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Get a package for symbolic Bayesian computations?

Posted 4 months ago
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Dear Mathematical Users Can anyone tell me if there is a package or codes available to carry out Bayesian symbolic computation ? I would like to do some symbolic derivation of posterior densities and similar any help / suggestions greatly appreciated kindest regards ifty khan university of oxford

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If you know the PDF of the prior you can compute the posterior distribution for univariate distributions manually using Bayes rule. For example, consider a normal distribution with fixed standard deviation \[Sigma] and unknown mean \[Mu] where \[Mu] has a normal prior with hyperparameters m and s. Given an observation x, we can compute the posterior distribution manually using Bayes rule

In[33]:= posterior=ProbabilityDistribution[PDF[NormalDistribution[m,s],\[Mu]]*Likelihood[NormalDistribution[\[Mu],\[Sigma]],{x}],{\[Mu],-\[Infinity],\[Infinity]},Method->"Normalize",Assumptions->(m|x)\[Element]Reals&&(\[Sigma]|s)\[Element]PositiveReals]
Out[33]= ProbabilityDistribution[(E^(-((\[FormalX]-m)^2/(2 s^2))-(-\[FormalX]+x)^2/(2 \[Sigma]^2)+(m-x)^2/(2 (s^2+\[Sigma]^2))) Sqrt[s^2+\[Sigma]^2])/(Sqrt[2 \[Pi]] s \[Sigma]),{\[FormalX],-\[Infinity],\[Infinity]},Assumptions->(m|x)\[Element]\[DoubleStruckCapitalR]&&(\[Sigma]|s)\[Element]\[DoubleStruckCapitalR]&&\[Sigma]>0&&s>0]

You'll notice that this result, while technically correct, is very messy.

In[39]:= {Mean,Variance}[posterior] // Through
Out[39]= {(s^2 x+m \[Sigma]^2)/(s^2+\[Sigma]^2),(s^2 \[Sigma]^2)/(s^2+\[Sigma]^2)}

The Wolfram Language doesn't implicitly know any conjugate distributions, nor does it seem to be able to identify when distributions have special known forms, outside of a few cases.

One feature that would make these kinds of calculations significantly easier, I believe, would be the ability to recognize exponential families automatically. This would allow the automatic calculation of many conjugate prior distributions, their update equations, and posterior predictive distributions.

Additionally, identifying conjugate priors would allow the system to fit posterior predictive distributions and more advanced clustering models efficiently using the EM algorithm automatically.

One final note is that the Wolfram Language cannot do symbolic matrix calculus when the number of dimensions aren't specified, which makes working with multivariate distributions somewhat of a chore. For example, the same calculation for a multivariate normal specified using matrix algebra doesn't work, since the Wolfram Language can't integrate with symbolic tensors to obtain the normalizing constant.

One little extra tip I can add, is that you can use DistributionParameterAssumptions and DistributionDomain to help you with the automatic generation of simplifying assumptions:

In[1]:= DistributionParameterAssumptions[NormalDistribution[m, s]] 
Out[1]= m \[Element] Reals && s > 0

In[3]:= DistributionDomain[NormalDistribution[m, s]]
Out[3]= Interval[{-\[Infinity], \[Infinity]}]

thanks for taking the time to send a reply. this is most helpful. Dr Khan

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