Documentation normal-inverse-gamma conjugate prior distribution and n…

…ormal model

doc/source/conjugate_normal_inverse_gamma.rst

@@ -1,6 +1,35 @@
 Normal-inverse-gamma distribution
 =================================
 
+The probability density function of the normal-inverse gamma distribution
+:math:`\mathcal{N}\Gamma^{-1}(\mu, \lambda, \alpha, \beta)` with location
+parameter :math:`\mu`, variance scale parameter :math:`\lambda > 0`, shape
+parameter :math:`\alpha > 0` and scale parameter :math:`\beta > 0,` for
+:math:`x \in \mathbb{R}` and :math:`\sigma^2 \in \mathbb{R}^+`, is given by
+
+.. math::
+
+   f(x,\sigma^2; \mu,\lambda,\alpha,\beta) =  \frac {\sqrt{\lambda}} {\sigma\sqrt{2\pi} } \, \frac{\beta^\alpha}{\Gamma(\alpha)} \, \left( \frac{1}{\sigma^2} \right)^{\alpha + 1}   \exp \left( -\frac { 2\beta + \lambda(x - \mu)^2} {2\sigma^2}  \right),
+
+and the cumulative distribution function is
+
+.. math::
+
+   F(x,\sigma^2; \mu,\lambda,\alpha,\beta) =  \frac{e^{-\frac{\beta}{\sigma^2}} \left(\frac{\beta }{\sigma ^2}\right)^\alpha
+      \left(\operatorname{erf}\left(\frac{\sqrt{\lambda} (x-\mu )}{\sqrt{2} \sigma }\right)+1\right)}{2
+      \sigma^2 \Gamma (\alpha)}.
+
+The expected value and variance are as follows
+
+   .. math::
+
+      \mathrm{E}[x] &= \mu, \quad \mathrm{E}[\sigma^2] = \frac{\beta}{\alpha-1}, \; \alpha > 1.
+
+      \mathrm{Var}[x] &= \frac{\beta}{(\alpha - 1)\lambda}, \; \alpha > 1,
+      \quad \mathrm{Var}[\sigma^2] = \frac{\beta^2}{(\alpha-1)^2(\alpha - 2)}, \; \alpha > 2.
+
+The normal-inverse-gamma distribution is used as a conjugate prior distribution for
+the normal distribution with unknown mean and variance.
 
 
 .. autoclass:: cprior.cdist.NormalInverseGammaModel

doc/source/formulas_models_normal.rst

@@ -4,14 +4,62 @@ Normal-normal-inverse-gamma conjugate model
 Posterior predictive distribution
 ---------------------------------
 
+If :math:`X| \mu, \sigma^2 \sim \mathcal{N}(\mu, \sigma^2)` with
+:math:`(\mu, \sigma) \sim \mathcal{N}\Gamma^{-1}(\mu_0, \lambda, \alpha, \beta)`,
+then the posterior predictive probability density function, the expected
+value and variance of :math:`X` are
+
+.. math::
+
+   f(x; \mu_0, \lambda, \alpha, \beta) = \frac{\alpha}{\beta(1 + \lambda^{-1})}
+   \frac{\left(1 + \frac{1}{2\alpha} \left(\frac{\alpha(x - \mu_0)}{\beta(1+\lambda^{-1})} \right)^2 \right)^{-\alpha - 1/2}}
+   {\sqrt{2\alpha}B(\alpha, 1/2)},
+
+.. math::
+
+   \mathrm{E}[X] = \mu_0, \quad \mathrm{Var}[X] = \frac{\left(\beta(1 +
+   \lambda^{-1})\right)^2}{\alpha(\alpha - 1)},
+
+where :math:`\mathrm{E}[X]` is defined for :math:`\alpha > 1/2` and
+:math:`\mathrm{Var}[X]` is defined for :math:`\alpha > 1`.
+
 
 Proofs
 ------
 
 Posterior predictive probability density function
 
+Note that this is the probability density function of the
+Student's t-distribution, thus
+
+.. math::
+
+   X \sim t_{2 \alpha}\left(\mu_0, \frac{\beta (1 + \lambda^{-1})}{\alpha}\right),
+
+see :cite:`Murphy2007`.
+
 
 Posterior predictive expected value
 
+Apply properties of the Student's t-distribution.
+
+.. math ::
+
+   \mathrm{E}[X] = \mu_0.
+
+
+Posterior predictive variance
+
+Apply properties of the Student's t-distribution.
+
+.. math::
+
+   \mathrm{Var}[X] = \frac{\left(\beta(1 +
+   \lambda^{-1})\right)^2}{\alpha(\alpha - 1)}.
+
+
+References
+----------
 
-Posterior predictive variance
+.. bibliography:: refs.bib
+   :filter: docname in docnames

doc/source/refs.bib

@@ -24,7 +24,7 @@ @article{Miller2015
 
 @article{Murphy2007,
 	title = {{Conjugate Bayesian analysis of the Gaussian distribution}},
-	author = {Murphy, Kevin P.},
+	author = {Murphy, K. P.},
 	journal = {},
 	year = {2007},
 	url = {https://www.cs.ubc.ca/~murphyk/Papers/bayesGauss.pdf}