   # [SF]Interpretation of a Stochastic Forecast

## Goal of a Stochastic Forecast

A population forecast is intended to be useful in the analysis and planning of social and economic systems such as pensions, health care, education, child care, marketing, investment etc. As a tool in decision making the forecast is a summary of our knowledge of past trends and likely future developments. Given that the predictive value of our current knowledge decreases rapidly as we look further into the future, a characterization of the increasing uncertainty is an essential part of the forecast.

Forecast users cannot, themselves, assess the uncertainty of forecasts. A general finding is that even sophisticated users tend to underestimate uncertainty. Stochastic forecasts are particularly useful when inform the users about the possibility of such future developments that are not the foremost in the minds of the planners and politicians.

## Predictive Distributions

The uncertainty of the UPE forecast is described in terms of probabilities. For each of the 47 forecast years, we have 101 age-groups for both sexes. Or, there are 47×101×2 random variables representing population size in a given year, at a given age, for a given sex. A predictive distribution of future population is the joint probability distribution of these random variables. This distribution takes the place of the conventional projection variants and scenarios.

In practice, predictive distributions are first specified for the vital processes of fertility, mortality, and migration. Then, cohort-component book-keeping is used to derive the induced predictive distribution for future population.

The term stochastic population forecast refers to the stochastic process aspects of the predictive distribution. The term probabilistic population forecast emphasizes the fact that we use probabilities rather than conventional variants or scenarios to describe uncertainty. Mathematically, they are equivalent to the notion of a predictive distribution.

## Subjective Interpretation and External Calibration

Since the events whose probabilities we consider only occur once (e.g., the female population of a country, in age 35, in the beginning of year 2025, obtains a unique value in the course of history), the probabilities cannot be interpreted as relative frequencies. They are best understood subjectively, as representing the degree of uncertainty we now feel concerning the size of future population.

On the other hand, a predictive distribution allows us to make (infinitely) many probability statements. When the future has unfolded, it is possible to evaluate whether or not they have given an appropriate description of uncertainty. We may refer to this as external calibration of predictive distributions. The practical difficulty in external calibration, is that errors in forecasting are typically highly correlated, so any statistical evaluation of past forecasts may have low statistical power. In addition, the waiting time until all forecast years have been followed up, may be prohibitively long.

Subjective probabilities can vary from one individual to another. However, for a predictive distribution to be generally useful, it is necessary that the probabilities can be characterized in relatively objective terms. Our goal has been to produce predictive distributions that have an empirical, statistical foundation.

## Medians and the Spread Around Them

In general, we use medians as the location parameter of the predictive distributions. They represent our best guess as to the future values of the vital processes, by age and sex. Thus, we should be able to interpret the medians as giving the best summary of future from what we have learned from the past. We formulate the medians based on estimates of current values, time series models and judgment.

The spread of the distribution around the median should give a realistic indication of forecast uncertainty the user of the forecast should expect. The starting point of the specification of the spread is the uncertainty observed in the past. We determine the level of past uncertainty using the errors of actual past forecasts, errors of statistical time-series models, and errors of simple baseline (or naive) forecasts. Judgment is used to adjust such estimates up or down so that the desired interpretation is achieved.

Last updated 17.9.2004

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