Scaled Inverse Chi-Squared Distribution#

Univariate, Continuous, Non-Negative

The Scaled Inverse chi-squared (Scale-Inv-\(\chi^2\)) is a continuous probability distribution characterized by two parameters, \(\nu\), usually called degrees of freedom, and \(\tau^2\), a scale parameter.

The Scaled Inverse chi-squared distribution is mainly used in Bayesian inference as a conjugate prior for variance parameters in normal models, particularly in Bayesian regression, hierarchical models, and time-series analysis.

Key properties and parameters#

Support

\(x \in [0, \infty)\)

Mean

\(\frac{\nu \tau^2}{\nu - 2}\) for \(\nu > 2\), else \(\infty\)

Variance

\(\frac{2 \nu^2 \tau^4}{(\nu - 2)^2 (\nu - 4)}\) for \(\nu > 4\), else \(\infty\)

Parameters:

  • \(\nu\) : (float) Degrees of freedom, \(\nu > 0\).

  • \(\tau^2\) : (float) Scale parameter, \(\tau^2 > 0\).

Probability Density Function (PDF)#

\[ f(x \mid \nu, \tau^2) = \frac{(\tau^2 \nu / 2)^{\nu/2}}{\Gamma(\nu/2)} \ \frac{\exp\left[-\dfrac{\nu \tau^2}{2x}\right]}{x^{1+\nu/2}} \]

where \(\Gamma(\nu)\) is the gamma function.

Scaled Inverse Chi-Squared Distribution PDF

Cumulative Distribution Function (CDF)#

\[ F(x \mid \nu, \tau^2) = \frac{\Gamma\left(\dfrac{\nu}{2}, \dfrac{\tau^2\nu}{2x}\right)}{\Gamma\left(\dfrac{\nu}{2}\right)} \]

where \(\Gamma(\nu, x)\) is the incomplete gamma function.

Scaled Inverse Chi-Squared Distribution CDF

See also

Related Distributions:

  • Chi-Squared - The Inverse Chi-Squared distribution can be thought of as the reciprocal of a scaled Chi-Squared random variable.

  • Gamma - the Inverse Chi-Squared distribution can be seen as the reciprocal of a Gamma-distributed variable with specific parameters.

References#