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| '''Statistical proof''' is the rational demonstration of degree of certainty for a [[proposition]], [[hypothesis]] or [[scientific theory|theory]] that is used to convince others subsequent to a [[statistical test]] of the supporting [[evidence]] and the types of [[inference]]s that can be drawn from the test scores. Statistical methods are used to increase the understanding of the facts and the proof demonstrates the [[Validity (statistics)|validity]] and logic of inference with explicit reference to a hypothesis, the [[experimental data]], the facts, the test, and the [[odds]]. [[Proof (truth)|Proof]] has two essential aims: the first is to convince and the second is to explain the proposition through peer and public review.<ref name="Gold08">{{cite book | last1=Gold | first1=B. | last2=Simons | first2=R. A. | title=Proof and other dilemmas: Mathematics and philosophy | year=2008 | publisher=Mathematics Association of America Inc. | isbn=0-88385-567-4 | url=http://books.google.ca/books?id=wPhwJdjI-dIC&printsec=frontcover&dq=Proof+and+other+dilemmas:+mathematics+and+philosophy&hl=en&ei=SVq7TvQ4g-KIAteutIkC&sa=X&oi=book_result&ct=book-thumbnail&resnum=1&ved=0CDEQ6wEwAA#v=onepage&q&f=false}}</ref>
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| The [[Philosophic burden of proof|burden of proof]] rests on the demonstrable application of the statistical method, the disclosure of the assumptions, and the relevance that the test has with respect to a genuine understanding of the data relative to the external world. There are adherents to several different statistical philosophies of inference, such as [[Bayes theorem]] versus the [[likelihood function]], or [[positivism]] versus [[critical rationalism]]. These methods of reason have direct bearing on statistical proof and its interpretations in the broader philosophy of science.<ref name="Gold08" /><ref name="Gattei08">{{cite book | title=Thomas Kuhn's "Linguistic Turn" and the Legacy of Logical Empiricism: Incommensurability, Rationality and the Search for Truth | last1=Gattei | first1=S. | year=2008 | pages=277 | publisher=Ashgate Pub Co | isbn=0-7546-6160-1 | url=http://books.google.ca/books?id=HDOuKjMc0AEC&printsec=frontcover&dq=Thomas+Kuhn's+%22linguistic+turn%22+and+the+legacy+of+logical+empiricism&hl=en&ei=Xz-8TpznC6aPiALPn6mdAw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CC4Q6AEwAA#v=onepage&q&f=false}}</ref>
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| A common demarcation between science and non-science is the [[hypothetico-deductive]] proof of falsification developed by [[Karl Popper]], which is a well-established practice in the tradition of statistics. Other modes of inference, however, may include the [[inductive reasoning|inductive]] and [[abductive reasoning|abductive]] modes of proof.<ref name="Pedemont07">{{cite journal |last1=Pedemont | first1=B. | year=2007 | title=How can the relationship between argumentation and proof be analysed? | journal=Educational Studies in Mathematics | volume=66 | issue=1 | pages=23–41 | doi=10.1007/s10649-006-9057-x | url=http://www.springerlink.com/content/424p8r420820316l/}}</ref> Scientists do not use statistical proof as a means to attain certainty, but to [[Falsifiability|falsify]] claims and explain theory. Science cannot achieve absolute certainty nor is it a continuous march toward an objective truth as the vernacular as opposed to the scientific meaning of the term "proof" might imply. Statistical proof offers a kind of proof of a theory's falsity and the means to learn [[heuristics|heuristically]] through repeated statistical trials and experimental error.<ref name="Gattei08" /> Statistical proof also has applications in legal matters with implications for the [[legal burden of proof]].<ref name="Meier86">{{cite journal | last1=Meier | first1=P. | title=Damned Liars and Expert Witnesses | journal=Journal of the American Statistical Association | volume=81 | issue=394 | year=1986 | pages=269–276 | url=http://users.stat.umn.edu/~sandy/courses/8801/articles/Law/meierlaw.pdf}}</ref>
| | If an existing Word - Press code is found vulnerable, Word - Press will immediately issue an update for that. This means you can setup your mailing list and auto-responder on your wordpress site and then you can add your subscription form to any other blog, splash page, capture page or any other site you like. Your parishioners and certainly interested audience can come in to you for further information from the group and sometimes even approaching happenings and systems with the church. They found out all the possible information about bringing up your baby and save money at the same time. Over a million people are using Wordpress to blog and the number of Wordpress users is increasing every day. <br><br>Choosing what kind of links you'll be using is a ctitical aspect of any linkwheel strategy, especially since there are several different types of links that are assessed by search engines. Best of all, you can still have all the functionality that you desire when you use the Word - Press platform. With the free Word - Press blog, you have the liberty to come up with your own personalized domain name. Apart from these, you are also required to give some backlinks on other sites as well. By using Word - Press, you can develop very rich, user-friendly and full-functional website. <br><br>You can down load it here at this link: and utilize your FTP software program to upload it to your Word - Press Plugin folder. It was also the very first year that the category of Martial Arts was included in the Parents - Connect nationwide online poll, allowing parents to vote for their favorite San Antonio Martial Arts Academy. You can now search through the thousands of available plugins to add all kinds of functionality to your Word - Press site. To turn the Word - Press Plugin on, click Activate on the far right side of the list. For any web design and development assignment, this is definitely one of the key concerns, specifically for online retail outlets as well as e-commerce websites. <br><br>Additionally Word - Press add a default theme named Twenty Fourteen. Here's more in regards to [http://ll.my/backupplugin181908 wordpress backup plugin] look into the site. In case you need to hire PHP developers or hire Offshore Code - Igniter development services or you are looking for Word - Press development experts then Mindfire Solutions would be the right choice for a Software Development partner. This allows for keeping the content editing toolbar in place at all times no matter how far down the page is scrolled. Giant business organizations can bank on enterprise solutions to incorporate latest web technologies such as content management system etc, yet some are looking for economical solutions. This includes enriching the content with proper key words, tactfully defining the tags and URL. <br><br>Every single module contains published data and guidelines, usually a lot more than 1 video, and when pertinent, incentive links and PDF files to assist you out. Mahatma Gandhi is known as one of the most prominent personalities and symbols of peace, non-violence and freedom. As a result, it is really crucial to just take aid of some experience when searching for superior quality totally free Word - Press themes, Word - Press Premium Themes for your web site. Thus, Word - Press is a good alternative if you are looking for free blogging software. Get started today so that people searching for your type of business will be directed to you. |
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| == Axioms ==
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| There are two kinds of [[axioms]], 1) conventions that are taken as true that should be avoided because they cannot be tested, and 2) hypotheses.<ref name="Wiley75">{{cite journal | last1=Wiley | first1=E. O. | title=Karl R. Popper, Systematics, and Classification: A Reply to Walter Bock and Other Evolutionary Taxonomists | journal=Systematic Biology | volume=24 | issue=2 | pages=233–243 | year=1974 | doi=10.1093/sysbio/24.2.233 | url=http://www.hum.utah.edu/~mhaber/Documents/Course%20Readings/Wiley-PopperSystematics-SystZoo1975.pdf}}</ref> Proof in the theory of probability was built on four axioms developed in the late 17th century:
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| #The probability of a hypotheses is a non-negative real number: <math>\bigg\{ \Pr(h) \geqq 0 \bigg\}</math>;
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| #The probability of necessary truth equals one: <math>\bigg\{ \Pr(t) = 1 \bigg\}</math>;
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| #If two hypotheses h<sub>1</sub> and h<sub>2</sub> are mutually exclusive, then the sum of their probabilities is equal to the probability of their [[Logical disjunction|disjunction]]: <math>\bigg\{ \Pr \left ( h_1 \right ) + \Pr \left (h_2 \right ) = \Pr \left (h_1 or h_2\right ) \bigg\}</math>;
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| #The conditional probability of h<sub>1</sub> given h<sub>2</sub> <math>\Bigg\{ \Pr(h_1|h_2) \Bigg\}</math> is equal to the unconditional probability <math>\bigg\{ \Pr(h_1 \And h_2) \bigg\}</math> of the conjunction h<sub>1</sub> and h<sub>2</sub>, divided by the unconditional probability <math>\bigg\{ \Pr(h_2) \bigg\}</math> of h<sub>2</sub> where that probability is positive <math>\bigg\{ \frac{\Pr(h_1|h_2) = \Pr(h_1 \And h_2)}{\Pr(h_2)} \bigg\}</math>, where <math>\bigg\{ \Pr (h_2) > 0 \bigg\}</math>.
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| The preceding axioms provide the statistical proof and basis for the [[scientific law|laws]] of randomness, or objective chance from where modern statistical theory has advanced. Experimental data, however, can never prove that the hypotheses (h) is true, but relies on an inductive inference by measuring the probability of the hypotheses relative to the empirical data. The proof is in the rational demonstration of using the [[Statistical inference|logic of inference]], [[mathematical proof|math]], [[Statistical hypothesis testing|testing]], and [[deductive]] [[reason]]ing of [[Statistical significance|significance]].<ref name="Gold08" /><ref name="Gattei08" /><ref name="Howson91">{{cite journal | last1=Howson | first1=C. | last2=Urbach | first2=P. | title=Bayesian reasoning in science | journal=Nature | volume=350 | issue=6317 | year=1991 | pages=371–374 | url=http://www.hum.utah.edu/~mhaber/Documents/Course%20Readings/Howson_Urbach_Nature1991.pdf}}</ref>
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| == Test and proof ==
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| {{main|Statistical tests}}
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| The term ''proof'' descended from its Latin roots (provable, probable, ''probare'' L.) meaning ''to test''.<ref name="Sundholm94">{{cite journal | last1=Sundholm | first1=G. | title=Proof-Theoretical Semantics and Fregean Identity Criteria for Propositions | journal=The Monist | volume=77 | issue=3 | pages=294–314 | url=https://openaccess.leidenuniv.nl/bitstream/handle/1887/11990/9_054_019.pdf?sequence=1}}</ref><ref name="Bissell96">{{cite journal | last1=Bissell | first1=D. | title=Statisticians have a Word for it | journal=Teaching Statistics | volume=18 | issue=3 | pages=87–89 | year=1996 | url=http://www.rsscse-edu.org.uk/tsj/wp-content/uploads/2011/03/bissell1.pdf}}</ref> Hence, proof is a form of inference by means of a statistical test. Statistical tests are formulated on models that generate [[probability distributions]]. Examples of probability distributions might include the [[Bernoulli distribution|binary]], [[normal distribution|normal]], or [[poisson distribution]] that give exact descriptions of variables that behave according to [[natural law]]s of [[Randomness|random chance]]. When a [[statistical test]] is applied to samples of a population, the test determines if the sample statistics are significantly different from the assumed [[null-hypothesis|null-model]]. True values of a population, which are unknowable in practice, are called parameters of the population. Researchers sample from populations, which provide estimates of the parameters, to calculate the mean or standard deviation. If the entire population is sampled, then the sample statistic mean and distribution will converge with the parametric distribution.<ref name="Sokal95">{{cite book | last1=Sokal | first1=R. R. | last2=Rohlf | first2=F. J. | title=Biometry | edition=3rd | year=1995 | isbn=0-7167-2411-1 | publisher=W.H. Freeman & Company | pages=887 | url=http://books.google.ca/books?id=N6KCNw5NHNkC&dq=biometry&hl=en&ei=_ka7TuLtCObmiAKfuKSDAg&sa=X&oi=book_result&ct=book-thumbnail&resnum=1&ved=0CDUQ6wEwAA}}</ref>
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| Using the scientific method of falsification, the [[P-value|probability value]] that the sample statistic is sufficiently different from the null-model than can be explained by chance alone is given prior to the test. Most statisticians set the prior probability value at 0.05 or 0.1, which means if the sample statistics diverge from the parametric model more than 5 (or 10) times out of 100, then the discrepancy is unlikely to be explained by chance alone and the null-hypothesis is rejected. Statistical models provide exact outcomes of the parametric and estimates of the sample statistics. Hence, the [[Philosophic burden of proof|burden of proof]] rests in the sample statistics that provide estimates of a statistical model. Statistical models contain the [[mathematical proof]] of the parametric values and their probability distributions.<ref name="Heath95">{{cite book | title=An introduction to experimental design and statistics for biology | last1=Heath | first1=David | year=1995 | isbn=1-85728-132-2 | publisher=CRC Press | url=http://books.google.ca/books?id=sJVHoJ2ML40C&printsec=frontcover&dq=statistics+for+biologists&hl=en&ei=9e-6ToGzMq3MiQKrkfiODA&sa=X&oi=book_result&ct=book-thumbnail&resnum=7&ved=0CEgQ6wEwBg#v=onepage&q=parametric&f=false}}</ref><ref name="Hald06">{{cite book | last1=Hald | first1=Anders | title=A History of Parametric Statistical Inference from Bernoulli to Fisher, 1713-1935 | year=2006 | publisher=Springer | pages=260 | isbn=0-387-46408-5 | url=http://books.google.ca/books?id=Uc9C90KKW_UC&printsec=frontcover&dq=A+history+of+parametric+statistical+inference+from+Bernoulli+to+Fisher&hl=en&ei=_fe6Tui3MIiuiQKhw5yMDA&sa=X&oi=book_result&ct=book-thumbnail&resnum=1&ved=0CDAQ6wEwAA#v=onepage&q&f=false}}</ref>
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| == Bayes theorem ==
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| {{main|Bayes theorem}} {{see also|Evidence under Bayes theorem}}
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| [[Bayesian statistics]] are based on a different philosophical approach for proof of [[Statistical inference|inference]]. The mathematical formula for Bayes's theorem is:
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| <math>Probability [Parameter | Data] = \frac{Probability [Data | Parameter ] X Probability [Parameter ]}{Pr[Data]}</math>
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| The formula is read as the probability of the parameter (or hypothesis ''=h'', as used in the notation on [[Statistical_proof#Axioms|axioms]]) “given” the data (or empirical observation), where the horizontal bar refers to "given". The right hand side of the formula calculates the prior probability of a statistical model (Pr [Parameter]) with the [[Likelihood function|likelihood]] (Pr [Data | Parameter]) to produce a posterior probability distribution of the parameter (Pr [Parameter | Data]). The posterior probability is the likelihood that the parameter is correct given the observed data or samples statistics.<ref name="Huelsenbeck01">{{cite journal | last1=Huelsenbeck | first1=J. P. | last2=Ronquist | first2=F. | last3=Bollback | first3=J. P. | year=2001 | title=Bayesian Inference of Phylogeny and Its Impact on Evolutionary Biology | journal=Science | volume=294 | issue=5550 | pages=2310=2314 | doi=10.1126/science.1065889 | url=http://bama.ua.edu/~molsyst/page2/assets/Huelsenbeck2001.pdf}}</ref> Hypotheses can be compared using Bayesian inference by means of the Bayes factor, which is the ratio of the posterior odds to the prior odds. It provides a measure of the data and if it has increased or decreased the likelihood of one hypotheses relative to another.<ref name="Wade00">{{cite journal | last1=Wade | first1=P. R. | title=Bayesian methods in conservation biology | journal=Conservation Biology | year=2000 | volume=14 | issue=5 | pages=1308–1316 | url=http://www.utm.utoronto.ca/~collinsn/1310/Nov%203/optional/Bayesian%20methods%20in%20conserv%20biol.pdf}}</ref>
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| The statistical proof is the Bayesian demonstration that one hypothesis has a higher (weak, strong, positive) likelihood.<ref name="Wade00" /> There is considerable debate if the Bayesian method aligns with Karl Poppers method of proof of falsification, where some have suggested that "...there is no such thing as "accepting" hypotheses at all. All that one does in science is assign degrees of belief..."<ref name="Sober91">{{cite book | last1=Sober | first1=E. | title=Reconstructing the Past: Parsimony, Evolution, and Inference | year=1991 | publisher=A Bradford Book | isbn=0-262-69144-2 | pages=284 | url=http://books.google.ca/books?id=WlUL_r06GnEC&pg=PA180&lpg=PA180&dq=%22All+that+one+does+in+science+is+assign+degrees+of+belief%22&source=bl&ots=ii2pXy2BzM&sig=FYFilXaMbM-9rRNrJLDs0rhBrD0&hl=en&ei=s328TrqZHKiZiQLA9sGIAw&sa=X&oi=book_result&ct=result&resnum=2&sqi=2&ved=0CCIQ6AEwAQ#v=onepage&q=%22All%20that%20one%20does%20in%20science%20is%20assign%20degrees%20of%20belief%22&f=false}}</ref>{{rp|180}} According to Popper, hypotheses that have withstood testing and have yet to be falsified are not verified but [[Corroborating evidence|corroborated]]. Some researches have suggested that Popper's quest to define corroboration on the premise of probability put his philosophy in line with the Bayesian approach. In this context, the likelihood of one hypothesis relative to another may be an index of corroboration, not confirmation, and thus statistically proven through rigorous objective standing.<ref name="Howson91" /><ref name="Helfenbein05">{{cite journal | last1=Helfenbein | first1=K. G. | last2=DeSalle | first2=R. | title=Falsifications and corroborations: Karl Popper's influence on systematics | journal=Molecular Phylogenetics and Evolution | volume=35 | year=2005 | pages=271–280 | doi=10.1016/j.ympev.2005.01.003 | url=http://research.amnh.org/users/desalle/pdf/Helfenbein.2005.MPE.pdf}}</ref>
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| == In legal proceedings ==
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| {{main|Legal burden of proof}}
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| {{quote box
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| | quote = "Where gross statistical disparities can be shown, they alone may in a proper case constitute ''prima facie'' proof of a pattern or practice of discrimination."{{#tag:ref|Supreme Court of the United States ''Castaneda v. Partida'', 1977 [http://www.law.cornell.edu/supct/html/historics/USSC_CR_0430_0482_ZS.html] cited in Meier (1986) Ibid. who states "Thus, in the space of less than half a year, the Supreme Court had moved from the traditional legal disdain for statistical proof to a strong endorsement of it as being capable, on its own, of establishing a prima facie case against a defendant."<ref name="Meier86" />|group="nb"}}{{Rp|271}}
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| Statistical proof in a legal proceeding can be sorted into three categories of evidence:
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| #The occurrence of an event, act, or type of conduct,
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| #The identity the individual(s) responsible
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| #The intent or psychological responsibility<ref name="Fienberg82">{{cite journal | last1=Fienberg | first2=S. E. | last2=Kadane | first2=J. B. | title=The presentation of Bayesian statistical analyses in legal proceedings | journal=Journal of the Royal Statistical Society, Series D | volume=32 | issue=1/2 | pages=88–98 | url=http://www.jstor.org/stable/2987595}}</ref>
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| Statistical proof was not regularly applied in decisions concerning United States legal proceedings until the mid 1970's following a landmark jury discrimination case in ''Castaneda v. Partida''. The US Supreme Court ruled that gross statistical disparities constitutes "''[[prima facie]]'' proof" of discrimination, resulting in a shift of the burden of proof from plaintiff to defendant. Since that ruling, statistical proof has been used in many other cases on inequality, discrimination, and DNA evidence.<ref name="Meier86" /><ref name="Garaud90">{{cite journal | last1=Garaud | first1=M. C. | title=Legal Standards and Statistical Proof in Title VII Litigation: In Search of a Coherent Disparate Impact Model | journal=University of Pennsylvania Law Review | volume=139 | issue=2 | pages=455–503 | year=1990 | url=http://www.jstor.org/stable/3312286}}</ref><ref name="HLRA95">{{cite journal | last1=The Harvard Law Review Association | title=Developments in the Law: Confronting the New Challenges of Scientific Evidence | journal=Harvard Law Review | volume=108 | issue=7 | year=1995 | pages=1481–1605 | url=http://www.jstor.org/stable/1341808}}</ref> However, there is not a one-to-one correspondence between statistical proof and the legal burden of proof. "The Supreme Court has stated that the degrees of rigor required in the fact finding processes of law and science do not necessarily correspond."<ref name="HLRA95" />{{rp|1533}}
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| In an example of a death row sentence (''McCleskey v. Kemp''{{#tag:ref|481 U.S. 279 (1987).<ref name="Faigman91" />|group="nb"}}) concerning racial discrimination, the petitioner, a black man named McCleskey was charged with the murder of a white police officer during a robbery. Expert testimony for McClesky introduced a statistical proof showing that "defendants charged with killing white victims were 4.3 times as likely to receive a death sentence as charged with killing blacks."<ref name="Faigman91">{{cite journal | last1=Faigman | first1=D. L. | title=Normative Constitutional Fact-Finding": Exploring the Empirical Component of Constitutional Interpretation | journal=University of Pennsylvania Law Review | volume=139 | issue=3 | year=1991 | pages=541–613 | url=http://www.jstor.org/stable/3312337}}</ref>{{rp|595}}. Nonetheless, the statistics was insufficient "to prove that the decisionmakers in his case acted with discriminatory purpose."<ref name="Faigman91" />{{rp|596}} It was further argued that there were "inherent limitations of the statistical proof"<ref name="Faigman91" />{{rp|596}}, because it did not refer to the specifics of the individual. Despite the statistical demonstration of an increased probability of discrimination, the legal burden of proof (it was argued) had to be examined on a case by case basis.<ref name="Faigman91" />
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| == See also ==
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| * [[Mathematical proof]]
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| * [[Data analysis]]
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| ==References==
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| <references/>
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| ==Notes==
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| {{reflist|group="nb"}}
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| [[Category:Statistical terminology]]
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| [[Category:Logic and statistics]]
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If an existing Word - Press code is found vulnerable, Word - Press will immediately issue an update for that. This means you can setup your mailing list and auto-responder on your wordpress site and then you can add your subscription form to any other blog, splash page, capture page or any other site you like. Your parishioners and certainly interested audience can come in to you for further information from the group and sometimes even approaching happenings and systems with the church. They found out all the possible information about bringing up your baby and save money at the same time. Over a million people are using Wordpress to blog and the number of Wordpress users is increasing every day.
Choosing what kind of links you'll be using is a ctitical aspect of any linkwheel strategy, especially since there are several different types of links that are assessed by search engines. Best of all, you can still have all the functionality that you desire when you use the Word - Press platform. With the free Word - Press blog, you have the liberty to come up with your own personalized domain name. Apart from these, you are also required to give some backlinks on other sites as well. By using Word - Press, you can develop very rich, user-friendly and full-functional website.
You can down load it here at this link: and utilize your FTP software program to upload it to your Word - Press Plugin folder. It was also the very first year that the category of Martial Arts was included in the Parents - Connect nationwide online poll, allowing parents to vote for their favorite San Antonio Martial Arts Academy. You can now search through the thousands of available plugins to add all kinds of functionality to your Word - Press site. To turn the Word - Press Plugin on, click Activate on the far right side of the list. For any web design and development assignment, this is definitely one of the key concerns, specifically for online retail outlets as well as e-commerce websites.
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Every single module contains published data and guidelines, usually a lot more than 1 video, and when pertinent, incentive links and PDF files to assist you out. Mahatma Gandhi is known as one of the most prominent personalities and symbols of peace, non-violence and freedom. As a result, it is really crucial to just take aid of some experience when searching for superior quality totally free Word - Press themes, Word - Press Premium Themes for your web site. Thus, Word - Press is a good alternative if you are looking for free blogging software. Get started today so that people searching for your type of business will be directed to you.