What do hour-by-hour weather probabilities mean?
The probabilistic forecasts I’ve read about work this way: Several weather forecasts are run in parallel. Each with slightly different settings, since weather measurements cannot completely capture the state of the atmosphere today and the models are anyway not finegrained enough to accomodate it. So there are many possible settings that for all we know are equally good. I’ve written a bit more about this in [1]. The resulting forecast is the forecast most often predicted. The uncertainty is small if almost all models arrived at the same answer and large if the forecast only won out by a small majority. As for your question: If a forecast of 20% probability of rain is given for each hour in a five hour timespan, what is the probability of rain over the whole time span? The answer is unfortunately that we can’t know. For example, assume that there’s five models running in parallel: • If four models predict no rain over the five hour interval and one model predict rain all the time, the
It really depends heavily on the underlying model of weather. If the rain events were independent across hours, with probabilities , then you would calculate to find out the probability that it will rain at some point during the day. But since the probability of rain is certainly not independent across hours, this is not a great model. At another extreme, if the weather model “knows” for sure that it will rain at most once, briefly, at some point during the day but is not sure which hour that rain will fall in, then it would be more accurate to use the sum of all the probabilities as the probability of any precipitation during the day. Indeed, we know (by the union bound) that the probability of rain at some point during the day is always at most the sum of the . Also, the probability of rain at some point during the day is at at least the maximum of all the , and for some models, it is exactly equal to the maximum of all the . In general, you can think of there being a bunch of stat
