Educational Tools for Introductory Bayesian Statistics using Mathematica

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Educational Toolsfor Introductory Bayesian
Statistics using Mathematica

• Shin-ichi MayekawaGraduate School of Decision
  Science and Technology.Tokyo Institute of
  Technology

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Purpose of this Research

• Find a way to use Mathematicaefficiently in
  Bayesian Statistics.
• Mathematica can do Symbolic Math.Especially,
  definite integration.

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Outline

• What Mathematica can and cannot do.
• What mathStatica can and cannot do.
• What my Bayespack can do.

Application of SuMOpack (2005)
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What Mathematica Can Do

• Knows(memorizes) PDF, CDF,mean, variance,
  skewness, kurtosisof many distributions.
• Knows(memorizes) Characteristic Function
  ofunivariate distribution.
• Can symbolically calculate/derive the
  expectation of a function of the random
  variable.
• And More.

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ltltStatistics
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What Mathematica Cannot Do

• Given an expression (full or kernel) and the
  name of the random variable, it cannot identify
  the distribution.
• Cannot directly calculate marginal/conditional
  distributions.
• Cannot handle fully symbolic multivariate
  distributions.
• No Bayesian distributions.

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What is mathStatica ?

• mathStatica is a package created by Colin Rose
  and Murray Smith(2002)
• Mathematical Statistics with MathematicaSpringer
  Texts in Statistics 2002with which we can
  study and practice mathematical statistics using
  Mathemtatica.
• http//www.mathstatica.com/reviews/

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What mathStatica Can Do

• Given PDF, mathStatica can do manysymbolic
  derivations using the followingfunctionsExpect
  , Var, Corr, Cov, Prob, Transform, Jacob,
  Sufficient,Conditional, Marginal, and more.

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What mathStatica Cannot Do

• Given an expression (full or kernel) and the
  name of the random variable, it cannot identify
  the distribution.
• Cannot handle fully symbolic multivariate
  distributions.
• No Bayesian distributions.
• No Bayesian statistics.

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Bayespack objectives

• Provide several Bayesian distributuionssuch as
  Inverted xxxx distribution.
• Given an expression (full or kernel) and the
  name of the random variable(RV), identify the
  distribution of RV.

• Find the kernel of the distributionby pattern
  matching,and find the normalizing constant.

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Bayespack objectives

• Should be able to handle fully symbolicmultivaria
  te random variables.

Use SuMOpack.
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SuMOpack for Mathematica (2005)
1) Fully Symbolic Matrix Operations
1. Simplification of Matrix Expressions 2.
Simplification of Partitioned Matrix Expressions
3. Conversion of Matrix Expressions to
Summation Expressions 4. Derivative of a Scalar
Function of Matrices w.r.t. a Matrix
2) Fully Symbolic Summation Operations
1. Simplification of Summation Expressions 2.
Conversion of Summation Expressions to Matrix
Expressions 3. Derivative of a Summation
Expression w.r.t. a Subscripted Variable
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Bayespack objectives

• Should be able to do the standard Bayesian
  Analysis.

• Identify the product of the Likelihood and the
  prior using the parametersas RV.

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Bayesian Distributions

• Chi and Chi-Squared distribution(with scale
  parametes)
• Inverted Chi, Chi-Squared distributionInverted
  Gamma distribution
• t distribution (with mean and scale parametes)
• Multivariate t distributionmatric t distribution
  (with mean and scale parametes)
• Inverted Wishart distribution

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Identifying the Distribution
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Bayesian Posterior Distributions

• Method 1 Using the tools such as
  completeSquare,transform the joint distribution
  to the standard form and identify.

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Completion of Square
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Completion of Square
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Completion of Square
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Normal (natural conjugate)
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Normal (natural conjugate)
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Normal (natural conjugate)
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Normal (natural conjugate)
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Normal (natural conjugate)
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Normal Regression (natural conjugate)
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Normal Regression (natural conjugate)
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Normal Regression (natural conjugate)
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Normal Regression (natural conjugate)
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Normal Regression (natural conjugate)
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Bayesian Posterior Distributions

• Method 2 Try to identify the distribution
  automatically if possible without transforming to
  the standard form.

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Normal Regression (natural conjugate)
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Normal Regression (natural conjugate)
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Normal Regression (natural conjugate)
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Normal Regression (natural conjugate)
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Normal Regression (natural conjugate)
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Normal Regression (natural conjugate)
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Conclusions

• Bayespack can be used as an Educational Tool.

It may be more suited
for those whowish to write a textbook on
Bayseian Statistics.
Thank you.
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Where to Download

• http//www.ms.hum.titech.ac.jp/sumopack/sumopack.
  zip
• http//www.ms.hum.titech.ac.jp/sumopack/Bayespack
  .zip (by the end of June)
• mayekawa_at_hum.titech.ac.jp