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26.6 Estimating event probabilities

Event probabilities involving either parameters or predictions or both may be coded in the generated quantities block. For example, to evaluate $\textrm{Pr}[\lambda > 5 \mid y]$ in the simple Poisson example with only a rate parameter $\lambda$, it suffices to define a generated quantity

generated quantities {
  int<lower=0, upper=1> lambda_gt_5 = lambda > 5;
  // ...
}

The value of the expression lambda > 5 is 1 if the condition is true and 0 otherwise. The posterior mean of this parameter is the event probability $$\begin{eqnarray*} \mbox{Pr}[\lambda > 5 \mid y] & = & \int \textrm{I}(\lambda > 5) \cdot p(\lambda \mid y) \, \textrm{d}\lambda \\[4pt] & \approx & \frac{1}{M} \sum_{m = 1}^M \textrm{I}[\lambda^{(m)} > 5], \end{eqnarray*}$$ where each $\lambda^{(m)} \sim p(\lambda \mid y)$ is distributed according to the posterior. In Stan, this is recovered as the posterior mean of the parameter lambda_gt_5.

In general, event probabilities may be expressed as expectations of indicator functions. For example, $$\begin{eqnarray*} \textrm{Pr}[\lambda > 5 \mid y] & = & \mathbb{E}[\textrm{I}[\lambda > 5] \mid y] \\[4pt] & = & \int \textrm{I}(\lambda > 5) \cdot p(\lambda \mid y) \, \textrm{d}\lambda \\[4pt] & \approx & \frac{1}{M} \sum_{m = 1}^M \textrm{I}(\lambda^{(m)} > 5). \end{eqnarray*}$$ The last line above is the posterior mean of the indicator function as coded in Stan.

Event probabilities involving posterior predictive quantities $\tilde{y}$ work exactly the same way as those for parameters. For example, if $\tilde{y}_n$ is the prediction for the $n$-th unobserved outcome (such as the score of a team in a game or a level of expression of a protein in a cell), then $$\begin{eqnarray*} \mbox{Pr}[\tilde{y}_3 > \tilde{y}_7 \mid \tilde{x}, x, y] & = & \mathbb{E}\!\left[I[\tilde{y}_3 > \tilde{y}_7] \mid \tilde{x}, x, y\right] \\[4pt] & = & \int \textrm{I}(\tilde{y}_3 > \tilde{y}_7) \cdot p(\tilde{y} \mid \tilde{x}, x, y) \, \textrm{d}\tilde{y} \\[4pt] & \approx & \frac{1}{M} \sum_{m = 1}^M \textrm{I}(\tilde{y}^{(m)}_3 > \tilde{y}^{(m)}_7), \end{eqnarray*}$$ where $\tilde{y}^{(m)} \sim p(\tilde{y} \mid \tilde{x}, x, y).$