Exercise 23.2 - Rejection sampling from a Gamma using a Cauchy proposal

Answers

The p.d.f. of a Cauchy is:

𝒯 (x | μ, σ^{2}, 1) = \frac{1}{Z} \cdot \frac{1}{1 + {(\frac{x - μ}{σ})}^{2}},

where $Z = 𝜋𝜎$ .

That of a Gamma is:

Ga (x | a, b) = \frac{b^{a}}{Γ (a)} x^{a - 1} e^{- 𝑥𝑏} .

Finally, we determine $M$ from:

M = \max_{x} {\frac{Ga (x | a, b)}{𝒯 (x | μ, σ^{2}, 1)}},

where $x$ takes value from the zeros of the polynomial:

f (x) = (a - 1 - 𝑏𝑥) [1 + {(\frac{x - μ}{σ})}^{2}] + \frac{2 x (x - μ)}{σ^{2}},

and can be solved analytically. The optimal value for $μ$ and $σ^{2}$ should be:

\arg \min_{μ, σ^{2}} {M (μ, σ^{2} | a, b)}

in order to increase the rate of acceptance.

solour_lfq

2021-03-24 13:42