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Concept
Normal-gamma distribution
Summary
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.
For a pair of random variables, (X,T), suppose that the conditional distribution of X given T is given by
meaning that the conditional distribution is a normal distribution with mean and precision — equivalently, with variance
Suppose also that the marginal distribution of T is given by
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
The joint probability density function of (X,T) is
By construction, the marginal distribution of is a gamma distribution, and the conditional distribution of given is a Gaussian distribution. The marginal distribution of is a three-parameter non-standardized Student's t-distribution with parameters .
The normal-gamma distribution is a four-parameter exponential family with natural parameters and natural statistics .
The following moments can be easily computed using the moment generating function of the sufficient statistic:
where is the digamma function,
If then for any is distributed as
Assume that x is distributed according to a normal distribution with unknown mean and precision .
and that the prior distribution on and , , has a normal-gamma distribution
for which the density pi satisfies
Suppose
i.e. the components of are conditionally independent given and the conditional distribution of each of them given is normal with expected value and variance The posterior distribution of and given this dataset can be analytically determined by Bayes' theorem explicitly,
where is the likelihood of the parameters given the data.
Since the data are i.i.d, the likelihood of the entire dataset is equal to the product of the likelihoods of the individual data samples:
This expression can be simplified as follows:
where , the mean of the data samples, and , the sample variance.
Official source
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