Probability the weather model is right tomorrow

Question

A new weather model has predicted rain correctly on 0 of its first 3 attempts. You start from a flat prior $\text{Beta}(1, 1)$ on its true accuracy $p$ , treating every value as equally plausible beforehand.

Using Bayesian updating, what is the probability that the model's next forecast is correct?

Accepted Answer

Intuition first. Zero for three looks bad, but three tries is not enough to declare the model hopeless. The Bayesian prediction averages the success probability across all still-plausible values of p, weighted by belief, and that average, the posterior mean, stays above zero to respect the tiny sample. The update rule. Beta is conjugate to correct/wrong data, so add correct forecasts to the first number and wrong ones to the second: 	ext{Beta}(a,b)\ +\ k 	ext{ correct and } (n-k) 	ext{ wrong}\ ightarrow\ 	ext{Beta}(a+k,\ b+n-k). Here a=1, b=1, with 0 correct and 3 wrong: 	ext{posterior…

Probability the weather model is right tomorrow

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