Solved – Gamma-gamma conjugacy for rate parameter of Gamma distribution

Question

the question is as follows.

Assume the shape $r$ is a known constant. For $x \sim$ Gamma(shape = $r$, rate = $v$), the p.d.f is:

$$p(x|r,v) = x^{r-1}e^{-vx}v^r/\Gamma(r)$$

a) Show that the $\theta \sim$ Gamma$(r_0, v_0)$ prior for $\theta$ is the conjugate prior for a Gamma($r,\theta$) likelihood. Do this by finding the exact posterior distribution of $\theta|y,$ where $y=(y_1, y_2,…y_n)$ and each $y_i \sim$ Gamma $(r, \theta)$.

b) Generalize this conjugacy result to a more general multiplicative relationship. Assume $\theta \sim$ Gamma$(r_0, v_0)$ prior distribution. Now, the rate parameter of the likelihood is $c\theta$, where $c$ is a known constant. That is, each $y_i \sim$ Gamma$(r, c\theta)$. Find the exact form of the posterior distribution of $\theta|y$ which will now also involve $c$.

c) What choices of $r_0$ and $v_0$ will minimize the effect of the prior distribution on the posterior distribution?

I'm unsure about how to solve this question. For part (a), how do I find the exact posterior distribution of $\theta|y$, and proceed further? I thought the posterior distribution was likelihood times prior, but I'm not sure how that applies here, and how to show that the asked for prior is a conjugate prior for the Gamma likelihood.

Similarly for part (b) and (c), I'm a bit confused as I'm new to Bayesian and proof-based Math. Would appreciate any detailed help – thank you in advance!

Answer 1

a) Show that the θ∼ Gamma(r0,v0) prior for θ is the conjugate prior for a Gamma(r,θ) likelihood.

As you mention, the proof only need consider the product "prior x likelihood"

$$\theta^{r_0-1} \exp\{-\theta\nu_0\} \times \theta^{nr} \exp\left\{-\sum_{i-1}^ny_i\theta\right\}$$
which factorises as
$$\theta^{nr+r_0-1} \exp\left\{-\theta\left[\nu_0+\sum_{i-1}^ny_i\right]\right\}$$which is a $$\text{Gamma}\left(r_0+nr,\nu_0+\sum_{i-1}^ny_i\right)$$distribution

which shows conjugacy.

b) Generalize this conjugacy result to the case when each $y_i∼ > \text{Gamma}(r,cθ)$.

This is a straightforward generalisation since

one only need replace the $y_i$'s with $c y_i$'s as$$y_i\sim \text{Gamma}(r,cθ)$$is equivalent to $$cy_i\sim \text{Gamma}(r,θ)$$ hence the posterior is now a $$\text{Gamma}\left(r_0+nr,\nu_0+c\sum_{i-1}^ny_i\right)$$distribution

As for the last question,

c) What choices of $r_0$ and $v_0$ will minimize the effect of the prior distribution on the posterior distribution?

it is an ill-posed question since the "effect" to be assessed is not formally defined and clearly spelled-out. Is the effect computed via the Kullback-Leibler distance [as in Bernardo's & Berger's reference priors]? or via a maximum risk (if it exists)?