Each night different meteorologists give us the probability...

Each night different meteorologists give us the probability that it will rain the next day. To judge how well these people predict, we will score each of them as follows: If a meteorologist says that it will rain with probability $p$, then he or she will receive a score of $$ \begin{array}{ll} 1-(1-p)^{2} & \text { if it does rain } \\ 1-p^{2} & \text { if it does not rain } \end{array} $$ We will then keep track of scores over a certain time span and conclude that the meteorologist with the highest average score is the best predictor of weather. Suppose now that a given meteorologist is aware of this and so wants to maximize his or her expected score. If this person truly believes that it will rain tomorrow with probability $p^{*}$, what value of $p$ should he or she assert so as to maximize the expected score?

Each night different meteorologists give us the probability that it will rain the next day. To judge how well these people predict, we will score each of them as follows: If a meteorologist says that it will rain with probability $p$, then he or she will receive a score of
$$
\begin{array}{ll}
1-(1-p)^{2} & \text { if it does rain } \\
1-p^{2} & \text { if it does not rain }
\end{array}
$$
We will then keep track of scores over a certain time span and conclude that the meteorologist with the highest average score is the best predictor of weather. Suppose now that a given meteorologist is aware of this and so wants to maximize his or her expected score. If this person truly believes that it will rain tomorrow with probability $p^{*}$, what value of $p$ should he or she assert so as to maximize the expected score?

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Key Concepts

Proper Scoring Rule

A proper scoring rule is a method of evaluating probabilistic forecasts such that the expected score is maximized when the forecaster reports their true belief. This property ensures that there is no incentive to skew the prediction to improve the score. By designing the scoring function in this way, the rule promotes honest and accurate reporting of subjective probabilities.

Expected Value Maximization

In decision theory, a forecast or prediction is evaluated by its expected value, which is computed as a weighted sum of the scores corresponding to different outcomes. A rational forecaster, aiming to maximize the expected score, will select the prediction that maximizes this expected value based on their true beliefs about the likelihood of events.

Truthful Reporting Incentive

Given that the scoring rule provided is proper, the best strategy for a meteorologist is to report the probability that aligns with their actual belief. If a meteorologist believes that it will rain with probability p*, then reporting any probability different from p* would lead to a lower expected score. This incentive for truthful reporting helps maintain the integrity of probabilistic forecasts.

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