> I haven’t found a good discussion of why these probabilistic methods are preferred or how we should interpret the associated probabilities
The reason is a bit subtle yet interesting: it's a practical trade-off and cleanliness of interpretation is not the primary goal! From what I understood from NWP folks, in ensemble forecasting the basic goal is not necessarily to be "distributionally accurate", although it can work in that sense as a quite rough first-order rule of thumb. Since each individual simulation ("member") is computationally expensive, you want to see as many diverse plausible scenarios as possible from that limited budget, rather than spend most of your CPU cycles just simulating minor deviations from a central scenario. Hence they use techniques like breeding / singular vectors (or, contemporarily, variational assimilation) to stimulate the fastest-growing perturbations and try to catch the main modes of the variability. They are more like "probes" of the dynamics of the system than a method that explicitly targets a rigorous probabilistic distribution of the system, so there is no natural probabilistic interpretation of raw ensemble output. Hence, as you say, the need for post-processing and calibration in most applications.
In other words, raw ensemble forecasts look like "sampling from a distribution" only as a first impression. The interpretation as a distribution is murkier than np.random.normal(size=50)
It may be remarkable how far we've come, but over here on the east coast we still need to squint at the weather radar, and sometimes even the clouds themselves, on the late summer afternoons and evenings to decide if that 40-60% chance of storm is happening or not. (It's our regular reminder of chaotic dynamics.)
> I haven’t found a good discussion of why these probabilistic methods are preferred or how we should interpret the associated probabilities
The reason is a bit subtle yet interesting: it's a practical trade-off and cleanliness of interpretation is not the primary goal! From what I understood from NWP folks, in ensemble forecasting the basic goal is not necessarily to be "distributionally accurate", although it can work in that sense as a quite rough first-order rule of thumb. Since each individual simulation ("member") is computationally expensive, you want to see as many diverse plausible scenarios as possible from that limited budget, rather than spend most of your CPU cycles just simulating minor deviations from a central scenario. Hence they use techniques like breeding / singular vectors (or, contemporarily, variational assimilation) to stimulate the fastest-growing perturbations and try to catch the main modes of the variability. They are more like "probes" of the dynamics of the system than a method that explicitly targets a rigorous probabilistic distribution of the system, so there is no natural probabilistic interpretation of raw ensemble output. Hence, as you say, the need for post-processing and calibration in most applications.
In other words, raw ensemble forecasts look like "sampling from a distribution" only as a first impression. The interpretation as a distribution is murkier than np.random.normal(size=50)
I appreciate this context! Many thanks.
Cool CA, AKA Hotter Than The Hinges To The Doors Of Hell, is the coolest town in California.
I grew up in Cool, CA. I would have forecasted a low probability of getting a shoutout in the comments on this post (or any, for that matter)
> 3-day forecasts are now remarkably prophetic.
It may be remarkable how far we've come, but over here on the east coast we still need to squint at the weather radar, and sometimes even the clouds themselves, on the late summer afternoons and evenings to decide if that 40-60% chance of storm is happening or not. (It's our regular reminder of chaotic dynamics.)
Separately, here's a fun ethnographic piece on how people (20 years ago) interpreted weather probabilities: https://onlinelibrary.wiley.com/doi/10.1111/j.1539-6924.2005.00608.x