normalinversegamma

Normal-Inverse-Gamma is a four-parameter family used as a conjugate prior for the mean and variance of a normal distribution. It models a joint distribution over the unknown mean mu and the precision tau (tau = 1/variance).

Parameterization and form. Let the parameters be mu0, lambda0 > 0, alpha0 > 0, beta0 > 0. If tau ~ Gamma(alpha0, beta0) with density proportional to tau^{alpha0-1} e^{-beta0 tau}, and conditional on tau, mu ~ Normal(mu0, (lambda0 tau)^{-1}), then the pair (mu, tau) has the Normal-Inverse-Gamma distribution. This is the standard conjugate prior for a normal model with unknown mean and unknown variance.

Posterior updating. Suppose data x1, ..., xn are independent given mu and tau, with x_i ~ Normal(mu, tau^{-1}). Define xbar as the sample mean and S^2 as the sum of squared deviations. The posterior parameters are:

- lambda_n = lambda0 + n

- mu_n = (lambda0 mu0 + n xbar) / lambda_n

- alpha_n = alpha0 + n/2

- beta_n = beta0 + 0.5 * sum_i (x_i - xbar)^2 + (lambda0 n (xbar - mu0)^2) / (2 lambda_n)

Then the posterior is mu | tau, data ~ Normal(mu_n, (lambda_n tau)^{-1}) and tau | data ~ Gamma(alpha_n, beta_n). Marginally, mu | data follows a Student-t distribution with degrees of freedom 2 alpha_n, location mu_n, and scale sqrt(beta_n / (alpha_n lambda_n)).

Applications. The Normal-Inverse-Gamma serves as a convenient prior in Bayesian inference for normal models with unknown mean and variance, enabling closed-form posterior updates and analytic marginals. It is related to the Normal-Gamma family and underpins many hierarchical and regression models.