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Educational Tools for Introductory Bayesian Statistics using Mathematica. Shin-ichi Mayekawa Graduate School of Decision Science and Technology. Tokyo Institute of Technology. Purpose of this Research. Find a way to use Mathematica e fficiently in Bayesian Statistics. - PowerPoint PPT Presentation
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IMPS200
6
Educational Toolsfor
Introductory Bayesian Statistics using Mathematica
Shin-ichi Mayekawa
Graduate School of Decision Science and Technology.
Tokyo Institute of Technology
IMPS2006
Purpose of this Research
Find a way to use Mathematicaefficiently in Bayesian Statistics.
Mathematica can do Symbolic Math.Especially, definite integration.
IMPS200
6
Outline
• What Mathematica can and cannot do.• What mathStatica can and cannot do.• What my Bayespack can do.
Application of SuMOpack (2005)
IMPS2006
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.
IMPS2006
In[67]:= dist BetaDistribution, PDFdist, XCDFdist, XMeandistVariancedistSkewnessdistKurtosisdistCharacteristicFunctiondist, t
Out[67]= BetaDistribution, Out[68]=1 X1 X1
, Out[69]= IX, Out[70]=
Out[71]= 2 1
Out[72]=2 1
2
Out[73]=3 12 2 6
2 3Out[74]= 1F1; ; tIn[75]:= ExpectedValueSqrtX, dist, X
Out[75]= 1
2
, 12
<<Statistics`
IMPS2006
In[100]:= dist MultinormalDistribution1, 2,s1,1, s1,2,s1,2, s2,2PDFdist,x1, x2CDFdist,x1, x2MeandistVariancedistSkewnessdistKurtosisdistCharacteristicFunctiondist, t
Out[100]= MultinormalDistribution1, 2,s1,1 s1,2
s1,2 s2,2
Out[101]=
E
12
x22x22s1,1s1,1s2,2s1,2
2 x11s1,2s1,1 s2,2s1,2
2
x11x11s2,2s1,1 s2,2s1,2
2 x22s1,2s1,1s2,2s1,2
2
2s1,1 s2,2 s1,2
2
Out[102]= CDFMultinormalDistribution
1, 2,s1,1 s1,2
s1,2 s2,2
,x1, x2Out[103]=1, 2Out[104]=s1,1, s2,2Out[105]=0, 0Out[106]=3, 3Out[107]= CharacteristicFunction
MultinormalDistribution1, 2,s1,1 s1,2
s1,2 s2,2
, tIn[45]:= ExpectedValue1, dist,x1, x2
ExpectedValuex1, dist,x1, x2Out[45]= 1
Out[46]=If1 0,
2 1s1,1 , IntegrateEx112
2s1,1 x1,x1, , , Assumptions 1 02s1,1
IMPS2006
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.
IMPS2006
Calculating the conditionaldistributions by NativeMathematica Conditional x given , and marginal f 12 2 Exp 1
2x ̂22
h0 1
2
Exp 222
x2
2222
2
222 Joint x and
g f h0
x2
22 222
22 Marginal x
f0 Integrateg,, , , Assumptions x, , , 0, 0 x2
222222 2
IMPS2006
Conditional given x h1 gf0h1 Simplify%, 0, 0, x ,
x2222x222 2
2222 2
22
2x2222
22 2222 22 Does it integrate to unity?
Integrateh1,, , , Assumptions x, , , 0, 01 Conditional Mean and Variance of given x meanMu Integrate h1,, , , Assumptions x, , , 0, 0varMu Integrate meanMu2 h1,, , , Assumptions x, , , 0, 02 x 2
2 2
2 2
2 2
IMPS2006
In[108]:= dist MultinormalDistribution, PDFdist, XCDFdist, XMeandistVariancedistSkewnessdistKurtosisdistCharacteristicFunctiondist, t
Out[108]= MultinormalDistribution, Out[109]= PDFMultinormalDistribution, , XOut[110]= CDFMultinormalDistribution, , XOut[111]=
Out[112]=
Out[113]= SkewnessMultinormalDistribution, Out[114]= KurtosisMultinormalDistribution, Out[115]= CharacteristicFunctionMultinormalDistribution, , tIn[116]:= ExpectedValue1, dist, X
Out[116]= ExpectedValue1, MultinormalDistribution, , X
IMPS2006
What is mathStatica ?mathStatica is a package created by: Colin Rose and Murray Smith(2002)
Mathematical Statistics with MathematicaSpringer Texts in Statistics 2002
with which we can study and practice mathematical statistics using Mathemtatica.
http://www.mathstatica.com/reviews/
IMPS2006
What mathStatica Can Do
Given PDF, mathStatica can do manysymbolic derivations using the followingfunctions:
Expect, Var, Corr, Cov, Prob, Transform, Jacob, Sufficient,Conditional, Marginal, and more.
IMPS2006
Calculating the conditional distributions by mathStatica
In[2]:= mathStatica.m
In[3]:= Conditional x given , and marginal f 12 2 Exp 1
2x ̂22
domainfx, , && Reals, 0Out[3]=
x22 222
Out[4]=x, , && Reals , 0In[5]:= h0
1
2
Exp 222
domainh0, , && Reals, 0Out[5]=
22 22
Out[6]=, , && Reals , 0In[7]:=
In[8]:= Joint x and g f h0domaingx, , ,, , && Reals, 0, 0
Out[8]=x22 2
22 2
22
Out[9]=x, , ,, , && Reals , 0, 0
IMPS2006
In[10]:= Conditional given x h1 Conditional, gh1 Simplify%, Reals, 0, 0, x , domainh1, , && Reals, 0Here is the conditional pdf gx:
Out[10]= 2x 2222
2 2 22222 22 2
Out[11]= 2x 2222
2 2 2222 22
Out[12]=, , && Reals , 0
IMPS2006
In[13]:= Conditional Mean and Variance of given x Expect, h1Simplify%, Reals, 0, 0Var, h1Simplify%, Reals, 0, 0This further assumes that: 1
2
1
2 0, 2
2 x 2
2 0Imx 2
2 0,
x 2
2 0
Out[13]=
2 x 21 2
22 232Out[14]=
2 x 2
2 2
This further assumes that: 12
1
2 0,
x 2
2 0Imx
2 0,
x 2
2 0
Out[15]= 2 x 222 22
2 4 22 x 22 x2 241
2 1
22 232
Out[16]=2 2
2 2
IMPS2006
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.
IMPS2006
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.
IMPS2006
Bayespack: objectives
Should be able to handle fully symbolicmultivariate random variables.
1
2x.1.xUse SuMOpack.
IMPS200
6
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
IMPS2006
Bayespack: objectives
Should be able to do the standard Bayesian Analysis.
Identify the product of the Likelihood and the prior using the parametersas RV.
IMPS2006
Bayesian Distributions
Chi and Chi-Squared distribution(with scale parametes)Inverted Chi, Chi-Squared distributionInverted Gamma distributiont distribution (with mean and scale parametes)
Multivariate t distributionmatric t distribution (with mean and scale parametes)
Inverted Wishart distribution
IMPS2006
In[79]:= list "ChiSquared", , spdfx, list, 1, 1;pdfx, list, 2, 1;
Out[79]=ChiSquared, , sThe density of ChiSquared distribution with
the Parameters , s22 s x
2 s2 x1 2
Gamma2
The kernel of ChiSquared distribution with
the Parameters , s s x
2 x12
IMPS2006
In[82]:= list "InvertedChiSquared", , spdfx, list, 1, 1;pdfx, list, 2, 1;
Out[82]=InvertedChiSquared, , sThe density of InvertedChiSquared distribution with
the Parameters , s22 s
2 x s2 x1 2
Gamma2
The kernel of InvertedChiSquared distribution with
the Parameters , s s2 x x1
2
IMPS2006
In[94]:= list "Multivariatet", , M, Spdfx, list, 1pdfx, list, 2
Out[94]=Multivariatet, , M , SOut[95]=
nrx
2 2x M.S1.x M12nrx1
2 nrxS
2
Out[96]=x M.S1.x M 2
nrx2
IMPS2006
In[109]:= list "MatrixT", M, S, V, f pdfX, listpdfX, list, 2
Out[109]=MatrixT, M , S , V , Out[110]=
1
2 ncXnrXS nrX2 V12ncXX M.S1.X MV112ncXnrX1
i1
nrX1
2 i 1
i1
nrX1
2 i ncX 1
Out[111]=X M.S1.X M V112ncXnrX1In[106]:= list "MatrixT", M, S, V, , 2
f pdfX, listpdfX, list, 2
Out[106]=MatrixT, M , S , V , , 2Out[107]=
1
2 ncXnrXS12nrXVncX2 S X M.V .X M12ncXnrX1
i1
ncX1
2 i 1
i1
ncX1
2 i nrX 1
Out[108]=S X M.V .X M12ncXnrX1
IMPS2006
In[115]:= list "InvertedWishart", , pdfX, list, 1pdfX, list, 2
Out[115]=InvertedWishart, , Out[116]=
2 1
2nrX1nrX 1
2 Tr1.X1 1
4nrX1nrXX212nrX1i1nrX
12 i nrX
Out[117]= 1
2 Tr1.X1X2
IMPS2006
Identifying the DistributionOut[276]=
E2x2222
22 22222 22 2
In[277]:= IdentifyDist%, f f1, where the argument of Exp and Power in f1is Expanded and Collected .
Input : f E 2x 2222
2 2 22222 22 2
Simplified Input : f1 12 E 2
2 222 122
2
22222 2
222 x22
22
x 2222 x2 2
2 222 x 22
2 2
2 2222 2
Pattern matched KNormal : E2 a_. b_.c_. d_.
The distribution of is the Normal Distribution with
mu 2 x 2
2 2, sigma2
2 2
2 2,
const
2 22
The result of the integration of f with respect to
1
Out[277]=1,Normal,2 x 2
2 2,
2 2
2 2
IMPS2006
In[165]:= IdentifyDist2 12 nr 141nrnrDet2i1
nrGamma1
21 i
DetX12nr1ExpTrInverse.X, X;Normal , MVt or Wishart ??
Input : f 2
12 nrETr1.X 14nr1nrX12nr12
i1
nr1
2i 1
Pattern matched KWishart : Ed_.c_. TrC_:Identity.XB_.DetXa_.b_ e_.The distribution of X is the Wishart Distribution withSigma 2 , df ,
const 2
12 nrETr0 1
4nr1nr2i1
nr1
2i 1
The result of the integration of f with respect to X 22 2
IMPS2006
Bayesian Posterior Distributions
Method 1 Using the tools such as completeSquare,transform the joint distribution to the standard form and identify.
IMPS2006
Completion of SquareIn[278]:= completeSquarecx a2 dx b2, x
Out[278]=c da b2c d
c dx a c b d
c d2
In[279]:= matCompleteSquaretx a.C.x a tx b.D.x b, xThe following matrices are assumed to be symmetric .C, DTwo quadratic forms of x found .
Out[279]=a b.C1 D11.a bxC D1.C.a D.b.C D.x C D1.C.a D.bIn[280]:= matCompleteSquaretx a.C.x a tx b.D.x b, x, 1, 0
The following matrices are assumed to be symmetric .C, DTwo quadratic forms of x found .
Out[280]=x C D1.C.a D.b.C D.x C D1.C.a D.ba b.C.C D1.D.a b
IMPS2006
Completion of SquareIn[308]:= exp ssqy X., W
matCompleteSquareRssexp, %.XT .W .X1.XT .W .y
Out[308]=y X ..W .y X .Out[309]=y X .X.W .X1.X.W .y.W .y X .X.W .X1.X.W .y X.W .X1.X.W .y.X.W .X .X.W .X1.X.W .yOut[310]=y X .
.W .y X .
.X.W .X .
In[311]:= matCompleteSquaressqy1 X1. ssqy2 X2., The following matrices are assumed to be symmetric .Identity , X1.X1, X2.X2Two quadratic forms of found .
Out[311]= X1.X1 X2.X21.X1.y1 X2.y2.X1.X1 X2.X2. X1.X1 X2.X21.X1.y1 X2.y2X1.X11.X1.y1X2.X21.X2.y2.X1.X11X2.X211.X1.X11.X1.y1 X2.X21.X2.y2y1 X1.X1.X11.X1.y1.X1.X1.y1 X1.X1.X11.X1.y1y2 X2.X2.X21.X2.y2.X2.X2.y2 X2.X2.X21.X2.y2
IMPS2006
Completion of SquareIn[313]:= matCompleteSquaressqy1 X1., W1 ssqy2 X2., W2,
The following matrices are assumed to be symmetric .W1, W2, X1.W1.X1, X2.W2.X2Two quadratic forms of found .
Out[313]= X1.W1.X1 X2.W2.X21.X1.W1.y1 X2.W2.y2.X1.W1.X1 X2.W2.X2. X1.W1.X1 X2.W2.X21.X1.W1.y1 X2.W2.y2X1.W1.X11.X1.W1.y1 X2.W2.X21.X2.W2.y2.X1.W1.X11 X2.W2.X211.X1.W1.X11.X1.W1.y1 X2.W2.X21.X2.W2.y2y1 X1.X1.W1.X11.X1.W1.y1.X1.W1.X1.y1 X1.X1.W1.X11.X1.W1.y1y2 X2.X2.W2.X21.X2.W2.y2.X2.W2.X2.y2 X2.X2.W2.X21.X2.W2.y2
IMPS2006
Normal (natural conjugate)
Bayesian Normal Model
Data xi N, 2, i=1,2,...,n Natural Conjugate Prior for and . | N(, 2n0) Inverted Sqrt-Gamma(alpha0,beta0) = Inverted Chi (,s)
IMPS2006
Normal (natural conjugate)In[365]:= The Likelihood
L Product1
2
Expx 2
2 2.x xi,i, 1, n The Natural Conjugate Prior
h1 1
2 2n0 Exp 2
2 2n0h2 Exp s
21
The Joint Density of and g L h1 h2
Out[365]=i1
nExi2
222
Out[366]=E2 n0
2222
n0
Out[367]= E s2 1
Out[368]=
En02
22 s2 1
i1
nExi2
222 22
n0
IMPS2006
Normal (natural conjugate)In[388]:= Simplify the joint distribution of and .
g0 gprodPowerToPowerSumg0 g0sumSimplify
Out[388]=2 n2
12 E
n02
22 i1
nxi2
22 s2 n21
n1
2
n0
Out[389]=2
12n1
En022si1
nxi222 n21
n1
2
n0
In[390]:= %.Expa_ ExpcompleteSqsumFullSimplifya, Out[390]=
212n1
Enn0i1
n xin2
22nn0 n22
n022 n0i1
n xinn0
2 i1n xi22nsni1
n xi2
2n2n21
n1
2
n0
IMPS2006
Normal (natural conjugate)In[391]:= g2 IdentifyDistM%, 1;
Input : f 212n1E nn0i1
n xin2
2 2nn0 n2 2
n02 2 n0i1n xi
nn02 i1n xi22 nsni1n xi
2
2 n 2 n21n 12
n0
Pattern matched KNormal : Eb_.a_.m_:02 d_.The distribution of is the Normal Distribution with
mu n0 i1n xi
n n0, sigma2
2
n n0,
const 212n1Ei1n xi2n2 si1n xi
2n0n 22i1n xi2 si1n xi2
2 2nn0 n21n 12
n0
The result of the integration of f with respect to
2n2 Ei1n xi2n2 si1n xi2n0n 22i1n xi2 si1n xi
22 2nn0
12 n2 nn0
n n0
IMPS2006
Normal (natural conjugate)In[394]:= IdentifyDistg2, ;
f f1, where the argument of Exp and Power in f1is Expanded and Collected .
Input : f 2n2 Ei1n xi2n2 si1n xi
2n0n 22i1n xi2 si1n xi2
2 2nn0 12 n2 nn0
n n0
Simplified Input : f1 2n2 E
nn0 22nn0 n0i1n xi
nn0i1n xi22nn0
ni1n xi2
2nn0 n0i1n xi2
2nn0 nsnn0
sn0nn0
2 12 n2 nn0
n n0
Pattern matched KInvertedSqrtGamma : Eb_.2
d_.a_ c_.
The distribution of is the Inverted Chi Distribution with
df n 1, s i1n xi2 n2 s i1n xi2 n0n 2 2i1n xi 2 s i1n xi2
n n0,
const 2n2 1
2 n2n0n n0s2is 2df. Alias Inverted Sqrt Gammadf2, 2s
The result of the integration of f with respect to
1
2 32
12 n2 1
2n 1n0n n012n2
n n0 2 2 n0i1n xi i1n xi2 2 n s 2 s n0 n n0
i1
n
xi212n1
IMPS2006
Normal Regression (natural conjugate)
Bayesian Multiple Regression Normal Model
Data y NX ., 2Identity, i=1,2,...,n Natural Conjugate Prior for and . N(, n0
2 )
1s,
IMPS2006
Normal Regression (natural conjugate)
In[404]:= The Likelihood wo constant Clear, , , , x, m;L
1n
Exp 12ty X..y X.2 The Natural Conjugate Prior
h1 1
DetExp n022
t .Inverse. h2 Exp s
2 21 The Joint Density of x and
g L h1 h2
Out[405]= EyX ..yX .
22 n
Out[406]=E.1.n0
22Out[407]= E
s22 1
Out[408]=E s
22 yX ..yX .
22 .1.n0
22 n1
IMPS2006
Normal Regression (natural conjugate)
In[409]:= matCompSq%, The following matrices are assumed to be symmetric .Identity , X.X, 1Two quadratic forms of found .
Out[409]=1E
sX.X1n01.X. y1.n0.X.X1n0.X.X1 n01.X. y1.n0X.X1.X. y.n0 X.X11.X.X1.X. yyX .X.X1.X. y.X.X .yX .X.X22
n1
IMPS2006
Normal Regression (natural conjugate)In[410]:= IdentifyDistM%, 1
Input : f 1
E sX.X1 n0
1.X.y1. n0.X.X1 n0.X.X1 n01.X.y1. n0X.X1.X.y.n0 X.X11.X.X1.X.yy 2 2
n1
Pattern matched KMVN : Ea_.Transposem_:0.Sinv_:Identity.m_:0b_.d_. e_.The distribution of is the Multivariate Normal Distribution with
Mu 2X.X 1 n01.X.y 1. n0, Sigma 2X.X 1 n01,const
EsX.X1.X.y.n0 X.X11.X.X1.X.yyX.X.X1.X.y.X.X.yX.X.X1.X.y
2 2 n1The result of the integration of f with respect to
1E sX.X
1.X.y.n0 X.X11.X.X1.X.yyX.X.X1.X.y.X.X.yX.X.X1.X.y2 2
2 nr2 nnrX.X1 n01X.X 1 n01
Out[410]=EsX.X1.X. y.n0 X.X11.X.X1.X. yyX .X.X1.X. y.X.X .yX .X.X1.X. y
22 2nr2 nnrX.X1n01X.X 1 n01
IMPS2006
Normal Regression (natural conjugate)
In[411]:= IdentifyDistM%, ;Input : f
1E sX.X
1.X.y.n0 X.X11.X.X1.X.yyX.X.X1.X.y.X.X.yX.X.X1.X.y2 2
2 nr2 nnrX.X1 n01X.X 1 n01
Pattern matched KInvertedSqrtGamma : Eb_.2
d_.a_ c_.
The distribution of is the Inverted Chi Distribution with
df n nrX.X 1 n0, s
s X.X1.X.y.n0
X.X11. X.X1.X.yy X.X.X1.X.y.X.X.y X.X.X1.X.y,const 2 nr2 X.X 1 n01s2is 2df. Alias Inverted Sqrt Gammadf2, 2s
The result of the integration of f with respect to
12 12nnrnrX.X1 n02 nr
2X.X 1 n01s X.X1.X.y.
n0X.X11. X.X1.X.y
y X.X.X1.X.y.X.X.y X.X.X1.X.y12nnrX.X1 n012n nrX.X 1 n0
IMPS2006
Bayesian Posterior Distributions
Method 2 Try to identify the distribution automatically if possible without transforming to the standard form.
IMPS2006
Normal Regression (natural conjugate)
Bayesian Multiple Regression Normal Model
Data y NX ., 2Identity, i=1,2,...,n Natural Conjugate Prior for and . N(, n0
2 )
1s,
IMPS2006
Normal Regression (natural conjugate)In[427]:= The Likelihood wo constant
Clear, , , , x, m;L
1n
Exp 12ty X..y X.2; The Natural Conjugate Prior
h1 1
DetExp n022
t .Inverse. ;h2 Exp s
2 21; The Joint Density of x and
g L h1 h2;printL, h1, h2, g;L E
yX..yX.
2 2 n
h1 E.1.n0
2 2h2 E
s2 2 1
g E s2 2
yX..yX.
2 2.1.n0
2 2 n1
IMPS2006
Normal Regression (natural conjugate)In[435]:= Identify the dist of and integrate g0 wrt to get the marginal of .
g2 IdentifyDistg, 1;Normal , MVt or Wishart ??
The following matrices are assumed to be symmetric .X.X2
, 1,1 n02
,X.X2
1 n02
f f1, where the argument of Exp and Power in f1is Expanded and Collected .
Input : f E s2 2
yX..yX.
2 2.1.n0
2 2 n1Simplified Input : f1
E s2 2
y.X.1 n0.
2y.y2 2
.X.X1 n0.
2 2.1. n0
2 2 n1Pattern matched KMVN : EB_. a_.Transpose.C_:Identity. c_.d_. e_.The distribution of is the Multivariate Normal Distribution with
Mu X.X 1 n01.X.y 1. n0, Sigma 2X.X 1 n01,const
1E sy.yy.X.X.X1 n01.X.y.1. n0.1.X.X1 n01.X.y n0n0y.X.X.X1 n01.1..1.X.X1 n0
2 2
n1
The result of the integration of f with respect to
1E sy.yy.X.X.X1 n01.X.y.1. n0.1.X.X1 n01.X.y n0n0y.X.X.X1 n01.1..1.X.X1 n0
2 2
2 nr2 nnrX.X1 n01X.X 1 n01
IMPS2006
Normal Regression (natural conjugate)In[436]:= Identify the marginal of .
IdentifyDistg2, , 0;f f1, where the argument of Exp and Power in f1is Expanded and Collected .
Input : f 1E sy
.yy.X.X.X1 n01.X.y.1. n0.1.X.X1 n01.X.y n0n0y.X.X.X1 n01.1..1.X.X1 n02 2
2 nr2 nnrX.X1 n01X.X 1 n01Simplified Input : f1
1X.X 1 n0E 1
2 .1.X.X1 n01.1. n02 12 .1. n0 1
2 y.X.X.X1 n01.1. n0 12 .1.X.X1 n01.X.y n0 s2 y.y2 1
2 y2
2 nr2 nnrX.X1 n01
Pattern matched KInvertedSqrtGamma : Eb_.2
d_.a_ c_.
The distribution of is the Inverted Chi Distribution with
df n nrX.X 1 n0, s .1.X.X 1 n01.1. n02 .1. n0
y.X.X.X 1 n01.1. n0 .1.X.X 1 n01.X.y n0 s y.y y.X.X.X 1 n01.X.y,const
2 nr2X.X 1 n0s2is 2df. Alias Inverted Sqrt Gammadf2, 2sThe result of the integration of f with respect to
1X.X 1 n02 12nnrnrX.X1 n02 nr
2 12n nrX.X 1 n0s y.y y.X.X.X 1 n01.X.y n0.1. y.X.X.X 1 n01.1.
.1.X.X 1 n01.X.y .1.X.X 1 n01.1. n012nnrX.X1 n0
IMPS2006
Normal Regression (natural conjugate)
In[437]:= Identify the dist of and integrate g0 wrt to get the marginal of . g1 IdentifyDistg, , 01;f f1, where the argument of Exp and Power in f1is Expanded and Collected .
Input : f E s2 2
yX..yX.
2 2.1.n0
2 2 n1Simplified Input : f1
E s2 12y.X.yX. 12.1.n0
2 n1Pattern matched KInvertedSqrtGamma : E
b_.2
d_.a_ c_.
The distribution of is the Inverted Chi Distribution with
df n , s s y .X.y X. .1. n0,const
1s2is 2df. Alias Inverted Sqrt Gammadf2, 2sThe result of the integration of f with respect to
212n2n
2s y .X.y X. .1. n012n
IMPS2006
Normal Regression (natural conjugate)In[438]:= Identify the marginal of .
IdentifyDistg1, ;Normal , MVt or Wishart ??
The following matrices are assumed to be symmetric .X.X, 1, 1 n0, X.X 1 n0f f1, where the argument of Exp and Power in f1is Expanded and Collected .
Input : f 212n2n
2s y .X.y X. .1. n012n
Simplified Input : f1
212n2n
2s 2y.X .1 n0. y.y .X.X 1 n0. .1. n012n
Pattern matched KMVt :B_. a_. Transpose.C_ : Identity. c_. d_.b_ e_.The distribution of is the Multivariate t Distribution with
Mu X.X 1 n01.X.y 1. n0, Sigma X.X 1 n01s y.y y.X .1 n0.X.X 1 n01.X.y 1. n0 .1. n0n nr
, df n nr,const
212n2n
2sy.yy.X.1 n0.X.X1 n01.X.y1. n0.1. n0nnr 12n
The result of the integration of f with respect to
12 12n2 nr
2X.X 1 n0112n nrn nr12nrnrX.X1 n0s y.y y.X .1 n0.X.X 1 n01.X.y 1. n0 .1. n012nnrX.X1 n0
IMPS2006
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.
IMPS200
6
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@hum.titech.ac.jp
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