jpg|thumb|300px|Replica of an early mass spectrometer]]
-In 1886, [[Eugen Goldstein]] observed rays in [[gas discharge]]s under low pressure that traveled away from the anode and through channels in a perforated [[cathode]], opposite to the direction of negatively charged [[cathode rays]] (which travel from cathode to anode). Goldstein called these positively charged [[anode rays]] "Kanalstrahlen"; the standard translation of this term into English is "[[canal rays]]". [[Wilhelm Wien]] found that strong electric or magnetic fields deflected the canal rays and, in 1899, constructed a device with parallel electric and magnetic fields that separated the positive rays according to their charge-to-mass ratio (''Q/m''). Wien found that the charge-to-mass ratio depended on the nature of the gas in the discharge tube. English scientist [[J.J. Thomson]] later improved on the work of Wien by reducing the pressure to create the mass spectrograph.
+==Examples==
-The first application of mass spectrometry to the analysis of amino acids and peptides was reported in 1958.<ref>''Carl-Ove Andersson'', Acta. Chem. Scand. '''1958''', 12, 1353</ref> Carl-Ove Andersson highlighted the main fragment ions observed in the ionization of methyl esters.<ref>''Mass Spec Anniversary'', [[Chemical & Engineering News]], '''87''', 6 (9 Feb. 2009), p. 4</ref>
+The textbook examples for the application of the principle of indifference are [[coin]]s, [[dice]], and [[playing cards|cards]].
-Some of the modern techniques of mass spectrometry were devised by [[Arthur Jeffrey Dempster]] and [[F.W. Aston]] in 1918 and 1919 respectively. In 1989, half of the [[Nobel Prize in Physics]] was awarded to [[Hans Dehmelt]] and [[Wolfgang Paul]] for the development of the ion trap technique in the 1950s and 1960s. In 2002, the [[Nobel Prize in Chemistry]] was awarded to [[John Bennett Fenn]] for the development of [[electrospray ionization]] (ESI) and [[Koichi Tanaka]] for the development of [[soft laser desorption]] (SLD) and their application to the ionization of biological macromolecules, especially proteins.<ref>{{cite news| title=The Nobel Prize in Chemistry 2002: Information for the Public | date=9 October 2002 | publisher=The Nobel Foundation | url =http://nobelprize.org/nobel_prizes/chemistry/laureates/2002/public.html | accessdate = 2007-08-29 }}</ref>
+In a [[macroscopic]] system, at least,
+it must be assumed that the physical laws which govern the system are not known well enough to predict the outcome.
+As observed some centuries ago by [[John Arbuthnot]] (in the preface of ''Of the Laws of Chance'', 1692),
-==Simplified example==
+:It is impossible for a Die, with such determin'd force and direction, not to fall on such determin'd side, only I don't know the force and direction which makes it fall on such determin'd side, and therefore I call it Chance, which is nothing but the want of art....
-[[Image:Mass spectrometer schematics.png|right|thumb|300px|Schematics of a simple mass spectrometer with sector type mass analyzer. This one is for the measurement of carbon dioxide [[isotope]] ratios ([[Isotope ratio mass spectrometry|IRMS]]) as in the [[carbon-13]] [[urea breath test]]]]
-The following example describes the operation of a spectrometer mass analyzer, which is of the [[sector instrument|sector]] type. (Other analyzer types are treated below.) Consider a sample of [[sodium chloride]] (table salt). In the ion source, the sample is [[vapor]]ized (turned into [[gas]]) and ionized (transformed into electrically charged particles) into [[sodium]] (Na<sup>+</sup>) and [[chloride]] (Cl<sup>-</sup>) ions. Sodium atoms and ions are [[Monoisotopic element|monoisotopic]], with a mass of about 23 amu. Chloride atoms and ions come in two isotopes with masses of approximately 35 amu (at a natural abundance of about 75 percent) and approximately 37 amu (at a natural abundance of about 25 percent). The analyzer part of the spectrometer contains [[Electric field|electric]] and [[Magnetic field|magnetic]] fields, which exert forces on ions traveling through these fields. The speed of a charged particle may be increased or decreased while passing through the electric field, and its direction may be altered by the magnetic field. The magnitude of the deflection of the moving ion's trajectory depends on its mass-to-charge ratio. Lighter ions get deflected by the magnetic force more than heavier ions (based on [[Newton's laws of motion|Newton's second law of motion]], ''F'' = ''ma''). The streams of sorted ions pass from the analyzer to the detector, which records the relative abundance of each ion type. This information is used to determine the chemical element composition of the original sample (i.e. that both sodium and chlorine are present in the sample) and the isotopic composition of its constituents (the ratio of <sup>35</sup>Cl to <sup>37</sup>Cl).
-==Creating ions==
+Given enough time and resources,
-{{main|Ion source}}
+there is no fundamental reason to suppose that suitably precise measurements could not be made,
-The ion source is the part of the mass spectrometer that ionizes the material under analysis (the analyte). The ions are then transported by [[magnetic field|magnetic]] or [[electric field]]s to the mass analyzer.
+which would enable the prediction of the outcome of coins, dice, and cards with high accuracy: [[Persi Diaconis]]'s work with [[coin flipping|coin-flipping]] machines is a practical example of this.
-Techniques for [[ion]]ization have been key to determining what types of samples can be analyzed by mass spectrometry.
+===Coins===
-[[Electron ionization]] and [[chemical ionization]] are used for [[gas]]es and [[vapor]]s. In''' [[chemical ionization]]''' sources, the analyte is ionized by chemical ion-molecule reactions during collisions in the source. Two techniques often used with [[liquid]] and [[solid]] biological samples include [[electrospray ionization]] (invented by [[John Bennett Fenn|John Fenn]]<ref>{{cite journal | author = Fenn, J. B.; Mann, M.; Meng, C. K.; Wong, S. F.; Whitehouse, C. M. | title = Electrospray ionization for mass spectrometry of large biomolecules | journal = [[Science (journal)|Science]] | year = 1989| volume = 246 | pages = 64–71 | doi = 10.1126/science.2675315 | pmid = 2675315 | issue = 4926|bibcode = 1989Sci...246...64F }}</ref>) and [[matrix-assisted laser desorption/ionization]] (MALDI, initially developed as a similar technique "Soft Laser Desorption (SLD)" by K. Tanaka<ref>{{cite journal |author=Tanaka, K.; Waki, H.; Ido, Y.; Akita, S.; Yoshida, Y.; Yoshida, T. |title=Protein and Polymer Analyses up to m/z 100 000 by Laser Ionization Time-of flight Mass Spectrometry |journal=[[Rapid Communications in Mass Spectrometry|Rapid Commun Mass Spectrom]] |volume=2 |issue=20 |pages=151–3 |year=1988|doi=10.1002/rcm.1290020802}}</ref> for which a Nobel Prize was awarded and as MALDI by M. Karas and F. Hillenkamp<ref>{{cite journal |author=Karas, M.; Bachman, D.; Bahr, U.; Hillenkamp, F. |title=Matrix-Assisted Ultraviolet Laser Desorption of Non-Volatile Compounds |journal=[[International Journal of Mass Spectrometry|Int J Mass Spectrom Ion Proc]] |volume=78 |pages=53–68 |year=1987|doi=10.1016/0168-1176(87)87041-6}}
-</ref>).
-===Inductively coupled plasma===
+A [[symmetry|symmetric]] coin has two sides, arbitrarily labeled ''heads'' and ''tails''.
-[[ICP-MS|Inductively coupled plasma]] (ICP) sources are used primarily for cation analysis of a wide array of sample types. In this type of Ion Source Technology, a 'flame' of plasma that is electrically neutral overall, but that has had a substantial fraction of its atoms ionized by high temperature, is used to atomize introduced sample molecules and to further strip the outer electrons from those atoms. The plasma is usually generated from argon gas, since the first ionization energy of argon atoms is higher than the first of any other elements except He, O, F and Ne, but lower than the second ionization energy of all except the most electropositive metals. The heating is achieved by a radio-frequency current passed through a coil surrounding the plasma.
+Assuming that the coin must land on one side or the other,
+the outcomes of a coin toss are mutually exclusive, exhaustive, and interchangeable.
+According to the principle of indifference, we assign each of the possible outcomes a probability of 1/2.
-===Other ionization techniques===
+It is implicit in this analysis that the forces acting on the coin are not known with any precision.
-Others include [[glow discharge]], [[field desorption]] (FD), [[fast atom bombardment]] (FAB), [[thermospray]], [[desorption/ionization on silicon]] (DIOS), [[DART ion source|Direct Analysis in Real Time (DART)]], [[atmospheric pressure chemical ionization]] (APCI), [[secondary ion mass spectrometry]] (SIMS), [[spark ionization]] and [[thermal ionization]] (TIMS).<ref>{{cite journal|author = A. P. Bruins|title = Mass spectrometry with ion sources operating at atmospheric pressure|year = 1991|journal = [[Mass Spectrometry Reviews]]|volume = 10|issue = 1|pages = 53–77|doi = 10.1002/mas.1280100104}}</ref>
+If the momentum imparted to the coin as it is launched were known with sufficient accuracy,
-[[IA-Mass|Ion attachment ionization]] is an ionization technique that allows for fragmentation free analysis.
+the flight of the coin could be predicted according to the laws of mechanics.
+Thus the uncertainty in the outcome of a coin toss is derived (for the most part) from the uncertainty with respect to initial conditions.
+This point is discussed at greater length in the article on [[Coin flipping#Physics|coin flipping]].
-==Mass selection==
+There is also a third possible outcome: the coin could land on its edge.
-<!-- "Mass accuracy" redirects here -->
+However,
-Mass analyzers separate the ions according to their [[mass-to-charge ratio]]. The following two laws govern the dynamics of charged particles in electric and magnetic fields in vacuum:
+the principle of indifference doesn't say anything about this outcome, as the labels ''head'', ''tail'', and ''edge'' are not interchangeable.
+One could argue, though, that ''head'' and ''tail'' remain interchangeable, and therefore Pr(''head'') and Pr(''tail'') are equal, and both are equal to 1/2 (1 - Pr(''edge'')).
-:<math>\mathbf{F} = Q (\mathbf{E} + \mathbf{v} \times \mathbf{B})</math> ([[Lorentz force law]]);
+===Dice===
-:<math>\mathbf{F}=m\mathbf{a}</math> ([[Newton's second law]] of motion in non-relativistic case, i.e. valid only at ion velocity much lower than the speed of light).
+A [[symmetry|symmetric]] [[dice]] has ''n'' faces, arbitrarily labeled from 1 to ''n''.
+Ordinary cubical dice have ''n'' = 6 faces,
+although symmetric dice with different numbers of faces can be constructed;
+see [[dice]].
+We assume that the die must land on one face or another,
+and there are no other possible outcomes.
+Applying the principle of indifference, we assign each of the possible outcomes a probability of 1/''n''.
-Here '''F''' is the force applied to the ion, ''m'' is the mass of the ion, '''a''' is the acceleration, ''Q'' is the ion charge, '''E''' is the electric field, and '''v''' × '''B''' is the [[vector cross product]] of the ion velocity and the magnetic field
+As with coins,
+it is assumed that the initial conditions of throwing the dice are not known
+with enough precision to predict the outcome according to the laws of mechanics.
+Dice are typically thrown so as to bounce on a table or other surface.
+This interaction makes prediction of the outcome much more difficult.
-Equating the above expressions for the force applied to the ion yields:
+===Cards===
-: <math>(m/Q)\mathbf{a} = \mathbf{E}+ \mathbf{v} \times \mathbf{B}.</math>
+A standard deck contains 52 cards, each given a unique label in an arbitrary fashion, i.e. arbitrarily ordered. We draw a card from the deck; applying the principle of indifference, we assign each of the possible outcomes a probability of 1/52.
-This [[differential equation]] <!-- It doesn't look like one, but I'm not an expert -->is the classic equation of motion for [[charged particle]]s. Together with the particle's initial conditions, it completely determines the particle's motion in space and time in terms of ''m/Q''. Thus mass spectrometers could be thought of as "mass-to-charge spectrometers". When presenting data, it is common to use the (officially) [[dimensionless]] ''m/z'', where z is the number of [[elementary charge]]s (''e'') on the ion (z=Q/e). This quantity, although it is informally called the mass-to-charge ratio, more accurately speaking represents the ratio of the mass number and the charge number, ''z''.
+This example, more than the others, shows the difficulty of actually applying the principle of indifference in real situations. What we really mean by the phrase "arbitrarily ordered" is simply that we don't have any information that would lead us to favor a particular card. In actual practice, this is rarely the case: a new deck of cards is certainly not in arbitrary order, and neither is a deck immediately after a hand of cards. In practice, we therefore [[shuffling playing cards|shuffle]] the cards; this does not destroy the information we have, but instead (hopefully) renders our information practically unusable, although it is still usable in principle. In fact, some expert blackjack players can track aces through the deck; for them, the condition for applying the principle of indifference is not satisfied.
-There are many types of mass analyzers, using either static or dynamic fields, and magnetic or electric fields, but all operate according to the above differential equation. Each analyzer type has its strengths and weaknesses. Many mass spectrometers use two or more mass analyzers for [[Tandem mass spectrometry|tandem mass spectrometry (MS/MS)]]. In addition to the more common mass analyzers listed below, there are others designed for special situations.
+==Application to continuous variables==
-There are several important analyser characteristics. The [[mass resolving power]] is the measure of the ability to distinguish two peaks of slightly different ''m/z''. The mass accuracy is the ratio of the ''m/z'' measurement error to the true m/z. Mass accuracy is usually measured in [[Parts per million|ppm]] or [[milli mass unit]]s. The mass range is the range of ''m/z'' amenable to analysis by a given analyzer. The linear dynamic range is the range over which ion signal is linear with analyte concentration. Speed refers to the time frame of the experiment and ultimately is used to determine the number of spectra per unit time that can be generated.
+Applying the principle of indifference incorrectly can easily lead to nonsensical results, especially in the case of multivariate, continuous variables.  A typical case of misuse is the following example.
-===Sector instruments===
+*Suppose there is a cube hidden in a box. A label on the box says the cube has a side length between 3 and 5&nbsp;cm.
-{{Details|sector instrument}}
+* We don't know the actual side length, but we might assume that all values are equally likely and simply pick the mid-value of 4&nbsp;cm.
-A sector field mass analyzer uses an electric and/or magnetic field to affect the path and/or [[velocity]] of the [[electric charge|charged]] particles in some way.
+* The information on the label allows us to calculate that the surface area of the cube is between 54 and 150&nbsp;cm². We don't know the actual surface area, but we might assume that all values are equally likely and simply pick the mid-value of 102&nbsp;cm².
-As shown above, [[sector instrument]]s bend the trajectories of the ions as they pass through the mass analyzer, according to their mass-to-charge ratios, deflecting the more charged and faster-moving, lighter ions more. The analyzer can be used to select a narrow range of ''m/z'' or to scan through a range of ''m/z'' to catalog the ions present.<ref>{{Cite journal|year = 1986|title = Extending the Mass Range of a Sector Mass Spectrometer|author = John S Cottrell, Roger J Greathead|journal = Mass Spectrometry Reviews|volume = 5|pages = 215–247|doi = 10.1002/mas.1280050302|issue = 3}}</ref>
+* The information on the label allows us to calculate that the volume of the cube is between 27 and 125&nbsp;cm<sup>3</sup>. We don't know the actual volume, but we might assume that all values are equally likely and simply pick the mid-value of 76&nbsp;cm<sup>3</sup>.
+* However, we have now reached the impossible conclusion that the cube has a side length of 4&nbsp;cm, a surface area of 102&nbsp;cm², and a volume of 76&nbsp;cm<sup>3</sup>!
-===Time-of-flight===
+In this example, mutually contradictory estimates of the length, surface area, and volume of the cube arise because we have assumed three mutually contradictory distributions for these parameters: a [[uniform distribution (continuous)|uniform distribution]] for any one of the variables implies a non-uniform distribution for the other two. (The same paradox arises if we make it discrete: the side is either exactly 3&nbsp;cm, 4&nbsp;cm, or 5&nbsp;cm, mutatis mutandis.)  In general, the principle of indifference does not indicate which variable (e.g. in this case, length, surface area, or volume) is to have a uniform epistemic probability distribution.
-{{Details|time-of-flight mass spectrometry}}
-The [[time-of-flight]] (TOF) analyzer uses an [[electric field]] to accelerate the ions through the same [[voltage|potential]], and then measures the time they take to reach the detector. If the particles all have the same [[electrical charge|charge]], the [[kinetic energy|kinetic energies]] will be identical, and their [[velocity|velocities]] will depend only on their [[mass]]es. Lighter ions will reach the detector first.<ref>  In the event that the ions do not start at identical kinetic energies, some ions may lag behind higher kinetic energy ions decreasing resolution.  Reflectron geometries are commonly employed to correct this problem.
-{{Cite journal|doi = 10.1002/mas.1280120202|title = Time-of-flight mass analyzers|year = 1993|author = Wollnik, H.|journal = Mass Spectrometry Reviews|volume = 12|page = 89|issue = 2
-}}</ref>
-===Quadrupole mass filter===
+Another classic example of this kind of misuse is [[Bertrand's paradox (probability)|Bertrand's paradox]].  [[Edwin T. Jaynes]] introduced the [[principle of transformation groups]], which can yield an epistemic probability distribution for this problem.  This generalises the principle of indifference, by saying that one is indifferent between ''equivalent problems'' rather than indifference between propositions.  This still reduces to the ordinary principle of indifference when one considers a permutation of the labels as generating equivalent problems (i.e. using the permutation transformation group).  To apply this to the above box example, we have three problems, with no reason to think one problem is "our problem" more than any other - we are indifferent between each.  If we have no reason to favour one over the other, then our prior probabilities must be related by the rule for changing variables in continuous distributions.  Let ''L'' be the length, and ''V'' be the volume.  Then we must have
-{{Details|Quadrupole mass analyzer}}
-[[Quadrupole mass analyzer]]s use oscillating electrical fields to selectively stabilize or destabilize the paths of ions passing through a [[radio frequency]] (RF) [[quadrupole]] field created between 4 parallel rods. Only the ions in a certain range of mass/charge ratio are passed through the system at any time, but changes to the potentials on the rods allow a wide range of m/z values to be swept rapidly, either continuously or in a succession of discrete hops. A quadrupole mass analyzer acts as a mass-selective filter and is closely related to the [[quadrupole ion trap]], particularly the linear quadrupole ion trap except that it is designed to pass the untrapped ions rather than collect the trapped ones, and is for that reason referred to as a transmission quadrupole.
-A common variation of the transmission quadrupole is the triple quadrupole mass spectrometer. The “triple quad” has three consecutive quadrupole stages, the first acting as a mass filter to transmit a particular incoming ion to the second quadrupole, a collision chamber, wherein that ion can be broken into fragments. The third quadrupole also acts as a mass filter, to transmit a particular fragment ion to the detector. If a quadrupole is made to rapidly and repetitively cycle through a range of mass filter settings, full spectra can be reported.  Likewise, a triple quad can be made to perform various scan types characteristic of [[tandem mass spectrometry]].
-===Ion traps===
+:<math>f(L)=|{\partial V \over \partial L}|f(V)=3 L^{2} f(L^{3})</math>
-====Three-dimensional quadrupole ion trap====
-{{Details|quadrupole ion trap}}
-The [[quadrupole ion trap]] works on the same physical principles as the quadrupole mass analyzer, but the ions are trapped and sequentially ejected. Ions are trapped in a mainly quadrupole RF field, in a space defined by a ring electrode (usually connected to the main RF potential) between two endcap electrodes (typically connected to DC or auxiliary AC potentials). The sample is ionized either internally (e.g. with an electron or laser beam), or externally, in which case the ions are often introduced through an aperture in an endcap electrode.
-There are many mass/charge separation and isolation methods but the most commonly used is the mass instability mode in which the RF potential is ramped so that the orbit of ions with a mass ''a &gt; b'' are stable while ions with mass ''b'' become unstable and are ejected on the ''z''-axis onto a detector. There are also non-destructive analysis methods.
+Which has a general solution: <math>f(L) = {K \over L}</math> Where ''K'' is an arbitrary constant, determined by the range of ''L'', in this case equal to:
-Ions may also be ejected by the resonance excitation method, whereby a supplemental oscillatory excitation voltage is applied to the endcap electrodes, and the trapping voltage amplitude and/or excitation voltage frequency is varied to bring ions into a resonance condition in order of their mass/charge ratio.<ref>{{cite journal|author = Paul W., Steinwedel H.|year = 1953|title = Ein neues Massenspektrometer ohne Magnetfeld|journal = Zeitschrift für Naturforschung A|volume = 8|issue = 7|pages = 448–450|bibcode = 1953ZNatA...8..448P|last2 = Steinwedel}}</ref><ref>{{cite journal|author = R. E. March|title = Quadrupole ion trap mass spectrometry: a view at the turn of the century|year = 2000|journal = [[International Journal of Mass Spectrometry]]|volume = 200|issue = 1–3|pages = 285–312|doi = 10.1016/S1387-3806(00)00345-6}}</ref>
+:<math>K^{-1}=\int_{3}^{5}{dL \over L} = \log({5 \over 3})</math>
-The [[cylindrical ion trap mass spectrometer]] is a derivative of the quadrupole ion trap mass spectrometer.
+To put this "to the test", we ask for the probability that the length is less than 4.  This has probability of:
-====Linear quadrupole ion trap====
+:<math>Pr(L<4)=\int_{3}^{4}{dL \over L \log({5 \over 3})}= {\log({4 \over 3}) \over \log({5 \over 3})} \approx 0.56</math>.
-A [[linear quadrupole ion trap]] is similar to a quadrupole ion trap, but it traps ions in a two dimensional quadrupole field, instead of a three-dimensional quadrupole field as in a 3D quadrupole ion trap. Thermo Fisher's LTQ ("linear trap quadrupole") is an example of the linear ion trap.<ref>{{cite journal |last = Schwartz |first = Jae C. |coauthors = Michael W. Senko and John E. P. Syka |title = A two-dimensional quadrupole ion trap mass spectrometer |journal = Journal of the American Society for Mass Spectrometry |volume = 13 |issue = 6 |pages = 659–669 |year = 2002 |doi = 10.1016/S1044-0305(02)00384-7 |pmid = 12056566 }}</ref>
-A toroidal ion trap can be visualized as a linear quadrupole curved around and connected at the ends or as a cross section of a 3D ion trap rotated on edge to form the toroid, donut shaped trap. The trap can store large volumes of ions by distributing them throughout the ring-like trap structure. This toroidal shaped trap is a configuration that allows the increased miniaturization of an ion trap mass analyzer. Additionally all ions are stored in the same trapping field and ejected together simplifying detection that can be complicated with array configurations due to variations in detector alignment and machining of the arrays.<ref>{{cite journal|author = Lammert SA, Rockwood AA, Wang M, and ML Lee|title = Miniature Toroidal Radio Frequency Ion Trap Mass Analyzer|year = 2006|journal = [[Journal of the American Society for Mass Spectrometry]]|volume = 17|issue = 7|pages = 916–922|doi = 10.1016/j.jasms.2006.02.009|pmid = 16697659}}</ref>
+For the volume, this should be equal to the probability that the volume is less than 4<sup>3</sup> = 64.  The pdf of the volume is
-====Orbitrap====
+:<math>f(V^{{1 \over 3}}) {1 \over 3} V^{-{2 \over 3}}={1 \over 3 V \log({5 \over 3})}</math>.
-{{Details|Orbitrap}}
-These are similar to [[Fourier transform ion cyclotron resonance]] mass spectrometers (see text below). Ions are [[electrostatic]]ally trapped in an orbit around a central, spindle shaped electrode. The electrode confines the ions so that they both orbit around the central electrode and oscillate back and forth along the central electrode's long axis. This oscillation generates an [[image current]] in the detector plates which is recorded by the instrument. The frequencies of these image currents depend on the mass to charge ratios of the ions. Mass spectra are obtained by [[Fourier transformation]] of the recorded image currents.
-Orbitraps have a high mass accuracy, high sensitivity and a good dynamic range.<ref>{{cite journal|author = Q. Hu, R. J. Noll, H. Li, A. Makarov, M. Hardman and R. G. Cooks|title = The Orbitrap: a new mass spectrometer|year = 2005|journal = [[Journal of Mass Spectrometry]]|volume = 40|issue = 4|pages = 430–443|doi = 10.1002/jms.856|pmid = 15838939}}</ref>
+And then probability of volume less than 64 is
-===Fourier transform ion cyclotron resonance===
+:<math>Pr(V<64)=\int_{27}^{64}{dV \over 3 V \log({5 \over 3})}={\log({64 \over 27}) \over 3 \log({5 \over 3})}={3 \log({4 \over 3}) \over 3 \log({5 \over 3})}={\log({4 \over 3}) \over \log({5 \over 3})} \approx 0.56</math>.
-{{Details|Fourier transform mass spectrometry}}
+Thus we have achieved invariance with respect to volume and length.  You can also show the same invariance with respect to surface area being less than 6(4<sup>2</sup>) = 96.  However, note that this probability assignment is not necessarily a "correct" one.  For the exact distribution of lengths, volume, or surface area will depend on how the "experiment" is conducted.  This probability assignment is very similar to the [[maximum entropy]] one, in that the frequency distribution corresponding to the above probability distribution is the most likely to be seen.  So, if one was to go to ''N'' people individually and simply say "make me a box somewhere between 3 and 5 cm, or a volume between 27 and 125 cm, or a surface area between 54 and 150 cm", then unless there is a systematic influence on how they make the boxes (e.g. they form a group, and choose one particular method of making boxes), about 56% of the boxes will be less than 4&nbsp;cm - and it will get very close to this amount very quickly.  So, for large N, any deviation from this basically indicates the makers of the boxes were "systematic" in how the boxes were made.
-[[Fourier transform mass spectrometry]] (FTMS), or more precisely [[Fourier transform ion cyclotron resonance]] MS, measures mass by detecting the [[image current]] produced by ions [[cyclotron]]ing in the presence of a magnetic field. Instead of measuring the deflection of ions with a detector such as an [[electron multiplier]], the ions are injected into a [[Penning trap]] (a static electric/magnetic [[ion trap]]) where they effectively form part of a circuit. Detectors at fixed positions in space measure the electrical signal of ions which pass near them over time, producing a periodic signal. Since the frequency of an ion's cycling is determined by its mass to charge ratio, this can be [[deconvolution|deconvoluted]] by performing a [[Fourier transform]] on the signal. [[FTMS]] has the advantage of high sensitivity (since each ion is "counted" more than once) and much higher [[resolution (mass spectrometry)|resolution]] and thus precision.<ref>{{cite journal|author = M. B. Comisarow and A. G. Marshall|title = Fourier transform ion cyclotron resonance spectroscopy|year = 1974|journal = [[Chemical Physics Letters]]|volume = 25|issue = 2|pages = 282–283|doi = 10.1016/0009-2614(74)89137-2|bibcode = 1974CPL....25..282C }}</ref><ref>{{cite journal|author = Marshall, A. G.; Hendrickson, C. L.; Jackson, G. S.|title = Fourier transform ion cyclotron resonance mass spectrometry: a primer|year = 1998|journal = [[Mass Spectrometry Reviews]]|volume = 17|issue = 1|pages = 1–34|doi = 10.1002/(SICI)1098-2787(1998)17:1<1::AID-MAS1>3.0.CO;2-K|pmid=9768511}}</ref>
-[[Ion cyclotron resonance]] (ICR) is an older mass analysis technique similar to FTMS except that ions are detected with a traditional detector. Ions trapped in a [[Penning trap]] are excited by an RF electric field until they impact the wall of the trap, where the detector is located. Ions of different mass are resolved according to impact time.
+The fundamental hypothesis of [[statistical physics]], that any two microstates of a system with the same total energy are equally probable at [[Thermodynamic equilibrium|equilibrium]], is in a sense an example of the principle of indifference. However, when the microstates are described by continuous variables (such as positions and momenta), an additional physical basis is needed in order to explain under ''which'' parameterization the probability density will be uniform.  [[Liouville's theorem (Hamiltonian)|Liouville's theorem]] justifies the use of canonically conjugate variables, such as positions and their conjugate momenta.
-==Detectors==
+==History of the principle of indifference==
-[[Image:Cont dynode detector.jpg|thumb|left|300 px|A continuous dynode particle multiplier detector.]]
-The final element of the mass spectrometer is the detector. The detector records either the charge induced or the current produced when an ion passes by or hits a surface. In a scanning instrument, the signal produced in the detector during the course of the scan versus where the instrument is in the scan (at what ''m/Q'') will produce a [[mass spectrum]], a record of ions as a function of ''m/Q''.
-Typically, some type of [[electron multiplier]] is used, though other detectors including [[Faraday cup]]s and [[ion-to-photon detector]]s are also used. Because the number of ions leaving the mass analyzer at a particular instant is typically quite small, considerable amplification is often necessary to get a signal. [[Microchannel plate detector]]s are commonly used in modern commercial instruments.<ref>{{cite journal|author = F. Dubois, R. Knochenmuss, R. Zenobi, A. Brunelle, C. Deprun and Y. L. Beyec|title = A comparison between ion-to-photon and microchannel plate detectors|year = 1999|journal = [[Rapid Communications in Mass Spectrometry]]|volume = 13|issue = 9|pages = 786–791|doi = 10.1002/(SICI)1097-0231(19990515)13:9<786::AID-RCM566>3.0.CO;2-3}}</ref> In [[FTMS]] and [[Orbitrap]]s, the detector consists of a pair of metal surfaces within the mass analyzer/ion trap region which the ions only pass near as they oscillate. No DC current is produced, only a weak AC image current is produced in a circuit between the electrodes. Other inductive detectors have also been used.<ref>{{cite journal|author = M. A. Park, J. H. Callahan and A. Vertes|title = An inductive detector for time-of-flight mass spectrometry|year = 1994|journal = [[Rapid Communications in Mass Spectrometry]]|volume = 8|issue = 4|pages = 317–322|doi = 10.1002/rcm.1290080407}}</ref>
+The original writers on probability, primarily [[Jacob Bernoulli]] and [[Pierre Simon Laplace]], considered the principle of indifference to be intuitively obvious and did not even bother to give it a name.  Laplace wrote:
-==Tandem mass spectrometry==
+:The theory of chance consists in reducing all the events of the same kind to a certain number of cases equally possible, that is to say, to such as we may be equally undecided about in regard to their existence, and in determining the number of cases favorable to the event whose probability is sought. The ratio of this number to that of all the cases possible is the measure of this probability, which is thus simply a fraction whose numerator is the number of favorable cases and whose denominator is the number of all the cases possible.
-{{Main|Tandem mass spectrometry}}
-A tandem mass spectrometer is one capable of multiple rounds of mass spectrometry, usually separated by some form of molecule fragmentation. For example, one mass analyzer can isolate one [[peptide]] from many entering a mass spectrometer. A second mass analyzer then stabilizes the peptide ions while they collide with a gas, causing them to fragment by [[collision-induced dissociation]] (CID). A third mass analyzer then sorts the fragments produced from the peptides. Tandem MS can also be done in a single mass analyzer over time, as in a [[quadrupole ion trap]]. There are various methods for [[Fragmentation (chemistry)|fragmenting]] molecules for tandem MS, including [[collision-induced dissociation]] (CID), [[electron capture dissociation]] (ECD), [[electron transfer dissociation]] (ETD), [[infrared multiphoton dissociation]] (IRMPD), [[blackbody infrared radiative dissociation]] (BIRD), [[electron-detachment dissociation]] (EDD) and [[surface-induced dissociation]] (SID). An important application using tandem mass spectrometry is in [[Mass spectrometry#Protein identification|protein identification]].<ref>{{cite journal|author = Robert K. Boyd|title = Linked-scan techniques for MS/MS using tandem-in-space instruments|year = 1994|journal = [[Mass Spectrometry Reviews]]|volume = 13|issue = 5–6|pages = 359–410|doi = 10.1002/mas.1280130502}}</ref>
-Tandem mass spectrometry enables a variety of experimental sequences. Many commercial mass spectrometers are designed to expedite the execution of such routine sequences as [[selected reaction monitoring|selected reaction monitoring (SRM)]] and precursor ion scanning. In SRM, the first analyzer allows only a single mass through and the second analyzer monitors for multiple user-defined fragment ions. SRM is most often used with scanning instruments where the second mass analysis event is [[duty cycle]] limited. These experiments are used to increase specificity of detection of known molecules, notably in pharmacokinetic studies. Precursor ion scanning refers to monitoring for a specific loss from the precursor ion. The first and second mass analyzers scan across the spectrum as partitioned by a user-defined ''m/z'' value. This experiment is used to detect specific motifs within unknown molecules.
+These earlier writers, Laplace in particular, naively generalized the principle of indifference to the case of continuous parameters, giving the so-called "uniform prior probability distribution", a function which is constant over all real numbers. He used this function to express a complete lack of knowledge as to the value of a parameter.  According to Stigler (page 135), Laplace's assumption of uniform prior probabilities was not a meta-physical assumption.  It was an implicit assumption made for the ease of analysis.
-Another type of tandem mass spectrometry used for [[radiocarbon dating]] is [[accelerator mass spectrometry]] (AMS), which uses very high voltages, usually in the mega-volt range, to accelerate negative ions into a type of tandem mass spectrometer.
+The '''principle of insufficient reason''' was its first name, given to it by later writers, possibly as a play on [[Gottfried Leibniz|Leibniz]]'s [[principle of sufficient reason]]. These later writers ([[George Boole]], [[John Venn]], and others) objected to the use of the uniform prior for two reasons. The first reason is that the constant function is not normalizable, and thus is not a proper probability distribution. The second reason is its inapplicability to continuous variables, as described above. (However, these paradoxical issues can be resolved. In the first case, a constant, or any more general finite polynomial, ''is'' normalizable within any finite range: the range [0,1] is all that matters here. Alternatively, the function may be modified to be zero outside that range, as with a [[continuous uniform distribution]]. In the second case, there is no ambiguity provided the problem is "well-posed", so that no unwarranted assumptions can be made, or have to be made, thereby fixing the appropriate prior [[probability density function]] or prior [[moment generating function]] (with variables fixed appropriately) to be used for the probability itself. See the [[Bertrand paradox (probability)]] for an analogous case.)
-==Common mass spectrometer configurations and techniques==
+The "Principle of insufficient reason" was renamed the "Principle of Indifference" by the economist {{harvs|first=John Maynard|last=Keynes|authorlink=John Maynard Keynes|year=1921|txt}}, who was careful to note that it applies only when there is no knowledge indicating unequal probabilities.
-When a specific configuration of source, analyzer, and detector becomes conventional in practice, often a compound [[List of mass spectrometry acronyms|acronym]] arises to designate it, and the compound acronym may be better known among nonspectrometrists than the component acronyms. The epitome of this is [[MALDI-TOF]], which simply refers to combining a [[matrix-assisted laser desorption/ionization]] source with a [[time-of-flight]] mass analyzer. The MALDI-TOF moniker is more widely recognized by the non-mass spectrometrists than MALDI or TOF individually. Other examples include [[ICP-MS|inductively coupled plasma-mass spectrometry (ICP-MS)]], [[accelerator mass spectrometry|accelerator mass spectrometry (AMS)]], [[thermal ionisation|thermal ionization-mass spectrometry (TIMS)]] and [[spark ionization|spark source mass spectrometry (SSMS)]]. Sometimes the use of the generic "MS" actually connotes a very specific mass analyzer and detection system, as is the case with AMS, which is always sector based.
+Attempts to put the notion on firmer [[philosophy|philosophical]] ground have generally begun with the concept of [[equipossibility]] and progressed from it to [[equiprobability]].
-Certain applications of mass spectrometry have developed monikers that although strictly speaking would seem to refer to a broad application, in practice have come instead to connote a specific or a limited number of instrument configurations. An example of this is [[isotope ratio mass spectrometry|isotope ratio mass spectrometry (IRMS)]], which refers in practice to the use of a limited number of sector based mass analyzers; this name is used to refer to both the application and the instrument used for the application.
+The principle of indifference can be given a deeper logical justification by noting that equivalent states of knowledge should be assigned equivalent epistemic probabilities.  This argument was propounded by [[E.T. Jaynes]]:  it leads to two generalizations, namely the [[principle of transformation groups]] as in the [[Jeffreys prior]], and the [[principle of maximum entropy]].
-==Chromatographic techniques combined with mass spectrometry==
+More generally, one speaks of [[non-informative prior]]s.
-An important enhancement to the mass resolving and mass determining capabilities of mass spectrometry is using it in tandem with [[chromatography|chromatographic]] separation techniques.
-===Gas chromatography===
+{{No footnotes|date=July 2010}}
-[[Image:GCMS open.jpg|thumb|right|300 px|A gas chromatograph (right) directly coupled to a mass spectrometer (left)]]
-{{See also|Gas chromatography-mass spectrometry}}
-A common combination is [[gas]] chromatography-mass spectrometry (GC/MS or GC-MS). In this technique, a [[gas chromatograph]] is used to separate different compounds. This stream of separated compounds is fed online into the [[ion]] source, a [[metal]]lic [[Electrical filament|filament]] to which [[voltage]] is applied. This filament emits electrons which ionize the compounds. The ions can then further fragment, yielding predictable patterns. Intact ions and fragments pass into the mass spectrometer's analyzer and are eventually detected.<ref>Eiceman, G.A. (2000). Gas Chromatography. In R.A. Meyers (Ed.), ''Encyclopedia of Analytical Chemistry: Applications, Theory, and Instrumentation'', pp. 10627. Chichester: Wiley. ISBN 0-471-97670-9</ref>
-===Liquid chromatography===
-{{See also|Liquid chromatography-mass spectrometry}}
-Similar to gas chromatography MS (GC/MS), liquid chromatography mass spectrometry (LC/MS or LC-MS) separates compounds chromatographically before they are introduced to the ion source and mass spectrometer. It differs from GC/MS in that the mobile phase is liquid, usually a mixture of [[water]] and organic [[solvent]]s, instead of gas and the ions fragments cannot yield predictable patterns. Most commonly, an [[electrospray ionization]] source is used in LC/MS. There are also some newly developed ionization techniques like [[laser spray]].
-===Ion mobility===
-{{See also|Ion mobility spectrometry-mass spectrometry}}
-[[Ion mobility spectrometry]]/mass spectrometry (IMS/MS or IMMS) is a technique where ions are first separated by drift time through some neutral gas under an applied electrical potential gradient before being introduced into a mass spectrometer.<ref name='GFVerbeck06012002'>{{cite journal|title=A fundamental introduction to ion mobility mass spectrometry applied to the analysis of biomolecules|journal=J Biomol Tech|year=2002|last=Verbeck, GF and Ruotolo, BT and Sawyer, HA and Gillig, KJ and Russell, DH|volume=13|issue=2|pages=56–61|pmc = 2279851|pmid=19498967|first1=G|last2=Ruotolo|first2=B|last3=Sawyer|first3=H|last4=Gillig|first4=K|last5=Russell|first5=D}}</ref> Drift time is a measure of the radius relative to the charge of the ion. The [[duty cycle]] of IMS (the time over which the experiment takes place) is longer than most mass spectrometric techniques, such that the mass spectrometer can sample along the course of the IMS separation. This produces data about the IMS separation and the mass-to-charge ratio of the ions in a manner similar to [[liquid chromatography-mass spectrometry|LC/MS]].<ref>{{cite journal|author = L. M. Matz, G. R. Asbury and H. H. Hill|title = Two-dimensional separations with electrospray ionization ambient pressure high-resolution ion mobility spectrometry/quadrupole mass spectrometry|year = 2002|journal = [[Rapid Communications in Mass Spectrometry]]|volume = 16|issue = 7|pages = 670–675|doi = 10.1002/rcm.623|pmid = 11921245}}</ref>
-The duty cycle of IMS is short relative to liquid chromatography or gas chromatography separations and can thus be coupled to such techniques, producing triple modalities such as LC/IMS/MS.<ref>{{cite journal|author = Rena A. Sowell, Stormy L. Koeniger, Stephen J. Valentine, Myeong Hee Moon and David E. Clemmer|title = Nanoflow LC/IMS-MS and LC/IMS-CID/MS of Protein Mixtures|year = 2004|journal = [[Journal of the American Society for Mass Spectrometry]]|volume = 15|issue = 9|pages = 1341–1353|doi = 10.1016/j.jasms.2004.06.014|pmid = 15337515}}</ref>
-==Data and analysis==
-[[Image:ObwiedniaPeptydu.gif|thumb|right|400 px|Mass spectrum of a peptide showing the isotopic distribution]]
-===Data representations===
-{{See also|Mass spectrometry data format}}
-Mass spectrometry produces various types of data. The most common data representation is the [[mass spectrum]].
-Certain types of mass spectrometry data are best represented as a [[mass chromatogram]]. Types of chromatograms include selected ion monitoring (SIM), total ion current (TIC), and selected reaction monitoring chromatogram (SRM), among many others.
-Other types of mass spectrometry data are well represented as a three-dimensional [[contour map]]. In this form, the mass-to-charge, ''m/z'' is on the ''x''-axis, intensity the ''y''-axis, and an additional experimental parameter, such as time, is recorded on the ''z''-axis.
-===Data analysis===
-'''Basics'''
-Mass spectrometry data analysis is a complicated subject that is very specific to the type of experiment producing the data. There are general subdivisions of data that are fundamental to understanding any data.
-Many mass spectrometers work in either ''negative ion mode'' or ''positive ion mode''. It is very important to know whether the observed ions are negatively or positively charged. This is often important in determining the neutral mass but it also indicates something about the nature of the molecules.
-Different types of ion source result in different arrays of fragments produced from the original molecules. An electron ionization source produces many fragments and mostly single-charged (1-) radicals (odd number of electrons), whereas an electrospray source usually produces non-radical quasimolecular ions that are frequently multiply charged. Tandem mass spectrometry purposely produces fragment ions post-source and can drastically change the sort of data achieved by an experiment.
-By understanding the origin of a sample, certain expectations can be assumed as to the component molecules of the sample and their fragmentations. A sample from a synthesis/manufacturing process will probably contain impurities chemically related to the target component. A relatively crudely prepared biological sample will probably contain a certain amount of salt, which may form [[adduct]]s with the analyte molecules in certain analyses.
-Results can also depend heavily on how the sample was prepared and how it was run/introduced. An important example is the issue of which matrix is used for MALDI spotting, since much of the energetics of the desorption/ionization event is controlled by the matrix rather than the laser power. Sometimes samples are spiked with sodium or another ion-carrying species to produce adducts rather than a protonated species.
-The greatest source of trouble when non-mass spectrometrists try to conduct mass spectrometry on their own or collaborate with a mass spectrometrist is inadequate definition of the research goal of the experiment. Adequate definition of the experimental goal is a prerequisite for collecting the proper data and successfully interpreting it. Among the determinations that can be achieved with mass spectrometry are molecular mass, molecular structure, and sample purity. Each of these questions requires a different experimental procedure. Simply asking for a "mass spec" will most likely not answer the real question at hand.
-'''Interpretation of mass spectra'''
-{{Main|Mass spectrum analysis}}
-Since the precise [[Chemical structure|structure]] or [[peptide sequence]] of a molecule is deciphered through the set of fragment masses, the interpretation of [[Mass spectrum|mass spectra]] requires combined use of various techniques. Usually the first strategy for identifying an unknown compound is to compare its experimental mass spectrum against a library of mass spectra. If the search comes up empty, then manual interpretation<ref>{{cite book|author=Tureček, František; McLafferty, Fred W.|year=1993|title=Interpretation of mass spectra|publisher=University Science Books|place=Sausalito|isbn=0-935702-25-3|url=http://books.google.com/?id=xQWk5WQfMQAC&printsec=frontcover}}</ref> or [[Mass spectrometry software|software assisted interpretation of mass spectra]] are performed. Computer simulation of [[ionization]] and fragmentation processes occurring in mass spectrometer is the primary tool for assigning structure or peptide sequence to a molecule. An ''[[A priori and a posteriori|a priori]]'' structural information is fragmented ''[[in silico]]'' and the resulting pattern is compared with observed spectrum. Such simulation is often supported by a fragmentation library<ref>Mistrik, R.(2004). [http://www.highchem.com/publications/a-new-concept-for-the-interpretation-of-mass-spectra.html A New Concept for the Interpretation of Mass Spectra Based on a Combination of a Fragmentation Mechanism Database and a Computer Expert System.] in Ashcroft, A.E., Brenton, G., Monaghan,J.J. (Eds.), ''Advances in Mass Spectrometry'', Elsevier, Amsterdam, vol. 16, pp. 821.</ref> that contains published patterns of known decomposition reactions. [[Mass spectrometry software|Software]] taking advantage of this idea has been developed for both small molecules and [[Peptide mass fingerprinting|proteins]].
-Another way of interpreting mass spectra involves spectra with [[accurate mass]]. A mass-to-charge ratio value (''m/z'') with only integer precision can represent an immense number of theoretically possible ion structures. More precise mass figures significantly reduce the number of candidate [[Molecule#Molecular formula|molecular formulas]], albeit each can still represent a large number of structurally diverse compounds. A computer algorithm called formula generator calculates all molecular formulas that theoretically fit a given [[Accurate mass|mass]] with specified tolerance.
-A recent technique for structure elucidation in mass spectrometry, called [[precursor ion fingerprinting]] identifies individual pieces of structural information by conducting a search of the [[Tandem mass spectrometry|tandem spectra]] of the molecule under investigation against a library of the [[Tandem mass spectrometry#Tandem MS experiments|product-ion spectra]] of structurally characterized precursor ions.
-==Applications==
-===Isotope ratio MS: isotope dating and tracking===
-[[Image:Mass-spectrometer awi hg.jpg|thumb|right|300 px|Mass spectrometer to determine the <sup>16</sup>O/<sup>18</sup>O and <sup>12</sup>C/<sup>13</sup>C isotope ratio on biogenous carbonate]]
-{{Main|Isotope ratio mass spectrometry}}
-Mass spectrometry is also used to determine the [[isotope|isotopic]] composition of elements within a sample. Differences in mass among isotopes of an element are very small, and the less abundant isotopes of an element are typically very rare, so a very sensitive instrument is required. These instruments, sometimes referred to as isotope ratio mass spectrometers (IR-MS), usually use a single magnet to bend a beam of ionized particles towards a series of [[Faraday cup]]s which convert particle impacts to [[electric current]]. A fast on-line analysis of [[deuterium]] content of water can be done using [[Flowing afterglow mass spectrometry]], FA-MS. Probably the most sensitive and accurate mass spectrometer for this purpose is the [[accelerator mass spectrometry|accelerator mass spectrometer]] (AMS). Isotope ratios are important markers of a variety of processes. Some isotope ratios are used to determine the age of materials for example as in [[carbon dating]]. Labeling with stable isotopes is also used for protein quantification. (see [[Mass spectrometry#Protein characterization|protein characterization]] below)
-===Trace gas analysis===
-Several techniques use ions created in a dedicated ion source injected into a flow tube or a drift tube: [[SIFT-MS selected ion flow tube mass spectrometry|selected ion flow tube]] (SIFT-MS), and [[Proton-transfer-reaction mass spectrometry|proton transfer reaction]] (PTR-MS), are variants of [[chemical ionization]] dedicated for trace gas analysis of air, breath or liquid headspace using well defined reaction time allowing calculations of analyte concentrations from the known reaction kinetics without the need for internal standard or calibration.
-===Atom probe===
-{{Main|Atom probe}}
-An [[atom probe]] is an instrument that combines [[time-of-flight]] mass spectrometry and [[field ion microscope|field ion microscopy]] (FIM) to map the location of individual atoms.
-===Pharmacokinetics===
-{{Main|Pharmacokinetics}}
-Pharmacokinetics is often studied using mass spectrometry because of the complex nature of the matrix (often blood or urine) and the need for high sensitivity to observe low dose and long time point data. The most common instrumentation used in this application is [[liquid chromatography-mass spectrometry|LC-MS]] with a [[quadrupole mass analyzer|triple quadrupole mass spectrometer]]. Tandem mass spectrometry is usually employed for added specificity. Standard curves and internal standards are used for quantitation of usually a single pharmaceutical in the samples. The samples represent different time points as a pharmaceutical is administered and then metabolized or cleared from the body. Blank or t=0 samples taken before administration are important in determining background and ensuring data integrity with such complex sample matrices. Much attention is paid to the linearity of the standard curve; however it is not uncommon to use [[curve fitting]] with more complex functions such as quadratics since the response of most mass spectrometers is less than linear across large concentration ranges.<ref>{{Cite journal|year = 2006|title = Systems for Drug Metabolism and Pharmacokinetic Screening, Y. Hsieh and W.A. Korfmacher, Current Drug Metabolism|volume = 7|issue = 5|pages = 479–489|doi =10.2174/138920006777697963|author = Hsieh, Yunsheng|journal = Current Drug Metabolism|pmid = 16787157|last2 = Korfmacher|first2 = WA}}</ref><ref>{{Cite journal|last1 = Covey|first1 = T.R.|last2 = Lee|first2 = E.D.|last3 = Henion|first3 = J.D.|year = 1986|title = Mass Spectrometry for the Determination of Drugs in Biological Samples|journal = [[Anal. Chem.]]|volume = 58|pages = 2453–2460|doi = 10.1021/ac00125a022|pmid=3789400|issue = 12}}</ref><ref>{{Cite journal|year = 1985|title = Mass Spectrometry Determination of Drugs and Their Metabolites in Biological Fluids|journal = [[Anal. Chem.]]|volume = 57|issue = 2|pages = 474–81|doi = 10.1021/ac50001a036|author = Covey, Tom R.|last2 = Crowther|first2 = Jonathan B.|last3 = Dewey|first3 = Elizabeth A.|last4 = Henion|first4 = Jack D.|pmid=3977076}}</ref>
-There is currently considerable interest in the use of very high sensitivity mass spectrometry for [[microdosing]] studies, which are seen as a promising alternative to [[animal experimentation]].
-===Protein characterization===
-{{Main|Protein mass spectrometry}}
-Mass spectrometry is an important emerging method for the characterization and [[protein sequencing|sequencing]] of proteins. The two primary methods for ionization of whole proteins are [[electrospray ionization]] (ESI) and [[matrix-assisted laser desorption/ionization]] (MALDI). In keeping with the performance and mass range of available mass spectrometers, two approaches are used for characterizing proteins. In the first, intact proteins are ionized by either of the two techniques described above, and then introduced to a mass analyzer. This approach is referred to as "[[top-down proteomics|top-down]]" strategy of protein analysis. In the second, proteins are enzymatically digested into smaller [[peptides]] using [[protease]]s such as [[trypsin]] or [[pepsin]], either in [[solution]] or [[in-gel digestion|in gel]] after [[electrophoresis|electrophoretic]] separation. Other proteolytic agents are also used. The collection of peptide products are then introduced to the mass analyzer. When the characteristic pattern of peptides is used for the identification of the protein the method is called [[peptide mass fingerprinting]] (PMF), if the identification is performed using the sequence data determined in [[Tandem mass spectrometry|tandem MS]] analysis it is called [[de novo sequencing]]. These procedures of protein analysis are also referred to as the "[[Bottom-up proteomics|bottom-up]]" approach.
-===Glycan analysis===
-Mass spectrometry (MS), with its low sample requirement and high sensitivity, has been predominantly used in [[glycobiology]] for characterization and elucidation of [[glycan|glycan structures]].<ref>{{Cite journal|last1 = Apte|first1 = A.|last2 = Meitei|first2 = N.S.|year = 2009|title = Bioinformatics in Glycomics: Glycan Characterization with Mass Spectrometric Data Using SimGlycan|work = Functional Glycomics: Methods and Protocols|journal = Methods in molecular biology (Clifton, N.J.)|pmid = 19882135|volume = 600|pages = 269–281|doi = 10.1007/978-1-60761-454-8_19|series = Methods in Molecular Biology|isbn = 978-1-60761-453-1}}</ref> Mass spectrometry provides a complementary method to [[High-performance liquid chromatography|HPLC]] for the analysis of glycans. Intact glycans may be detected directly as singly charged ions by [[matrix-assisted laser desorption/ionization|matrix-assisted laser desorption/ionization mass spectrometry]] (MALDI-MS) or, following permethylation or peracetylation, by [[fast atom bombardment|fast atom bombardment mass spectrometry]] (FAB-MS).<ref>{{Cite journal|last1 =Harvey|first1 = D.|last2 = Dwek|first2 = R.A.|last3 = Rudd|first3 = P.M.|year = 2000|title = Determining the Structure of Glycan Moieties by Mass Spectrometry|pmid =18429296|journal = [[Current Protocols]] in Protein Science|volume =Chapter 12|pages = 12.7–12.7.15|doi = 10.1002/0471140864.ps1207s43|isbn =0-471-14086-4|unused_data =Unit Number: UNIT 12.7}}</ref> [[Electrospray ionization mass spectrometry]] (ESI-MS) also gives good signals for the smaller glycans.<ref>{{Cite journal|last1 = Blow|first = N.|year = 2009|title = Glycobiology: A spoonful of sugar|pmid = 19177129|journal = [[Nature (journal)|Nature]]|volume = 457|issue = 7229|pages = 617–620|doi = 10.1038/457617a|bibcode = 2009Natur.457..617B|unused_data = first N. }}</ref> Various [[Glycobiology#Software for MS Data Interpretation|free and commercial software]] are now available which interpret MS data and aid in Glycan structure characterization.
-===Space exploration===
-As a standard method for analysis, mass spectrometers have reached other planets and moons. Two were taken to [[Mars]] by the [[Viking program]]. In early 2005 the [[Cassini–Huygens]] mission delivered a specialized [[GC-MS]] instrument aboard the [[Huygens probe]] through the atmosphere of [[Titan (moon)|Titan]], the largest moon of the planet [[Saturn]]. This instrument analyzed atmospheric samples along its descent trajectory and was able to vaporize and analyze samples of Titan's frozen, hydrocarbon covered surface once the probe had landed. These measurements compare the abundance of isotope(s) of each particle comparatively to earth's natural abundance.<ref>{{cite journal|author = S. Petrie and D. K. Bohme|title = Ions in space|year = 2007|journal = [[Mass Spectrometry Reviews]]|volume = 26|issue = 2|pages = 258–280|doi = 10.1002/mas.20114|pmid = 17111346}}</ref> Also on board the [[Cassini–Huygens]] spacecraft is an ion and neutral mass spectrometer which has been taking measurements of Titan's atmospheric composition as well as the composition of [[Enceladus (moon)|Enceladus]]' plumes. A [[Thermal and Evolved Gas Analyzer]] mass spectrometer was carried by the [[Phoenix (spacecraft)|Mars Phoenix Lander]] launched in 2007.<ref>{{Cite journal| doi = 10.1016/j.jasms.2008.07.015| title = Phoenix Mars Mission—The Thermal Evolved Gas Analyzer| year = 2008| last1 = Hoffman| first1 = J| last2 = Chaney| first2 = R| last3 = Hammack| first3 = H| journal = Journal of the American Society for Mass Spectrometry| volume = 19| pages = 1377–83| postscript = <!--None-->| pmid = 18715800| issue = 10 }}</ref>
-Mass spectrometers are also widely used in space missions to measure the composition of plasmas. For example, the Cassini spacecraft carries the Cassini Plasma Spectrometer (CAPS),<ref>{{cite web|url=http://caps.space.swri.edu/ |title=Cassini Plasma Spectrometer |accessdate=2008-01-04 |publisher=[[Southwest Research Institute]]}}</ref> which measures the mass of ions in Saturn's [[magnetosphere]].
-===Respired gas monitor===
-Mass spectrometers were used in hospitals for respiratory gas analysis beginning around 1975 through the end of the century. Some are probably still in use but none are currently being manufactured.<ref>{{cite journal |author=Riker JB, Haberman B |title=Expired gas monitoring by mass spectrometry in a respiratory intensive care unit |journal=Crit. Care Med. |volume=4 |issue=5 |pages=223–9 |year=1976 |pmid=975846 |doi= 10.1097/00003246-197609000-00002}}</ref>
-Found mostly in the [[operating room]], they were a part of a complex system, in which respired gas samples from patients undergoing [[anesthesia]] were drawn into the instrument through a valve mechanism designed to sequentially connect up to 32 rooms to the mass spectrometer. A computer directed all operations of the system. The data collected from the mass spectrometer was delivered to the individual rooms for the anesthesiologist to use.
-The uniqueness of this magnetic sector mass spectrometer may have been the fact that a plane of detectors, each purposely positioned to collect all of the ion species expected to be in the samples, allowed the instrument to simultaneously report all of the gases respired by the patient. Although the mass range was limited to slightly over 120 [[atomic mass unit|u]], fragmentation of some of the heavier molecules negated the need for a higher detection limit.<ref>{{cite journal|author = J. W. W. Gothard, C. M. Busst, M. A. Branthwaite, N. J. H. Davies and D. M. Denison|title = Applications of respiratory mass spectrometry to intensive care|year = 1980|journal = [[Anaesthesia (journal)|Anaesthesia]]|volume = 35|issue = 9|pages = 890–895|doi = 10.1111/j.1365-2044.1980.tb03950.x|pmid=6778243}}</ref>
-== See also ==
-* [[Mass spectrometry software]]
-* [[Calutron]]
-* [[Helium mass spectrometer]]
-* [[Mass spectrometry imaging]]
-* [[Reflectron]]
-* [[Dumas method of molecular weight determination]]
-* [[MassBank (database)]], a Japanese spectral database
-{{colend}}
 == References ==
-{{reflist|2}}
-== Bibliography ==
-{{refbegin}}
-*{{cite book |author=Tureček, František; McLafferty, Fred W. |title=Interpretation of mass spectra |publisher=University Science Books |location=Sausalito, Calif |year=1993 |isbn=0-935702-25-3|url=http://books.google.com/?id=xQWk5WQfMQAC&printsec=frontcover}}
-*{{cite book|author = Edmond de Hoffman|coauthors = Vincent Stroobant|title = Mass Spectrometry: Principles and Applications|edition = 2nd|publisher = John Wiley and Sons|year = 2001|isbn = 0-471-48566-7}}
-*{{cite book |author=Downard, Kevin |title=Mass Spectrometry – A Foundation Course |publisher=Royal Society of Chemistry |location=Cambridge UK |year=2004 |url=http://books.google.com/?id=-8LtzxKrSwkC|isbn=0-85404-609-7}}
-*{{cite book |author=Siuzdak, Gary |title=Mass spectrometry for biotechnology |publisher=Academic Press |location=Boston |year=1996 |isbn=0-12-647471-0}}
-*{{cite book |author=Dass, Chhabil |title=Principles and practice of biological mass spectrometry |publisher=John Wiley |location=New York |year=2001 |isbn=0-471-33053-1}}
-*{{cite book |author=Jnrgen H. Gross |title=Mass Spectrometry: A Textbook |publisher=Springer-Verlag |location=Berlin |year= 2006|isbn=3-540-40739-1|url=http://books.google.com/?id=e10yKTODUzoC&printsec=frontcover}}
-*{{cite journal|author=Muzikar, P., ''et al.''|title=Accelerator Mass Spectrometry in Geologic Research|journal=Geological Society of America Bulletin|volume=115|year=2003|pages=643–654|doi=10.1130/0016-7606(2003)115<0643:AMSIGR>2.0.CO;2|issn=0016-7606}}
-*{{cite book |author=O. David Sparkman |title=Mass Spectrometry Desk Reference |publisher=Global View Pub |location=Pittsburgh |year= 2006|isbn=0-9660813-9-0}}
-*{{cite book |author=J. Throck Watson and O. David Sparkman |title=Introduction to Mass Spectrometry: Instrumentatio, Applications, and Strategies for Data Interpretation, 4th Ed. |publisher=Jonh Wiley & Sons |location=Chichester |year= 2007|isbn=978-0-470-51634-8}}
-*{{cite book |author=Tuniz, C. |title=Accelerator mass spectrometry: ultrasensitive analysis for global science |publisher=CRC Press |location=Boca Raton |year=1998|isbn=0-8493-4538-3|url=http://books.google.com/?id=RhW2k4u70ZcC&printsec=frontcover}}
-{{refend}}
-== External links ==
-{{Wikibooks}}
-{{Commons}}
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-*[http://asms.org ASMS] American Society for Mass Spectrometry
-*[http://www.magnet.fsu.edu/education/tutorials/java/massspectra/index.html Interactive tutorial on mass spectra] National High Magnetic Field Laboratory
-*[http://www.vias.org/simulations/simusoft_msscope.html Mass spectrometer simulation] An interactive application simulating the console of a mass spectrometer
-*[http://www.gazard.com/stephen/chemistry/ Realtime Mass Spectra simulation] Tool to simulate mass spectra in the browser
-{{Analytical chemistry}}
-{{pathology}}
-{{Mass spectrometry}}
-{{DEFAULTSORT:Mass Spectrometry}}
-[[Category:Chemical pathology]]
-[[Category:Mass spectrometry]]
-[[Category:Measuring instruments]]
-[[Category:Scientific techniques]]
-{{Link FA|pl}}
+{{reflist}}
-<!-- interwiki -->
+* Edwin Thompson Jaynes. ''Probability Theory: The Logic of Science''. [[Cambridge University Press]], 2003. ISBN 0-521-59271-2.
+* Persi Diaconis and Joseph B. Keller. "Fair Dice". ''The American Mathematical Monthly'', 96(4):337-339, 1989. ''(Discussion of dice that are fair "by symmetry" and "by continuity".)''
+*{{citation|last=Keynes|first=John Maynard|authorlink=John Maynard Keynes|contribution=Chapter IV. The Principle of Indifference|title=A Treatise on Probability|volume=4|publisher=Macmillan and Co.|year=1921|url=http://books.google.com/books?id=YmCvAAAAIAAJ&pg=PA41&dq=%22principle+of+indifference%22#v=onepage&q=%22principle%20of%20indifference%22&f=false|pages=41–64}}.
+*{{cite book | last = Stigler | first = Stephen M.
+| title = The history of statistics : the measurement of uncertainty before 1900
+| publisher = Belknap Press of Harvard University Press
+| location = Cambridge, Mass | year = 1986 | isbn = 0-674-40340-1}}
-[[ar:مطيافية الكتلة]]
+[[Category:Probability theory]]
-[[bg:Масспектрометрия]]
+[[Category:Statistical principles]]
-[[ca:Espectrometria de massa]]
-[[cs:Hmotnostní spektrometrie]]
-[[da:Massespektrometri]]
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-[[et:Massispektromeetria]]
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-[[eo:Mas-spektrogramo]]
-[[fa:طیف‌سنجی جرمی]]
-[[fr:Spectrométrie de masse]]
-[[ko:질량 분석]]
-[[hr:Masena spektrometrija]]
-[[io:Mas-spektrogramo]]
-[[id:Spektrometri massa]]
-[[it:Spettrometria di massa]]
-[[he:ספקטרומטר מסה]]
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