The normal-gamma distribution jointly models a Gaussian mean parameter μ and a precision parameter λ. Its density is the product of a normal distribution and a gamma distribution, with parameters μ′, λ′, α′, and β′. section 13.7.1
For a Gaussian likelihood with unknown mean and precision, a normal-gamma prior produces another normal-gamma posterior. This conjugacy keeps Bayesian updating in the same distribution family and supports maximum-a-posteriori estimation. section 13.7 section 13.7.2
The parameter roles and computational updates are:
- λ represents Gaussian precision, the inverse of variance; α and β parameterize the gamma component. section 13.7.2
- Given sample count n, sample mean, sample standard deviation, and prior parameters, compute μₙ, λₙ, αₙ, and βₙ for the posterior. section 13.7.2
- Read the posterior mode as an estimate of μ and precision, then convert precision to standard deviation using σ = √(1/λ). section 13.7.2
Estimating adult height
Prior information from a neighboring town sets μ₀ = 150, λ₀ = 100, α₀ = 10.5, and β₀ = 810. Applying the posterior update to height data yields MAP estimates μ = 149.98 and σ = 9.56, compared with maximum-likelihood estimates of 149.68 and 11.52. section 13.8.2
