In probability theory and statistics, the normal-gamma distribution (or Gaussian-gamma distribution) is a bivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and precision.
Definition For a pair of random variables, (X,T), suppose that the conditional distribution of X given T is given by
X ∣ T ∼ N ( μ , 1 / ( λ T ) ) , {\displaystyle X\mid T\sim N(\mu ,1/(\lambda T))\,\!,}
meaning that the conditional distribution is a normal distribution with mean μ {\displaystyle \mu } and precision λ T {\displaystyle \lambda T} — equivalently, with variance 1 / ( λ T ) . {\displaystyle 1/(\lambda T).}
Suppose also that the marginal distribution of T is given by
T ∣ α , β ∼ Gamma ( α , β ) , {\displaystyle T\mid \alpha ,\beta \sim \operatorname {Gamma} (\alpha ,\beta ),}
where this means that T has a gamma distribution. Here λ, α and β are parameters of the joint distribution. Then (X,T) has a normal-gamma distribution, and this is denoted by
( X , T ) ∼ NormalGamma ( μ , λ , α , β ) . {\displaystyle (X,T)\sim \operatorname {NormalGamma} (\mu ,\lambda ,\alpha ,\beta ).}
Properties
Probability density function The joint probability density function of (X,T) is
f ( x , τ ∣ μ , λ , α , β ) = β α λ Γ ( α ) 2 π τ α − 1 2 e − β τ exp ( − λ τ ( x − μ ) 2 2 ) , {\displaystyle f(x,\tau \mid \mu ,\lambda ,\alpha ,\beta )={\frac {\beta ^{\alpha }{\sqrt {\lambda }}}{\Gamma (\alpha ){\sqrt {2\pi }}}}\,\tau ^{\alpha -{\frac {1}{2}}}\,e^{-\beta \tau }\exp \left(-{\frac {\lambda \tau (x-\mu )^{2}}{2}}\right),}
where the conditional probability for f ( x , τ ∣ μ , λ , α , β ) = f ( x ∣ τ , μ , λ , α , β ) f ( τ ∣ μ , λ , α , β ) {\displaystyle f(x,\tau \mid \mu ,\lambda ,\alpha ,\beta )=f(x\mid \tau ,\mu ,\lambda ,\alpha ,\beta )f(\tau \mid \mu ,\lambda ,\alpha ,\beta )}
was used.
Marginal distributions By construction, the marginal distribution of τ {\displaystyle \tau } is a gamma distribution, and the conditional distribution of x {\displaystyle x} given τ {\displaystyle \tau } is a Gaussian distribution. The marginal distribution of x {\displaystyle x} is a three-parameter non-standardized Student's t-distribution with parameters ( ν , μ , σ 2 ) = ( 2 α , μ , β / ( λ α ) ) {\displaystyle (\nu ,\mu ,\sigma ^{2})=(2\alpha ,\mu ,\beta /(\lambda \alpha ))} .
Exponential family The normal-gamma distribution is a four-parameter exponential family with natural parameters α − 1 / 2 , − β − λ μ 2 / 2 , λ μ , − λ / 2 {\displaystyle \alpha -1/2,-\beta -\lambda \mu ^{2}/2,\lambda \mu ,-\lambda /2} and natural statistics ln τ , τ , τ x , τ x 2 {\displaystyle \ln \tau ,\tau ,\tau x,\tau x^{2}} .
Moments of the natural statistics The following moments can be easily computed using the moment generating function of the sufficient statistic:
E ( ln T ) = ψ ( α ) − ln β , {\displaystyle \operatorname {E} (\ln T)=\psi \left(\alpha \right)-\ln \beta ,}
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