User:Paulpeeling/Sandbox

Stubs

 * Generalized gamma distribution
 * Scale mixture of normals
 * Probability hypothesis density filter
 * Variable rate particle filter

Scratchpad
Conjugate prior of a Gamma distribution when $$n \neq 1$$

$$ p(\beta) \propto \beta^{\alpha_0-1} \exp(\beta \beta_0) \prod_n \left( \exp(x_n \beta) \beta^\alpha \right) $$ $$ \propto \beta^{\alpha_0 + n\alpha - 1} \exp\left(\beta \left( \beta_0+\sum_n x_n \right) \right) = \Gamma\left(\alpha_0+n\alpha,\beta_0+\sum_n x_n\right) $$

The Kullback-Leibler divergence between inverse-gamma distributions with scale parameterization is the same as that for Gamma distributions with inverse-scale parameterization.