Normal-gamma distribution

In probability theory and statistics, the normal-gamma distribution is a four-parameter family of continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and variance.

Definition
Suppose


 * $$ x|\tau, \mu, \lambda \sim N(\mu,\lambda / \tau) \,\! $$

has a normal distribution with mean $$ \mu$$ and variance $$ \lambda / \tau$$, where


 * $$\tau|\alpha, \beta \sim \mathrm{Gamma}(\alpha,\beta) \!$$

has a gamma distribution. Then $$(x,\tau) $$ has a normal-gamma distribution, denoted as
 * $$ (x,\tau) \sim \mathrm{NormalGamma}(\mu,\lambda,\alpha,\beta) \!.

$$

Probability density function

 * $$f(x,\tau|\mu,\lambda,\alpha,\beta) = \frac{\beta^\alpha}{\Gamma(\alpha)\sqrt{2\pi\lambda}} \, \tau^{\alpha-\frac{1}{2}}\,e^{-\beta\tau}\,e^{ -\frac{\tau(x- \mu)^2}{2\lambda}}$$

Scaling
For any t > 0, tX is distributed $${\rm NormalGamma}(t\mu, \lambda, \alpha, t^2\beta)$$