Conditional density following Gamma distribution

Question

We know that if a random variable say x ~ Gamma(a, b), then its probability density function is $ \propto x^{a-1} exp^{-bx}$.

In a Bayesian hierarchical model, for example

$Z_1, \cdots, Z_n |\theta \sim iid \; Gamma(r, \theta)$, (r known)$ ;

$\theta \sim \; Gamma(a, b)$,

the full conditional distribution of $\theta$ is

$p(\theta | Z_{1:n}) \propto p(\theta, Z_{1:n}) \propto p(Z_{1:n}|\theta) \times p(\theta)$,

by $Z_{1:n} | \theta \sim iid \; Gamma(r, \theta)$, I think $p(Z_{1:n}|\theta)$ shall be $\Pi_i Z_i^{r-1} exp^{- \theta Z_i}$, but the reading material suggests

$p(Z_{1:n}|\theta) = \Pi_i \theta^{r} exp^{-\theta Z_i}$,

I'm not sure I fully understand why it writes $\theta^{r}$ instead of $Z_i^{r-1}$ in front of $exp^{-\theta Z_i}$.

Can anyone explain? Thanks.

Answer 1

The full joint density of $Z_{1:n}$ conditional on $\theta$ is $$ p(z_{1:n}|\theta)= \prod_{i=1}^n \frac{\theta^r}{\Gamma(r)} z_i^{r-1} \exp(-\theta z_i) \,. $$ If we consider this as a function of $z_{1:n}$, the term $\theta^{r}/\Gamma(r)$ is just a normalising constant and can be dropped.

As you point out, the conditional distribution of $\theta$ given $z_{1:n}$ (i.e. the posterior of $\theta$ given the observed sample) is $$ p(\theta|z_{1:n}) \propto p(\theta) p(z_{1:n}|\theta) = p(\theta) \prod_{i=1}^n \frac{\theta^r}{\Gamma(r)} z_i^{r-1} \exp(-\theta z_i) $$ This is a function of $\theta$, so we need to keep the terms involving $\theta$ (such as $\theta^{r}$), but we can drop the other terms (such as $z_i^{r-1}$). We can thus write $$ p(\theta|z_{1:n}) \propto p(\theta) \prod_{i=1}^n \theta^r \exp(-\theta z_i) \propto \theta^{a-1} \exp(-b\theta)\,\, \theta^{nr} \exp(-\theta n \bar{z}) $$ which we identify as another Gamma density.