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'''Bayesian experimental design''' provides a general probability-theoretical framework from which other theories on [[Design of experiments|experimental design]] can be derived. It is based on [[Bayesian inference]] to interpret the observations/data acquired during the experiment. This allows accounting for both any prior knowledge on the parameters to be determined as well as uncertainties in observations.
 
The theory of Bayesian experimental design is to a certain extent based on the theory for making [[optimal decision|optimal decisions under uncertainty]]. The aim when designing an experiment is to  maximize the expected utility of the experiment outcome. The utility is most commonly defined in terms of a measure of the accuracy of the information provided by the experiment (e.g. the [[Shannon information]] or the negative [[variance]]), but may also involve factors such as the financial cost of performing the experiment. What will be the optimal experiment design depends on the particular utility criterion chosen.
 
==Relations to more specialized optimal design theory==
===Linear theory===
If the model is linear, the prior [[probability density function]] (PDF) is homogeneous and observational errors are [[Multivariate normal distribution|normally distributed]], the theory simplifies to the classical [[optimal design|optimal experimental design theory]].
 
===Approximate normality===
In numerous publications on Bayesian experimental design, it is (often implicitly) assumed that all posterior PDFs will be approximately normal. This allows for the expected utility to be calculated using linear theory, averaging over the space of model parameters, an approach reviewed in {{Harvtxt|Chaloner|Verdinelli|1995}}. Caution must however be taken when applying this method, since approximate normality of all possible posteriors is difficult to verify, even in cases of normal observational errors and uniform prior PDF.
 
===Posterior distribution===
Recently, increased computational resources allow inference of the [[posterior distribution]] of model parameters, which can directly be used for experiment design. {{Harvtxt|Vanlier|Tiemann|Hilbers|van Riel|2012}} proposed an approach that uses the [[posterior predictive distribution]] to assess the effect of new measurements on prediction uncertainty, while {{Harvtxt|Liepe|Filippi|Komorowski|Stumpf|2013}} suggest maximizing the mutual information between parameters, predictions and potential new experiments.
 
==Mathematical formulation==
 
{| border=1 align="right" cellpadding="0" cellspacing="0" bordercolor="#111111" width="270px"
|
{| border=0
|+ '''Notation'''
|-valign="top"
|nowrap| <math>\theta\,</math>
| parameters to be determined
|-valign="top"
|nowrap| <math>y\,</math>
| observation or data
|-valign="top"
|nowrap| <math>\xi\,</math>
| design
|-valign="top"
|nowrap| <math>p(y|\theta,\xi)\,</math>
| PDF for making observation <math>y</math>, given parameter values <math>\theta</math> and design <math>\xi</math>
|-valign="top"
|nowrap| <math>p(\theta)\,</math>
| prior PDF
|-valign="top"
|nowrap| <math>p(y|\xi)\,</math>
| marginal PDF in observation space
|-valign="top"
|nowrap| <math>p(\theta | y, \xi)\,</math> &nbsp;&nbsp;
| posterior PDF
|-valign="top"
|nowrap| <math>U(\xi)\,</math> &nbsp;&nbsp;
| utility of the design <math>\xi</math>
|-valign="top"
|nowrap| <math>U(y, \xi)\,</math> &nbsp;&nbsp;
| utility of the experiment outcome after observation <math>y</math> with design <math>\xi</math>
|}
|}
 
Given a vector <math>\theta</math> of parameters to determine, a [[Prior probability|prior PDF]] <math>p(\theta)</math> over those parameters and a PDF <math>p(y|\theta,\xi)</math> for making observation <math>y</math>, given parameter values <math>\theta</math> and an experiment design <math>\xi</math>, the posterior PDF can be calculated using [[Bayes' theorem]]
:<math>p(\theta | y, \xi) = \frac{p(y | \theta, \xi) p(\theta)}{p(y | \xi)}  \, ,</math>
 
where <math>p(y|\xi)</math> is the marginal probability density in observation space
:<math>p(y|\xi) = \int{p(\theta)p(y|\theta,\xi)d\theta}\, .</math>
 
The expected utility of an experiment with design <math>\xi</math> can then be defined
:<math>U(\xi)=\int{p(y|\xi)U(y,\xi)dy}\, ,</math>
where <math>U(y,\xi)</math> is some real-valued functional of the [[Posterior probability|posterior PDF]] <math>p(\theta | y, \xi)</math> after making observation <math>y</math> using an experiment design <math>\xi</math>.
 
===Gain in Shannon information as utility===
 
Utility may be defined as the prior-posterior gain in [[Differential entropy|Shannon information]]
:<math> U(y, \xi) = \int{\log(p(\theta | y, \xi))p(\theta | y, \xi)d\theta} - \int{\log(p(\theta))p(\theta)d\theta} \, .</math>
Note also that
:<math>U(y, \xi) = D_{KL}(p(\theta|y,\xi) \| p(\theta|\xi)) \, ,</math>
the [[Kullback–Leibler divergence]] of the prior from the posterior distribution.
{{Harvtxt|Lindley|1956}} noted that the expected utility will then be coordinate-independent and can be written in two forms
:<math>
\begin{alignat}{2}
U(\xi) & = \int{\int{\log(p(\theta | y,\xi))p(\theta, y | \xi)d\theta}dy} - \int{\log(p(\theta))p(\theta)d\theta} \\
      & = \int{\int{\log(p(y | \theta,\xi))p(\theta, y | \xi)dy}d\theta} - \int{\log(p(y| \xi))p(y| \xi)dy} ,
\end{alignat}
\, </math>
 
of which the latter can be evaluated without the need for evaluating individual posterior PDFs
<math>p(\theta | y,\xi)</math> for all possible observations <math>y</math>. Worth noting is that the first term on the second equation line will not depend on the design <math>\xi</math>, as long as the observational uncertainty doesn't. On the other hand, the integral of <math>p(\theta) \log p(\theta)</math> in the first form is constant for all <math>\xi</math>, so if the goal is to choose the design with the highest utility, the term need not be computed at all. Several authors have considered numerical techniques for evaluating and optimizing this criterion, e.g. {{Harvtxt|van den Berg|Curtis|Trampert|2003}} and {{Harvtxt|Ryan|2003}}.  Note that
:<math>U(\xi) = I(\theta;y)\, ,</math>
the expected '''information gain''' being exactly the '''[[mutual information]]'''  between the parameter ''θ'' and the observation ''y''.  {{Harvtxt|Kelly|1956}} also derived just such a utility function for a gambler seeking to [[Kelly criterion|profit maximally]] from [[Gambling and information theory|side information in a horse race]]; Kelly's situation is identical to the foregoing, with the side information, or "private wire" taking the place of the experiment.
 
==See also==
 
*[[Optimal design|Optimal Designs]]
*[[Active_learning_(machine_learning)|Active Learning]]
 
{{No footnotes|date=March 2011}}
==References==
 
* {{Citation
  | last1 = Vanlier
  | last2 = Tiemann
  | last3 = Hilbers
  | last4 = van Riel
  | year = 2012
  | title = A Bayesian approach to targeted experiment design
  | journal = Bioinformatics
  | volume = 28
  | pages = 1136–1142
  | url = http://bioinformatics.oxfordjournals.org/content/28/8/1136.full.pdf
  | doi = 10.1093/bioinformatics/bts092
  | issue = 8
}}
 
* {{Citation
  | last1 = Liepe
  | last2 = Filippi
  | last3 = Komorowski
  | last4 = Stumpf
  | year = 2013
  | title = Maximizing the Information Content of Experiments in Systems Biology
  | journal = PLoS computational biology
  | volume = 9
  | pages = e1002888
  | url = http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1002888
  | doi = 10.1371/journal.pcbi.1002888
  | issue = 1
}}
 
* {{Citation
  | last1 = van den Berg
  | last2 = Curtis
  | last3 = Trampert
  | year = 2003
  | title = Optimal nonlinear Bayesian experimental design: an application to amplitude versus offset experiments
  | journal = Geophysical Journal International
  | volume = 55
  | pages = 411–421
  | url = http://www.geos.ed.ac.uk/homes/acurtis/Papers/vandenBerg_etal2003.pdf
  | issue = 2
}}
 
* {{Citation
  | last1 = Chaloner
  | last2 = Verdinelli
  | year = 1995
  | title = Bayesian experimental design: a review
  | journal = Statistical Science
  | volume = 10
  | pages = 273–304
  | url = http://www.stat.uiowa.edu/~gwoodwor/AdvancedDesign/Chaloner%20Verdinelli.pdf
  | doi = 10.1214/ss/1177009939
  | issue = 3
  | first1 = Kathryn
  | first2 = Isabella
}}
 
*  {{Citation| author=DasGupta, A.
  | chapter=Review of optimal Bayes designs
  | url = http://www.stat.purdue.edu/~dasgupta/publications/tr95-04.pdf
  | pages=1099–1148
  | title=Design and Analysis of Experiments
  | series=Handbook of Statistics
  | volume=13
  |editor=Ghosh, S. and [[Calyampudi Radhakrishna Rao|Rao, C. R.]]
  | publisher=North-Holland| year=1996| isbn=0-444-82061-2
}}
 
* {{Citation
  | last = Lindley
  | year = 1956
  | title = On a measure of information provided by an experiment
  | journal = Annals of Mathematical Statistics
  | volume = 27
  | pages = 986–1005
  | url = http://projecteuclid.org/handle/euclid.aoms/1177728069
  | doi = 10.1214/aoms/1177728069
  | issue = 4
  | first1 = D. V.
}}
 
* {{Citation
  | last = Ryan
  | first= K. J.
  | year = 2003
  | title = Estimating Expected Information Gains for Experimental Designs With Application to the Random Fatigue-Limit Model
  | journal = Journal of Computational and Graphical Statistics
  | volume = 12
  | pages = 585–603
  | doi = 10.1198/1061860032012
  | issue = 3
}}
 
{{Experimental design|state=expanded}}
{{Statistics|collection|state=collapsed}}
 
[[Category:Bayesian statistics|Experimental design]]
[[Category:Design of experiments]]
[[Category:Statistical methods]]
[[Category:Optimal decisions]]
[[Category:Operations research]]
[[Category:Industrial engineering]]
[[Category:Systems engineering]]
[[Category:Information, knowledge, and uncertainty]]

Revision as of 13:40, 27 January 2014

Bayesian experimental design provides a general probability-theoretical framework from which other theories on experimental design can be derived. It is based on Bayesian inference to interpret the observations/data acquired during the experiment. This allows accounting for both any prior knowledge on the parameters to be determined as well as uncertainties in observations.

The theory of Bayesian experimental design is to a certain extent based on the theory for making optimal decisions under uncertainty. The aim when designing an experiment is to maximize the expected utility of the experiment outcome. The utility is most commonly defined in terms of a measure of the accuracy of the information provided by the experiment (e.g. the Shannon information or the negative variance), but may also involve factors such as the financial cost of performing the experiment. What will be the optimal experiment design depends on the particular utility criterion chosen.

Relations to more specialized optimal design theory

Linear theory

If the model is linear, the prior probability density function (PDF) is homogeneous and observational errors are normally distributed, the theory simplifies to the classical optimal experimental design theory.

Approximate normality

In numerous publications on Bayesian experimental design, it is (often implicitly) assumed that all posterior PDFs will be approximately normal. This allows for the expected utility to be calculated using linear theory, averaging over the space of model parameters, an approach reviewed in Template:Harvtxt. Caution must however be taken when applying this method, since approximate normality of all possible posteriors is difficult to verify, even in cases of normal observational errors and uniform prior PDF.

Posterior distribution

Recently, increased computational resources allow inference of the posterior distribution of model parameters, which can directly be used for experiment design. Template:Harvtxt proposed an approach that uses the posterior predictive distribution to assess the effect of new measurements on prediction uncertainty, while Template:Harvtxt suggest maximizing the mutual information between parameters, predictions and potential new experiments.

Mathematical formulation

Notation
parameters to be determined
observation or data
design
PDF for making observation , given parameter values and design
prior PDF
marginal PDF in observation space
   posterior PDF
   utility of the design
   utility of the experiment outcome after observation with design

Given a vector of parameters to determine, a prior PDF over those parameters and a PDF for making observation , given parameter values and an experiment design , the posterior PDF can be calculated using Bayes' theorem

where is the marginal probability density in observation space

The expected utility of an experiment with design can then be defined

where is some real-valued functional of the posterior PDF after making observation using an experiment design .

Gain in Shannon information as utility

Utility may be defined as the prior-posterior gain in Shannon information

Note also that

the Kullback–Leibler divergence of the prior from the posterior distribution. Template:Harvtxt noted that the expected utility will then be coordinate-independent and can be written in two forms

of which the latter can be evaluated without the need for evaluating individual posterior PDFs for all possible observations . Worth noting is that the first term on the second equation line will not depend on the design , as long as the observational uncertainty doesn't. On the other hand, the integral of in the first form is constant for all , so if the goal is to choose the design with the highest utility, the term need not be computed at all. Several authors have considered numerical techniques for evaluating and optimizing this criterion, e.g. Template:Harvtxt and Template:Harvtxt. Note that

the expected information gain being exactly the mutual information between the parameter θ and the observation y. Template:Harvtxt also derived just such a utility function for a gambler seeking to profit maximally from side information in a horse race; Kelly's situation is identical to the foregoing, with the side information, or "private wire" taking the place of the experiment.

See also

Template:No footnotes

References

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    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

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    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

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