Thursday, June 20, 2013

A little addendum to the compendium of conjugate priors

The excellent "Compendium of Conjugate Priors" by Daniel Fink contains a little typo in equation (122) for the posterior of Athreya's conjugate distribution to the normal. Made a note here lest it trip someone else up. This arises when a modified inverse gauss distribution looking something like a gamma distribution $$ P(\nu; \alpha, \beta, \gamma) \propto \nu^\alpha e^{-\frac{\beta}{\nu} - \gamma \nu} $$ is used as the marginal distribution for the variance parameter of a normal distribution whose mean and variance are unknown; and when the conditional mean of that distribution is given by $$ P(m ; \nu, \alpha,\beta, \gamma, \tau, \mu) \propto \frac{1}{\sqrt{\nu}} e^{ \frac{1}{\nu \tau}(m-\mu) } $$ to create a five parameter prior distribution for both mean \(m\) and variance \(\nu\) of the normal distribution parameters proportional to $$ P(m,\nu ; \alpha,\beta, \gamma, \tau, \mu ) \propto \nu^{\alpha-\frac{1}{2}} e^{-\frac{\beta}{\nu} - \gamma \nu} \ e^{ \frac{1}{\nu \tau}(m-\mu) } $$ Upon observation of \(n\) independent samples \(\{x_i\}_{i=1}^n\) presumably drawn from said distribution the posterior distribution takes the same form provided we update the parameters as follows \begin{eqnarray*} \alpha' & = & \alpha - \frac{n}{2} \\ \beta' & = & \beta + \frac{1}{2} \sum_{i=1}^n (x_i-\bar{x})^2 + + \frac{n}{2}\frac{ (\bar{x} -\mu)^2}{1 + n \tau} \\ \mu' & = & \frac{\mu + \tau n \bar{x}}{1 + n\tau} \\ \tau' & = & \frac{\tau}{1+n\tau}\\ \gamma' & = & \gamma \end{eqnarray*} where \(\bar{x}\) is the mean of \(\{x_i\}_{i=1}^n\). The mentioned paper contains a mistake in the update for \(\beta\).

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