Constructing Individual Prediction Intervals

Why an Individual Prediction Needs an Interval

A point forecast gives one summary of a future concentration, but it does not show how widely plausible future values are spread. For an individual with observed concentration data, the forecast also depends on uncertainty about that individual’s parameters and on future residual variability. A prediction interval summarizes the resulting uncertainty at a specified future time, dose, and covariate profile.

The interval is conditional on the individual’s available data and the forecasting conditions. It is not an interval for a population mean or for a model parameter. Its target is the future concentration itself, as represented by the individual posterior predictive distribution.

Definition: Let $\mathcal D_i$ denote the observed data for individual $i$, and let $\tilde C$ denote that individual’s future concentration under specified forecasting conditions. The individual posterior predictive distribution is $$ p(\tilde c\mid \mathcal D_i) = \int p(\tilde c\mid \theta_i)\,p(\theta_i\mid \mathcal D_i)\,d\theta_i, $$ where $\theta_i$ denotes the individual-specific parameters. The conditional distribution $p(\tilde c\mid \theta_i)$ includes future residual variability as specified by the model. A prediction interval with predictive content $1-\alpha$ is a set $A$ for which $P(\tilde C\in A\mid\mathcal D_i)=1-\alpha$, or approximately that value when computed numerically.

This integral averages the future-concentration distribution over the individual’s posterior parameter uncertainty. In practice, the distribution may be evaluated by simulation or numerical integration. The introductory material on posterior predictive distributions establishes this forecasting target; here the focus is how to turn that distribution into an interval.

Two Ways to Choose Predictive Interval Endpoints

A central equal-tail interval allocates the omitted probability equally between the lower and upper tails. If $F$ is the conditional predictive cumulative distribution function and $Q(p)=\inf\{c:F(c)\ge p\}$ is its quantile function, the central equal-tail interval is

[Q(\alpha/2),\,Q(1-\alpha/2)].

For a continuous predictive distribution, this interval has probability $1-\alpha$, with probability $\alpha/2$ below its lower endpoint and $\alpha/2$ above its upper endpoint. The endpoints are quantiles of the predictive distribution, not estimates of the lower and upper limits of a parameter.

A highest-posterior-density (HPD) region instead includes values with the greatest predictive density. Choose a density threshold $k$ so that the set of concentrations with density at least $k$ has probability $1-\alpha$, when such a threshold exists. If the boundary level set has positive probability, a suitable subset of it may be needed to attain exactly $1-\alpha$; the resulting HPD region need not be unique:

A_{\mathrm{HPD}}=\{c:f(c\mid\mathcal D_i)>k\}\cup B, \qquad B\subseteq\{c:f(c\mid\mathcal D_i)=k\}, \qquad \int_{A_{\mathrm{HPD}}} f(c\mid\mathcal D_i)\,dc=1-\alpha.

When the predictive density is unimodal, this set is usually a single interval. Its endpoints have equal density, and the interval is a shortest interval among intervals with the same probability content. For a multimodal distribution, the HPD region can be a union of separated intervals; reporting only the span from its smallest to largest value would include low-density gaps and may change its probability content. In that case, report the region’s components or explain explicitly that a single enclosing interval is being used.

Why Equal Density Identifies the Shortest Interval

For a continuous unimodal density $f$, consider an interval $[a,b]$ having fixed probability content $1-\alpha$. A small change to its lower endpoint must be accompanied by a change to its upper endpoint that keeps the content fixed. Differentiating $F(b)-F(a)=1-\alpha$ gives

f(b)\,db-f(a)\,da=0, \qquad \frac{db}{da}=\frac{f(a)}{f(b)}.

The interval width is $b-a$, so its rate of change under this adjustment is

\frac{d(b-a)}{da}=\frac{f(a)}{f(b)}-1.

At an interior minimum of width where both endpoints can vary freely, this derivative is zero, requiring $f(a)=f(b)$. If an endpoint is constrained by the support boundary, a shortest interval may have an endpoint at that boundary and unequal endpoint densities. The central equal-tail interval is not generally the shortest interval: it is defined by equal tail probabilities, a different criterion.

Worked Example: A Symmetric Concentration Forecast

Suppose a hypothetical individual posterior predictive distribution for concentration at a specified future time is $N(10,4)$, in which the mean is $10$ mg/L and the variance is $4$ $(\mathrm{mg/L})^2$. The standard deviation is therefore $2$ mg/L. Construct a central 90% interval.

For a normal distribution, the 95th percentile of the standard normal distribution is approximately $1.6449$. The central 90% interval is the predictive mean plus or minus $1.6449$ predictive standard deviations:

10\pm 1.6449(2) =10\pm 3.2898 =[6.710,\,13.29]\ \mathrm{mg/L}.

Under this symmetric normal forecast, 5% of predictive probability lies below 6.710 mg/L and 5% lies above 13.29 mg/L. Because the normal density is symmetric and unimodal, these endpoints also have equal density. The 90% HPD interval is therefore the same interval. This agreement is a consequence of symmetry, not a general identity between the two interval rules.

How Skewness Changes the Interval

Concentrations are nonnegative, and predictive distributions can be right-skewed. For such a distribution, the two interval rules can give different endpoints even when they contain the same predictive probability. An equal-tail interval preserves balanced tail probabilities; an HPD interval instead concentrates its length around regions of high density. Neither rule removes uncertainty or makes the forecast more accurate by itself.

Worked Example: Comparing Intervals for a Skewed Forecast

Suppose a hypothetical individual predictive distribution is lognormal: $\log(\tilde C)\sim N(\log 8,0.6^2)$, with concentration measured in mg/L. Its median is $8$ mg/L. Construct and compare 90% central equal-tail and HPD intervals.

For the central interval, the standard-normal 95th percentile is $1.6449$. Transforming the log-scale quantiles back to the concentration scale gives

[8\exp(-1.6449(0.6)),\,8\exp(1.6449(0.6))] \approx [2.982,\,21.46]\ \mathrm{mg/L}.

This interval leaves 5% of predictive probability below its lower endpoint and 5% above its upper endpoint. Its width is $21.46-2.982\approx18.48$ mg/L.

For the HPD interval, write the log-scale endpoint values as $z_L$ and $z_U$ standard deviations from $\log 8$. Equal density at the concentration-scale endpoints requires $z_L+z_U=-2(0.6)=-1.2$. The 90% probability condition is $\Phi(z_U)-\Phi(z_L)=0.90$, where $\Phi$ is the standard normal CDF. Solving these two equations gives approximately $z_L=-2.516$ and $z_U=1.316$. Thus

[8\exp(0.6(-2.516)),\,8\exp(0.6(1.316))] \approx [1.768,\,17.62]\ \mathrm{mg/L}.

The HPD interval has approximately 90% predictive content and width $17.62-1.768\approx15.85$ mg/L, shorter than the equal-tail interval. Its tails are not balanced: approximately 0.6% of predictive probability is below the lower endpoint and 9.4% is above the upper endpoint. The low-concentration endpoint extends farther because the lognormal density’s high-density region is shifted relative to its long upper tail. The interval rules answer different questions: balanced tail allocation versus concentration of a specified amount of probability in the highest-density region.

Constructing Intervals from Predictive Simulations

When the predictive distribution is available only through simulation, draw future concentrations from the individual posterior predictive distribution. Each draw should reflect both uncertainty in the individual parameters and future residual variability. Equal-tail endpoints can be estimated by sorting the simulated concentrations and taking the corresponding empirical quantiles. An HPD interval for a continuous distribution can be approximated by finding the shortest contiguous interval containing the desired fraction of draws, or by estimating the predictive density and finding its threshold region.

Finite simulation creates approximation error. The number of draws below an estimated lower endpoint and above an estimated upper endpoint need not be exactly equal, even when those endpoints estimate equal-tail predictive quantiles. Likewise, an empirical shortest interval can vary with simulation size and quantile or window conventions. Increase simulation size and check endpoint stability when the decision depends materially on the interval.

Worked Example: A Discrete Predictive Approximation

Suppose a hypothetical simulation summary is represented by the following discrete predictive distribution for a future concentration: 2 mg/L with probability 0.05, 4 mg/L with probability 0.15, 6 mg/L with probability 0.35, 10 mg/L with probability 0.30, and 18 mg/L with probability 0.15. Construct an 80% central equal-tail interval and an 80% highest-probability set.

The cumulative probabilities at 2, 4, 6, 10, and 18 mg/L are 0.05, 0.20, 0.55, 0.85, and 1.00. Using $Q(p)=\inf\{c:F(c)\ge p\}$, $Q(0.10)=4$ mg/L and $Q(0.90)=18$ mg/L. The central interval is therefore $[4,18]$ mg/L. Because the distribution has an atom at 4 mg/L, its closed-interval probability is $0.15+0.35+0.30+0.15=0.95$, rather than exactly 0.80. This is a feature of discrete quantiles, not a calculation error.

To form an 80% highest-probability set, include the support values with greatest probability first. The masses at 6, 10, and 4 mg/L sum to $0.35+0.30+0.15=0.80$. The resulting set is $\{4,6,10\}$ mg/L, whose enclosing interval is $[4,10]$ mg/L and has probability 0.80 under this distribution. This discrete mass-ranking construction illustrates the HPD principle; for a continuous density, use density rather than point probability to rank values.

Diagnostics, Choices, and Common Pitfalls

Before interpreting an interval, check that it corresponds to the intended individual, future time, dose, and covariates. Also verify that predictive simulation includes the relevant future variability; intervals based only on parameter draws can be too narrow if the target is an observed future concentration and residual error has been omitted. A useful numerical check is to compute the fraction of predictive draws inside the reported interval and compare it with the intended content, allowing for Monte Carlo error and discrete endpoint effects.

Unequal simulated tail counts do not, by themselves, rule out an equal-tail interval. Check how the endpoints were constructed. Predictive 5th and 95th percentiles define a central 90% equal-tail interval, although a finite set of draws may fall unevenly in the tails. If a different endpoint rule was used, the interval is not established as equal-tail by its observed tail counts.
Do not transform endpoints as though every interval rule were invariant to transformation. Quantiles transform monotonically, so equal-tail quantiles on a log scale map to equal-tail quantiles on the concentration scale. An HPD region depends on density measured on the chosen scale; the highest-density region on the log scale need not map to the highest-density region on the concentration scale.
Check for multimodality and gaps. A single interval spanning separated HPD components can include low-density values. Report separate components when they are important, or state that the reported interval is an enclosing interval rather than the HPD region itself.
Respect the support and units. A concentration forecast should not yield negative values if the model’s support is strictly positive. An interval that violates a scientifically required support may indicate an unsuitable predictive model or an incorrect transformation.

Equal-tail intervals are straightforward to compute from quantiles and remain easy to interpret when the predictive distribution is skewed. HPD regions can be shorter for a unimodal distribution, but require density estimation or optimization and can be harder to summarize when the distribution is multimodal. The choice should be stated alongside the interval, because the same nominal predictive content does not imply the same endpoints or tail allocation.

Key takeaway: An individual prediction interval is constructed from the individual posterior predictive distribution. Equal-tail intervals divide omitted probability evenly between the tails; HPD regions include the highest-density values. Their endpoints coincide for symmetric unimodal forecasts but can differ substantially for skewed forecasts.

Check Your Understanding

Answer each question using the interval rule and predictive distribution specified.

A central 90% interval is constructed from the predictive 5th and 95th percentiles. A finite simulation has unequal numbers of draws below and above the endpoints. Does this show that the interval is not equal-tail?
For a symmetric unimodal predictive distribution, why do the central equal-tail and HPD intervals coincide?
For the discrete distribution in the worked example, what is the probability of the central interval $[4,18]$ mg/L?
Why might an HPD region for a multimodal predictive distribution be disconnected?
What predictive variability may be missing if an interval is computed from parameter draws alone?

Solutions. (1) No. The endpoints were constructed as predictive 5th and 95th percentiles, so the interval is equal-tail by construction; finite draws can produce uneven counts. (2) Symmetry makes the equal-tail endpoints equidistant from the center, and the unimodal density has equal values at those endpoints, which identifies the shortest interval of that content. (3) Its probability is $0.15+0.35+0.30+0.15=0.95$, because the closed interval includes atoms at both 4 and 18 mg/L. (4) High-density values can occur in separated modes, with lower-density values between them, so the density-threshold set can have multiple components. (5) Future residual variability may be omitted; if the target is an observed future concentration, that omission can make the prediction interval too narrow.