83 lines
2.5 KiB
Plaintext
83 lines
2.5 KiB
Plaintext
@q $Id$ @>
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@q Copyright 2007, Ondra Kamenik @>
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@*2 Conjugate family for normal distribution. Start of {\tt
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normal\_conjugate.h} file.
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The main purpose here is to implement a class representing conjugate
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distributions for mean and variance of the normal distribution. The
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class has two main methods: the first one is to update itself with
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respect to one observation, the second one is to update itself with
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respect to anothe object of the class. In the both methods, the
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previous state of the class corresponds to the prior distribution, and
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the final state corresponds to the posterior distribution.
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The algrebra can be found in Gelman, Carlin, Stern, Rubin (p.87). It
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goes as follows: Prior conjugate distribution takes the following form:
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$$\eqalign{
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\Sigma \sim& {\rm InvWishart}_{\nu_0}(\Lambda_0^{-1}) \cr
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\mu\vert\Sigma \sim& N(\mu_0,\Sigma/\kappa_0)
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}$$
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If the observations are $y_1\ldots y_n$, then the posterior distribution has the
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same form with the following parameters:
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$$\eqalign{
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\mu_n = &\; {\kappa_0\over \kappa_0+n}\mu_0 + {n\over \kappa_0+n}\bar y\cr
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\kappa_n = &\; \kappa_0 + n\cr
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\nu_n = &\; \nu_0 + n\cr
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\Lambda_n = &\; \Lambda_0 + S + {\kappa_0 n\over\kappa_0+n}(\bar y-\mu_0)(\bar y-\mu_0)^T,
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}$$
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where
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$$\eqalign{
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\bar y = &\; {1\over n}\sum_{i=1}^ny_i\cr
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S = &\; \sum_{i=1}^n(y_i-\bar y)(y_i-\bar y)^T
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}$$
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@s NormalConj int
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@c
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#ifndef NORMAL_CONJUGATE_H
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#define NORMAL_CONJUGATE_H
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#include "twod_matrix.h"
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@<|NormalConj| class declaration@>;
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#endif
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@ The class is described by the four parameters: $\mu$, $\kappa$, $\nu$ and $\Lambda$.
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@<|NormalConj| class declaration@>=
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class NormalConj {
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protected:@;
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Vector mu;
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int kappa;
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int nu;
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TwoDMatrix lambda;
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public:@;
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@<|NormalConj| constructors@>;
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virtual ~NormalConj() @+{}
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void update(const ConstVector& y);
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void update(const ConstTwoDMatrix& ydata);
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void update(const NormalConj& nc);
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int getDim() const
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{@+ return mu.length();@+}
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const Vector& getMean() const
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{@+ return mu;@+}
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void getVariance(TwoDMatrix& v) const;
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};
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@ We provide the following constructors: The first constructs diffuse
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(Jeffrey's) prior. It sets $\kappa$, and $\Lambda$ to zeros, $nu$ to
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$-1$ and also the mean $\mu$ to zero (it should not be
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referenced). The second constructs the posterior using the diffuse
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prior and the observed data (columnwise). The third is a copy
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constructor.
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@<|NormalConj| constructors@>=
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NormalConj(int d);
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NormalConj(const ConstTwoDMatrix& ydata);
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NormalConj(const NormalConj& nc);
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@ End of {\tt normal\_conjugate.h} file.
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