I'm new to probabilistic programming, and have run into problems of this kind a few times now. Simply put: I often find myself wanting to model a random variable that mostly has some nice, continuous distribution, except that some of the time it falls on a particular fixed value, or is undefined.
Here is a toy example: We have a machine that predicts imminent tsunamis. For each actual tsunami we are interested in both whether or not a tsunami was predicted, as well as how far in advance it was predicted.
I could define a random variable $T$ which models the prediction lead time for any given tsunami. $T=2$ indicates that a tsunami was predicted two minutes in advance. If $T=0$, it means the tsunami was not predicted. I might use an exponentially distributed random variable to do this, but there will be a lot of weight at exactly zero that will break this model.
It seems like I'm after some kind of mixture, but it's not clear to me how to actually construct it either on paper, or in PyMC3. Should I be considering a completely different approach to modelling $T$?