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24.2 Computing the posterior predictive distribution

The posterior predictive density (or mass) of a prediction \(\tilde{y}\) given observed data \(y\) can be computed using Monte Carlo draws

\[ \theta^{(m)} \sim p(\theta \mid y) \] from the posterior as \[ p(\tilde{y} \mid y) \approx \frac{1}{M} \sum_{m = 1}^M p(\tilde{y} \mid \theta^{(m)}). \]

Computing directly using this formula will lead to underflow in many situations, but the log posterior predictive density, \(\log p(\tilde{y} \mid y)\) may be computed using the stable log sum of exponents function as \[\begin{eqnarray*} \log p(\tilde{y} \mid y) & \approx & \log \frac{1}{M} \sum_{m = 1}^M p(\tilde{y} \mid \theta^{(m)}). \\[4pt] & = & - \log M + \textrm{log-sum-exp}_{m = 1}^M \log p(\tilde{y} \mid \theta^{(m)}), \end{eqnarray*}\]

where \[ \textrm{log-sum-exp}_{m = 1}^M v_m = \log \sum_{m = 1}^M \exp v_m \] is used to maintain arithmetic precision. See the section on log sum of exponentials for more details.