Uncertainty For Future Counts At Fixed Exposure

What Is Uncertain About a Future Count?

Suppose a Poisson rate model estimates an event rate, and we want to predict the count over a future, specified amount of exposure. There are two sources of variation to keep distinct. Even if the mean count were known exactly, a future count would vary according to its Poisson distribution. In addition, the mean count itself is estimated from data and is therefore uncertain.

A plug-in prediction accounts for the first source by treating the fitted mean as fixed. A predictive distribution that also represents uncertainty in that mean spreads probability across possible means and then averages the corresponding Poisson distributions. This distinction matters because a prediction interval is intended to cover a future count, not just an unknown expected count.

In an offset model, the fitted log mean at a fixed exposure can be written as \(\hat{\eta}^*=\log(E^*)+z^{*T}\hat{\beta}\), where \(z^*\) is the future observation’s predictor vector and \(\hat{\beta}\) is the fitted coefficient vector. Thus \(\hat{\Lambda}^*=\exp(\hat{\eta}^*)\). The exposure is held fixed here; uncertainty in the fitted coefficients affects the mean, while Poisson variation affects the future count conditional on that mean.

Plug-In Predictions and Predictive Mixtures

For a plug-in prediction, the model assigns probability \(P(Y^*=y)=\exp(-\hat{\Lambda}^*)\hat{\Lambda}^{*y}/y!\), for integers \(y\geq 0\). Its mean and variance are both \(\hat{\Lambda}^*\). This is a useful baseline, but it does not include uncertainty in the fitted rate.

To include mean uncertainty, let \(G\) be a probability distribution on nonnegative values of \(\Lambda^*\), representing uncertainty about the future mean count. Conditional on \(\Lambda^*=\lambda\), suppose \(Y^*\) has a Poisson distribution with mean \(\lambda\). The resulting mixture probabilities are

This formula averages the conditional probability of each count over possible mean counts. It is a genuine probability distribution: the integrand is nonnegative, and summing over the nonnegative integers gives the integral of the Poisson probabilities, which is one. In applications, \(G\) might come from an approximate sampling distribution for fitted coefficients, a parametric bootstrap, or a posterior distribution. In a frequentist analysis it is an uncertainty representation produced by the chosen procedure, not a claim that the fixed model parameter itself is random.

Proof. Conditional on \(\Lambda^*\), the Poisson mean and variance are both \(\Lambda^*\). By the law of total expectation,

The finite second-moment assumption ensures that the terms in the law of total variance are finite. Applying that law gives

The first term is the average Poisson variation; the second is the extra variation from uncertainty in the mean. If the mean is fixed, its variance is zero and the usual Poisson variance is recovered. \(\square\)

Worked Examples

Using a Lognormal Approximation from a Fitted Model

A common practical approximation starts with an estimated linear predictor \(\hat{\eta}^*\) and its estimated variance \(s_*^2\). For example, with estimated coefficient covariance matrix \(\widehat{\operatorname{Var}}(\hat{\beta})\), one may use \(s_*^2=z^{*T}\widehat{\operatorname{Var}}(\hat{\beta})z^*\). If the log mean is approximated as normal with mean \(\hat{\eta}^*\) and variance \(s_*^2\), then the mean count is approximated as lognormal. This uses the uncertainty in the fitted linear predictor at the specified predictors and exposure; it does not add Poisson variation to \(s_*^2\).

This approximation can be used to calculate mixture moments directly, as in the preceding example. To obtain a prediction interval, one can also evaluate or simulate the mixture probabilities: draw a possible mean from the uncertainty distribution, draw a Poisson count conditional on that mean, and repeat. Alternatively, a parametric bootstrap can refit the model in repeated simulated data sets and generate future counts at the fixed exposure. The interval should be based on the resulting predictive count distribution, not merely on the endpoints of an interval for the mean.

A discrete equal-tail prediction interval is formed from the predictive count cumulative distribution function \(F(y)=P(Y^*\leq y)\): its lower endpoint is the smallest integer \(y\) with \(F(y)\geq0.025\), and its upper endpoint is the smallest integer \(y\) with \(F(y)\geq0.975\). Other interval conventions are possible, but the convention should be stated. Because counts are discrete, nominal tail probabilities are generally not attained exactly.

Common Pitfall: Treating a Mean Interval as a Count Interval

A common incorrect argument is: “The fitted mean count is \(4\), and its 95% confidence interval is \([3,5]\), so \([3,5]\) is a 95% prediction interval for the future count.” This ignores that, even if the true mean were exactly \(4\), a Poisson future count could be well below \(3\) or above \(5\). The interval for the mean describes uncertainty about a parameter or expected count; it does not describe the distribution of a future observation.

The correction is to represent uncertainty in the mean and then include the conditional count distribution. In a mixture, this is done by averaging Poisson probabilities over possible means. The total variance identity shows the two contributions explicitly: \(E[\Lambda^*]\) from conditional Poisson variation and \(\operatorname{Var}(\Lambda^*)\) from mean uncertainty. A plug-in Poisson prediction is appropriate when mean uncertainty is intentionally ignored, but it should be identified as such.