Someone replied, but deleted, asking about forecasting and their values and I'm wondering the same. Is the weather forecast an expected value, like say an average given a forecasted upper and lower bounds?
This gives you a distribution of predictions, which you can use to produce an average, as well as uncertainty, confidence bands, etc.
The quality of these predictions and ensembles is empirically validated using proper scoring rules (https://en.wikipedia.org/wiki/Scoring_rule), which is a robust way to measure the quality of a probabilistic forecast. Negative-log-likelihood, a ubiquitous loss function in machine learning, is an example of a proper scoring rule (with a minus in front of it).
The science behind forecasting has gotten pretty sophisticated, it's an interesting field.
Then someone else told me that was bogus, and I suspected as much.
It may depend on the source of the forecast
https://forecast.weather.gov/MapClick.php?lat=40.6924&lon=-8...
(the narrative forecasts are more compact than the hourly forecast, but it seems like they would at least have overlapping methodology)
Chance of rain = (chance you will be somewhere in the forecast area where it is raining) x (chance it will rain at all) [1]
The two are independent variables, the overall probability of two independent events both occurring is the product of either event occurring. The chance of rain is the product of two independent events.
So, 50% could mean a 50% chance it will rain everywhere, or a 100% chance it will rain in 50% of the forecast area, or it could be anything in between. (Eg, a 1 mile square cloud that is dropping rain travelling through a 10 square mile area, or a 10% chance that a rain forming cloud will develop over an entire area, or anything in between is a 10% chance of rain. The first case can often be observed with radar and visually, an hourly forecast can be very accurate to know that one cloud will still be raining in an hour. Hence the size of rain area and size of forecast area are variables that come into play)
As a few counterpoints:
- Multiple models do not mean each model is equally likely. Nor do multiple models mean we cover the set of all possibilities. For chance of rain to be the fraction of models predicting rain, those two statements would have to be true
- models themselves each give a chance of rain. These different values are not combined in any way, forecasters pick a model
Some implications and interesting points about forecast area: - if a forecast area is "zoomed" out, should that alter the chance of rain? [Yes, this implies that chance of rain has forecast area as a dependent variable]. Hence, if you "zoom in" on a forecast area far enough, the chance of rain developing becomes the dominant variable for chance of rain.
One interesting thing is a forecast assumes equal probability that you will be at any given location within that forecast area. Which means the percent area forecasted to receive rain is the same probability that you will be at any given location within a forecast area receiving rain
They pick a point and base it on the chance of rain there.
Suppose the hourly forecast has 50% chance of precipitation each hour for two hours. In addition to the points you raise about how to interpret 50% at any moment in time, how are we supposed to know if this means either of the following (or something else):
1. A big storm is passing near a point for those 2 hours, and the only question is whether it passes over it or to the side. I.e., both hours have rain or both don't.
2. A small storm is definitely passing over that point, and the only question is when it will arrive. You either have rain in the first hour or the second but not both or neither.