Normal-scaled inverse gamma distribution

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Normal-scaled inverse gamma
Probability density function
Cumulative distribution function
Parameters \nu\, location (real)
\lambda > 0\, (real)
\alpha > 0\, (real)
\beta > 0\, (real)
Support \mu \in (-\infty, \infty)\,\!, \; \sigma^2 \in (0,\infty)
Probability density function (pdf)
Cumulative distribution function (cdf)
Mean
Median
Mode
Variance
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf)
Characteristic function

In probability theory and statistics, the normal-scaled 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.

Contents

[edit] Definition

Suppose

\mu | \sigma^2, \nu, \lambda \sim \mathrm{N}(\nu,\sigma^2 / \lambda) \,\!

has a normal distribution with mean ν and variance σ2, where

\sigma^2|\alpha, \beta \sim \mbox{Inv-Gamma}(\alpha,\beta) \!

has an inverse gamma distribution. Then (μ,σ2) has a normal-scaled inverse gamma distribution, denoted as

(\mu,\sigma^2) \sim \mbox{Normal-Inv-Gamma}(\nu,\lambda,\alpha,\beta) \! .

[edit] Characterization

[edit] Probability density function

f(\mu,\sigma^2|\nu,\lambda,\alpha,\beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} \frac {\sqrt{\lambda}} {\sigma\sqrt{2\pi} } \, \left( \lambda / \sigma^2 \right)^{\alpha + 1} \, e^{ -\frac {\lambda \left( \beta + (\mu - \nu)^2 / 2\right)} {\sigma^2} }

[edit] Alternative Parameterization

It is also possible to let γ = 1 / λ in which case the pdf becomes

f(\mu,\sigma^2|\nu,\gamma,\alpha,\beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} \frac {1} {\sigma\sqrt{2\pi\gamma} } \, \left( 1 / \gamma \sigma^2 \right)^{\alpha + 1} \, \, e^{ -\frac{\beta + (\mu - \nu)^2 / 2}{\gamma \sigma^2}}


[edit] Cumulative distribution function

[edit] Properties

[edit] Summation

[edit] Scaling

[edit] Exponential family

[edit] Information entropy

[edit] Kullback-Leibler divergence

[edit] Maximum likelihood estimation

[edit] Generating normal-gamma random variates

Generation of random variates is straightforward:

  1. Sample σ2 from an inverse gamma distribution with parameters α and β
  2. Sample μ from a normal distribution with mean ν and variance σ2 / λ

[edit] Related distributions

  • \mu | \sigma^2, \nu, \lambda \sim N(\nu,\sigma^2 / \lambda) \,\! is normally distributed
  • \sigma^2|\alpha, \beta \sim \mathrm{InvGamma}(\alpha,\beta) \! is distributed as an inverse gamma
  • 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 positive-definite covariance matrix is the Multivariate normal-inverse Wishart distribution

[edit] References

Dominici, Francesca; Giovanni Parmigiani, Merlise Clyde (2000-09). "Conjugate Analysis of Multivariate Normal Data with Incomplete Observations". The Canadian Journal of Statistics / La Revue Canadienne de Statistique 28 (3): 533-550. ISSN 03195724. 

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