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In probability theory and statistics, the normal-inverse-gamma distribution (or Gaussian-inverse-gamma distribution) is a four-parameter family of multivariate continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and variance.
Definition[edit]
Suppose
has a normal distribution with mean and variance , where
has an inverse-gamma distribution. Then has a normal-inverse-gamma distribution, denoted as
( is also used instead of )
The normal-inverse-Wishart distribution is a generalization of the normal-inverse-gamma distribution that is defined over multivariate random variables.
Characterization[edit]
Probability density function[edit]
For the multivariate form where is a random vector,
where is the determinant of the matrix . Note how this last equation reduces to the first form if so that are scalars.
Alternative parameterization[edit]
It is also possible to let in which case the pdf becomes
In the multivariate form, the corresponding change would be to regard the covariance matrix instead of its inverse as a parameter.
Cumulative distribution function[edit]
Properties[edit]
Marginal distributions[edit]
Given as above, by itself follows an inverse gamma distribution:
while follows a t distribution with degrees of freedom.[1]
For probability density function is
Marginal distribution over is
Except for normalization factor, expression under the integral coincides with Inverse-gamma distribution
with , , .
Since , and
Substituting this expression and factoring dependence on ,
Shape of generalized Student's t-distribution is
.
Marginal distribution follows t-distribution with degrees of freedom
.
In the multivariate case, the marginal distribution of is a multivariate t distribution:
Summation[edit]
Scaling[edit]
Suppose
Then for ,
Proof: To prove this let and fix . Defining , observe that the PDF of the random variable evaluated at is given by times the PDF of a random variable evaluated at . Hence the PDF of evaluated at is given by :
The right hand expression is the PDF for a random variable evaluated at , which completes the proof.
Exponential family[edit]
Normal-inverse-gamma distributions form an exponential family with natural parameters , , , and and sufficient statistics , , , and .
Information entropy[edit]
Kullback–Leibler divergence[edit]
Measures difference between two distributions.
Maximum likelihood estimation[edit]
Posterior distribution of the parameters[edit]
See the articles on normal-gamma distribution and conjugate prior.
Interpretation of the parameters[edit]
See the articles on normal-gamma distribution and conjugate prior.
Generating normal-inverse-gamma random variates[edit]
Generation of random variates is straightforward:
- Sample from an inverse gamma distribution with parameters and
- Sample from a normal distribution with mean and variance
Related distributions[edit]
- The normal-gamma distribution is the same distribution parameterized by precision rather than variance
- A generalization of this distribution which allows for a multivariate mean and a completely unknown positive-definite covariance matrix (whereas in the multivariate inverse-gamma distribution the covariance matrix is regarded as known up to the scale factor ) is the normal-inverse-Wishart distribution
See also[edit]
References[edit]
- ^ Ramírez-Hassan, Andrés. 4.2 Conjugate prior to exponential family | Introduction to Bayesian Econometrics.