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The posterior predictive probability averages the conditional probability over the Bayesian posterior. Using its gamma distribution density,For fixed $a,b$ and $n/T\to \hat{\lambda }<\infty$,Consequently $p_0/e^{-n/T}\to 1$, and both tend to $e^{-\hat{\lambda }}$. This is reasonable because the posterior mean tends to the observed rate and the posterior variance tends to zero. Averaging $e^{-\lambda }$ then approaches evaluating it at the estimated rate: with abundant observations, predictive uncertainty about the rate vanishes.

Let $q_0=n_0$, $q_1=n_0/c^2$ and $y=\overline{y}$. These are the two prior precision parameters. Completing the square in $n(\beta -y{)}^2+q_i{\beta }^2$ gives the Normal-normal conjugacy updateThus the posterior mean is a precision-weighted average of the observed mean and the prior center zero. The posterior variance is the reciprocal of the total precision.

A posterior variance is the variance of a parameter under its Bayesian posterior, namely $\mathbb{E}[(\theta -\mathbb{E}[\theta \mid y]{)}^2\mid y]$ when finite. It describes remaining uncertainty after observing data and is distinct from the repeated-sampling variance of a fitted estimator.