Probabilistic forecasting summarises what is known, or opinions about, future events. In contrast to a single-valued forecasts (such as forecasting that the maximum temperature at given site on a given day will be 23 degrees Celsius or that the result in a given football match will be a no-score draw), probabilistic forecasts assign a probability to each of a number of different outcomes, and the complete set of probabilities represents a probability forecast.
Weather forecasting represents a service in which probability forecasts are sometimes published for public consumption, although they may also be used by weather forecasters as the basis of a simpler type of forecast. For example forecasters may combine their own experience together with computer-generated probability forecasts to construct a forecast of the type "we expect heavy rainfall".
Sports betting is another field of application where probabilistic forecasting can play a role. The pre-race odds published for a horse race can be considered to correspond to a summary of bettors' opinions about the likely outcome of a race, although this needs to be tempered with caution as bookmakers' profits needs to be taken into account. In sports betting, probability forecasts may not be published as such, but may underlie bookmakers' activities in setting pay-off rates, etc..
The probability information is typically derived by using several numerical model runs, with slightly varying initial conditions. This technique is usually referred to as ensemble forecasting by an Ensemble Prediction System (EPS). EPS does not produce a full forecast probability distribution over all possible events, and it is possible to use purely statistical or hybrid statistical/numerical methods to do this. For example, temperature can take on a theoretically infinite number of possible values (events); a statistical method would produce a distribution assigning a probability value to every possible temperature. Implausibly high or low temperatures would then have close to zero probability values.
If it were possible to run the model for every possible set of initial conditions, each with an associated probability, then according to how many members (i.e., individual model runs) of the ensemble predict a certain event, one could compute the actual conditional probability of the given event. In practice, forecasters try to guess a small number of perturbations (usually around 20) that they deem are most likely to yield distinct weather outcomes. Two common techniques for this purpose are breeding vectors (BV) and singular vectors (SV). This technique is not guaranteed to yield an ensemble distribution identical to the actual forecast distribution, but attaining such probabilistic information is one goal of the choice of initial perturbations. Other variants of ensemble forecasting systems that have no immediate probabilistic interpretation include those that assemble the forecasts produced by different numerical weather prediction systems.
Canada has been one of the first countries to broadcast their probabilistic forecast by giving chances of precipitation in percentages. As an example of fully probabilistic forecasts, recently, distribution forecasts of rainfall amounts by purely statistical methods have been developed whose performance is competitive with hybrid EPS[clarification needed]/statistical rainfall forecasts of daily rainfall amounts.
Probability forecasts have also been used in the field of population forecasting.
Assessing probabilistic forecasts is more complex than assessing deterministic forecasts. If an ensemble-based approach is being used, the individual ensemble members need first to be combined and expressed in terms of a probability distribution. There exist probabilistic (proper) scoring rules such as the continuous-ranked probability score for evaluating probabilistic forecasts. One example of such a rule is the Brier score.
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