1.2 Likelihood

We use the standard Gamma parameterization (McCullagh and Nelder 1989; Agresti 2015).

\[ Y_i \mid v \sim \mathrm{Gamma}(\text{shape}=v,\ \text{rate}=v/\mu_i), \]

so that

\[ \mathbb{E}(Y_i)=\mu_i, \qquad \mathrm{Var}(Y_i)=\frac{\mu_i^2}{v}. \]

The log‑likelihood contribution of observation \(i\) is

\[ \ell_i(v) = v\log v - v\log \mu_i + (v-1)\log y_i - \log\Gamma(v) - \frac{v}{\mu_i} y_i. \]

With weights \(w_i\), the total log‑likelihood is

\[ \ell(v) = \sum_{i=1}^n w_i \left[ v\log v - v\log \mu_i + (v-1)\log y_i - \log\Gamma(v) - \frac{v}{\mu_i} y_i \right]. \]

1.3 Prior

We place a Gamma prior on the precision:

\[ v \sim \mathrm{Gamma}(a_0, b_0), \qquad \log p(v) = a_0 \log b_0 \;-\; \log\Gamma(a_0) \;+\; (a_0 - 1)\log v \;-\; b_0 v. \]

1.4. Posterior Log‑Density

The posterior log‑density is

\[ f(v) = \ell(v) + \log p(v) \;-\; \log (C_{1}) \]

where \(C_{1}\) is a normalizing constant for the posterior density.

2.1 Proposal Distribution

Let \(\bar{c}(v_{\text{tangent}})\) denote the gradient of \(-\ell(v)\) evaluated at the tangency point: \[ \bar{c}(v_{\text{tangent}}) = -\ell'(v_{\text{tangent}}). \]

We then define a gamma proposal distribution by:

\[ v \sim \mathrm{Gamma}_{\text{prop}}\!\left(a_0 + \tfrac{1}{2}\sum_i w_i,\; b_0 - \bar{c}(v_{\text{tangent}})\right). \]

where the log of the proposal distribution specifically takes the form

\[ \begin{aligned} \log p_{prop}(v) & = (a_0+ \frac{1}{2}\sum_i w_i )\log [ b_0 - \bar{c}(v_{\text{tangent}})] \;-\; \log\Gamma(a_0+ \frac{1}{2}\sum_i w_i ) \\[4pt] & \;+\; ((a_0+ \frac{1}{2}\sum_i w_i ) - 1)\log v \;-\; [ b_0 - \bar{c}(v_{\text{tangent}})] v. \end{aligned} \] —

In the implementation, this proposal is used in truncated form on the interval \([v_{\min}, v_{\max}]\), with \(v_{\min}, v_{\max}\) derived from a curvature-based Gamma surrogate and optional dispersion bounds.

2.2 Bounding function

Let \(v_{\min}\) be lower bound for the precision \(v\). We can then define a bounding function \(\log h(\cdot)\) by

\[ \log h(v) = \ell(v) - \bigl[\ell(v_{\text{tangent}}) - \bar{c}(v_{\text{tangent}})(v - v_{\text{tangent}})\bigr] - \frac{1}{2}\Bigl(\sum_i w_i\Bigr)\log\!\left(\frac{v}{v_{\min}}\right). \]

It follows from the concavity of \(\ell(\cdot)\) and the fact that \(\log(v/v_{\min}) \ge 0\) for \(v \ge v_{\min}\) that \(\log h(v) \le 0\) for all \(v \ge v_{\min}\).

2.3 Equivalence Claim

Throughout this section, all densities are understood on the truncated domain \(v \in [v_{\min}, v_{\max}]\), with normalizing constants absorbed into \(C_1\) and \(C_2\).

Claim 1:

The posterior log density \(f(.)\) satisfies the property that

\[ f(v)=\log p_{prop}(v)+\log h(v) + C_{2} \]

where \[\begin{aligned} C_{2} & = \Bigl( - \log (C_{1}) \\[4pt] & - [(a_0+ \frac{1}{2}\sum_i w_i ) \log\!\left(b_0 - \bar{c}(v_{\text{tangent}})\right) \;-\; \log\Gamma(a_0+ \frac{1}{2}\sum_i w_i )] \\[4pt] & + [a_0 \log b_0 \;-\; \log\Gamma(a_0) ] \\[4pt] & +[\ell(v_{\text{tangent}})+\bar{c}(v_{\text{tangent}})v_{\text{tangent}} ]\\[4pt] & - [\frac{1}{2}\sum_i w_i] \log v_{\text{min}}\Bigr)\end{aligned} \].

Proof:

\[ \begin{aligned} f(v) & = \log p(v) +\ell(v) - \log (C_{1}) \\[4pt] & = \Bigl( [a_0 \log b_0 \;-\; \log\Gamma(a_0) \;+\; (a_0 - 1)\log v \;-\; b_0 v] \\[4pt] & + [[\frac{1}{2}\sum_i w_i] \log v - [-\bar{c}(v_{\text{tangent}}) v] \\[4pt] & + [(a_0+ \frac{1}{2}\sum_i w_i )\log [ b_0 - \bar{c}(v_{\text{tangent}})] \;-\; \log\Gamma(a_0+ \frac{1}{2}\sum_i w_i )] \\[4pt] & - [a_0 \log b_0 \;-\; \log\Gamma(a_0) ] \Bigr) \\[4pt] & + \Bigl(\ell(v) - [[\frac{1}{2}\sum_i w_i] \log v -[- \bar{c}(v_{\text{tangent}}) v]] \Bigr) \\[4pt] & + \Bigl( - \log (C_{1}) \\[4pt] & - [(a_0+ \frac{1}{2}\sum_i w_i )\log [ b_0 - \bar{c}(v_{\text{tangent}})] \;-\; \log\Gamma(a_0+ \frac{1}{2}\sum_i w_i )] \\[4pt] & + [a_0 \log b_0 \;-\; \log\Gamma(a_0) ] \Bigr) \\[4pt] & = \Bigl( [(a_0+ \frac{1}{2}\sum_i w_i )\log [ b_0 - \bar{c}(v_{\text{tangent}})] \;-\; \log\Gamma(a_0+ \frac{1}{2}\sum_i w_i )] \\[4pt] & + ((a_0+ \frac{1}{2}\sum_i w_i ) - 1)\log v \;-\; [ b_0 - \bar{c}(v_{\text{tangent}})]v \Bigr) \\[4pt] & + \Bigl(\ell(v) - [[\frac{1}{2}\sum_i w_i] \log v -[- \bar{c}(v_{\text{tangent}}) v]] \\[4pt] & -[\ell(v_{\text{tangent}})+\bar{c}(v_{\text{tangent}})v_{\text{tangent}} ] \\[4pt] & + [\frac{1}{2}\sum_i w_i] \log v_{\text{min}} \Bigr) \\[4pt] & + \Bigl( - \log (C_{1}) \\[4pt] & - [(a_0+ \frac{1}{2}\sum_i w_i )\log [ b_0 - \bar{c}(v_{\text{tangent}})] \;-\; \log\Gamma(a_0+ \frac{1}{2}\sum_i w_i )] \\[4pt] & + [a_0 \log b_0 \;-\; \log\Gamma(a_0) ] \\[4pt] & +[\ell(v_{\text{tangent}})+\bar{c}(v_{\text{tangent}})v_{\text{tangent}} ]\\[4pt] & - [\frac{1}{2}\sum_i w_i] \log v_{\text{min}} \Bigr) \\[4pt] & = \log p_{prop}(v) \\[4pt] & + \Bigl([\ell(v)-[\ell(v_{\text{tangent}})-\bar{c}(v_{\text{tangent}})(v - v_{\text{tangent}})]] \\[4pt] & - \frac{1}{2}\Bigl(\sum_i w_i\Bigr)\log\!\left(\frac{v}{v_{\min}}\right) \Bigr) \\[4pt] & + \Bigl( - \log (C_{1}) \\[4pt] & - [(a_0+ \frac{1}{2}\sum_i w_i )\log [ b_0 - \bar{c}(v_{\text{tangent}})] \;-\; \log\Gamma(a_0+ \frac{1}{2}\sum_i w_i )] \\[4pt] & + [a_0 \log b_0 \;-\; \log\Gamma(a_0) ] \\[4pt] & +[\ell(v_{\text{tangent}})+\bar{c}(v_{\text{tangent}})v_{\text{tangent}} ]\\[4pt] & - [\frac{1}{2}\sum_i w_i] \log v_{\text{min}}\Bigr) \\[4pt] & = \log p_{prop}(v) + \log h(v) + C_{2} \end{aligned} \]

2.4 Implementation Outline

To implement the sampler, we proceed as follows:

1. Find the posterior mode \(v_\star\) using a Gamma‑surrogate fixed‑point iteration.
2. Approximate the posterior mean \(v_{\text{mean}}\) using curvature at \(v_\star\) and a Gamma surrogate \(\Gamma(\alpha_{\!bar}, \beta_{\!bar})\).
3. Set the tangency point to \[ v_{\text{tangent}} = v_{\text{mean}}. \]
4. Build the proposal and bounding function using \(v_{\text{tangent}}\) as above.
5. Sample as follows:
   • 1. Generate a candidate \(v_{\text{cand}}\) from the (possibly truncated) proposal \(\mathrm{Gamma}(\text{shape}_\text{prop}, \text{rate}_\text{prop})\).
   • 2. Draw \(u \sim \mathrm{Uniform}(0,1)\).
   • 3. Accept if \[ \log h(v_{\text{cand}}) - \log u > 0. \]

The accepted draws for \(v\) are then transformed to dispersion by \(\phi = 1/v\).