Index: The Book of Statistical ProofsStatistical Models ▷ Univariate normal data ▷ Univariate Gaussian ▷ Posterior distribution

Theorem: Let

$\label{eq:ug} y = \left\lbrace y_1, \ldots, y_n \right\rbrace, \quad y_i \sim \mathcal{N}(\mu, \sigma^2), \quad i = 1, \ldots, n$

be a univariate Gaussian data set with unknown mean $\mu$ and unknown variance $\sigma^2$. Moreover, assume a normal-gamma prior distribution over the model parameters $\mu$ and $\tau = 1/\sigma^2$:

$\label{eq:UG-NG-prior} p(\mu,\tau) = \mathcal{N}(\mu; \mu_0, (\tau \lambda_0)^{-1}) \cdot \mathrm{Gam}(\tau; a_0, b_0) \; .$

Then, the posterior distribution is also a normal-gamma distribution

$\label{eq:UG-NG-post} p(\mu,\tau|y) = \mathcal{N}(\mu; \mu_n, (\tau \lambda_n)^{-1}) \cdot \mathrm{Gam}(\tau; a_n, b_n)$

and the posterior hyperparameters are given by

$\label{eq:UG-NG-post-par} \begin{split} \mu_n &= \frac{\lambda_0 \mu_0 + n \bar{y}}{\lambda_0 + n} \\ \lambda_n &= \lambda_0 + n \\ a_n &= a_0 + \frac{n}{2} \\ b_n &= b_0 + \frac{1}{2} (y^\mathrm{T} y + \lambda_0 \mu_0^2 - \lambda_n \mu_n^2) \; . \end{split}$

Proof: According to Bayes’ theorem, the posterior distribution is given by

$\label{eq:UG-NG-BT} p(\mu,\tau|y) = \frac{p(y|\mu,\tau) \, p(\mu,\tau)}{p(y)} \; .$

Since $p(y)$ is just a normalization factor, the posterior is proportional to the numerator:

$\label{eq:UG-NG-post-JL} p(\mu,\tau|y) \propto p(y|\mu,\tau) \, p(\mu,\tau) = p(y,\mu,\tau) \; .$

Equation \eqref{eq:ug} implies the following likelihood function

$\label{eq:UG-LF-class} \begin{split} p(y|\mu,\sigma^2) &= \prod_{i=1}^{n} \mathcal{N}(y_i; \mu, \sigma^2) \\ &= \prod_{i=1}^{n} \frac{1}{\sqrt{2 \pi} \sigma} \cdot \exp\left[ -\frac{1}{2} \left( \frac{y_i-\mu}{\sigma} \right)^2 \right] \\ &= \frac{1}{(\sqrt{2 \pi \sigma^2})^n} \cdot \exp\left[ -\frac{1}{2 \sigma^2} \sum_{i=1}^{n} \left( y_i-\mu \right)^2 \right] \end{split}$

which, for mathematical convenience, can also be parametrized as

$\label{eq:UG-LF-Bayes} \begin{split} p(y|\mu,\tau) &= \prod_{i=1}^{n} \mathcal{N}(y_i; \mu, \tau^{-1}) \\ &= \prod_{i=1}^{n} \sqrt{\frac{\tau}{2 \pi}} \cdot \exp\left[ -\frac{\tau}{2} \left( y_i-\mu \right)^2 \right] \\ &= \left( \sqrt{\frac{\tau}{2 \pi}} \right)^n \cdot \exp\left[ -\frac{\tau}{2} \sum_{i=1}^{n} \left( y_i-\mu \right)^2 \right] \end{split}$

using the inverse variance or precision $\tau = 1/\sigma^2$.

Combining the likelihood function \eqref{eq:UG-LF-Bayes} with the prior distribution \eqref{eq:UG-NG-prior}, the joint likelihood of the model is given by

$\label{eq:UG-NG-JL-s1} \begin{split} p(y,\mu,\tau) = \; & p(y|\mu,\tau) \, p(\mu,\tau) \\ = \; & \left( \sqrt{\frac{\tau}{2 \pi}} \right)^n \cdot \exp\left[ -\frac{\tau}{2} \sum_{i=1}^{n} \left( y_i-\mu \right)^2 \right] \cdot \\ & \sqrt{\frac{\tau \lambda_0}{2 \pi}} \cdot \exp\left[ -\frac{\tau \lambda_0}{2} \left( \mu-\mu_0 \right)^2 \right] \cdot \\ & \frac{ {b_0}^{a_0} }{\Gamma(a_0)} \, \tau^{a_0-1} \exp[-b_0 \tau] \; . \end{split}$

Collecting identical variables gives:

$\label{eq:UG-NG-JL-s2} \begin{split} p(y,\mu,\tau) = \; & \sqrt{\frac{\tau^{n+1} \lambda_0}{(2 \pi)^{n+1}}} \, \frac{ {b_0}^{a_0}}{\Gamma(a_0)} \, \tau^{a_0-1} \exp[-b_0 \tau] \cdot \\ & \exp\left[ -\frac{\tau}{2} \left( \sum_{i=1}^{n} \left( y_i-\mu \right)^2 + \lambda_0 \left( \mu-\mu_0 \right)^2 \right) \right] \; . \end{split}$ $\label{eq:UG-NG-JL-s3} \begin{split} p(y,\mu,\tau) = \; & \sqrt{\frac{\tau^{n+1} \lambda_0}{(2 \pi)^{n+1}}} \, \frac{ {b_0}^{a_0}}{\Gamma(a_0)} \, \tau^{a_0-1} \exp[-b_0 \tau] \cdot \\ & \exp\left[ -\frac{\tau}{2} \left( (y^\mathrm{T} y - 2 \mu n \bar{y} + n \mu^2) + \lambda_0 (\mu^2 - 2 \mu \mu_0 + \mu_0^2) \right) \right] \end{split}$

where $\bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i$ and $y^\mathrm{T} y = \sum_{i=1}^{n} y_i^2$, such that

$\label{eq:UG-NG-JL-s4} \begin{split} p(y,\mu,\tau) = \; & \sqrt{\frac{\tau^{n+1} \lambda_0}{(2 \pi)^{n+1}}} \, \frac{ {b_0}^{a_0}}{\Gamma(a_0)} \, \tau^{a_0-1} \exp[-b_0 \tau] \cdot \\ & \exp\left[ -\frac{\tau}{2} \left( \mu^2 (\lambda_0 + n) - 2 \mu (\lambda_0 \mu_0 + n \bar{y}) + (y^\mathrm{T} y + \lambda_0 \mu_0^2) \right) \right] \end{split}$

Completing the square over $\mu$, we finally have

$\label{eq:UG-NG-JL-s5} \begin{split} p(y,\mu,\tau) = \; & \sqrt{\frac{\tau^{n+1} \lambda_0}{(2 \pi)^{n+1}}} \, \frac{ {b_0}^{a_0}}{\Gamma(a_0)} \, \tau^{a_0-1} \exp[-b_0 \tau] \cdot \\ & \exp\left[ -\frac{\tau \lambda_n}{2} \left( \mu - \mu_n \right)^2 -\frac{\tau}{2} \left( y^\mathrm{T} y + \lambda_0 \mu_0^2 - \lambda_n \mu_n^2 \right) \right] \end{split}$

with the posterior hyperparameters

$\label{eq:UG-NG-post-mu-par} \begin{split} \mu_n &= \frac{\lambda_0 \mu_0 + n \bar{y}}{\lambda_0 + n} \\ \lambda_n &= \lambda_0 + n \; . \end{split}$

Ergo, the joint likelihood is proportional to

$\label{eq:UG-NG-JL-s6} p(y,\mu,\tau) \propto \tau^{1/2} \cdot \exp\left[ -\frac{\tau \lambda_n}{2} \left( \mu - \mu_n \right)^2 \right] \cdot \tau^{a_n-1} \cdot \exp\left[ -b_n \tau \right]$

with the posterior hyperparameters

$\label{eq:UG-NG-post-tau-par} \begin{split} a_n &= a_0 + \frac{n}{2} \\ b_n &= b_0 + \frac{1}{2} \left( y^\mathrm{T} y + \lambda_0 \mu_0^2 - \lambda_n \mu_n^2 \right) \; . \end{split}$

From the term in \eqref{eq:UG-NG-JL-s5}, we can isolate the posterior distribution over $\mu$ given $\tau$:

$\label{eq:UG-NG-post-mu} p(\mu|\tau,y) = \mathcal{N}(\mu; \mu_n, (\tau \lambda_n)^{-1}) \; .$

From the remaining term, we can isolate the posterior distribution over $\tau$:

$\label{eq:UG-NG-post-tau} p(\tau|y) = \mathrm{Gam}(\tau; a_n, b_n) \; .$

Together, \eqref{eq:UG-NG-post-mu} and \eqref{eq:UG-NG-post-tau} constitute the joint posterior distribution of $\mu$ and $\tau$.

Sources:

Metadata: ID: P202 | shortcut: ug-post | author: JoramSoch | date: 2021-03-03, 09:53.