NormalGamma

NormalGamma is a continuous probability distribution that models situations where both a normal and a gamma distribution are relevant, often used in Bayesian statistics as a conjugate prior for the mean and precision of a normal distribution. It is characterized by four parameters: a shape parameter (α), a rate parameter (β), a mean parameter (μ), and a scale parameter (λ).

Mathematically, the NormalGamma distribution combines a normal distribution for the mean (μ) with a gamma distribution for the precision (τ), where precision is the reciprocal of variance. The joint probability density function (pdf) of the NormalGamma distribution is expressed as a product of a normal and a gamma pdf, capturing the dependency between the mean and precision.

This distribution is advantageous in Bayesian inference because it provides a conjugate prior for the mean and variance of a normal distribution with unknown parameters. When the prior is set as a NormalGamma, the posterior distribution after observing data remains a NormalGamma, facilitating analytical calculations.

The NormalGamma distribution is widely applicable in hierarchical modeling, time series analysis, and other statistical models where uncertainty about variance and mean needs to be jointly estimated. Its flexibility and conjugate properties make it a valuable tool in probabilistic modeling and Bayesian updating.

Despite its mathematical utility, the NormalGamma can be complex to interpret visually due to its four parameters, but understanding its components helps in setting appropriate priors and interpreting posterior distributions in Bayesian analysis.