Why probability models
If you want a mathematical model to incorporate uncertainty, you create a probability model. Probability models uncertainty. An application of probability need not have anything to do with randomness; randomness is only one kind of uncertainty.
If something is perfectly deterministic in theory but not accurately known, it’s often useful to model it as if it were random, even if we know full well it is not. As I like to say, random is as random does.
Challenging problems cannot be solved by applying existing probability models. Instead they require constructing new probability models to capture the nuances of the particular problem. Conventional statistical models fail when they shoehorn reality into a model that does not fit.
Statistical models explicitly depend on probability; one of the distinctions between statistics and machine learning is that the former insists on having an underlying probability model while the latter does not. (Machine learning often uses probability models, implicitly if not explicitly, but it does not insist that a method have a probability model at its foundation.)
Mathematical models that do not appear to require probability, such as differential equations, often do in practice. When I was studying differential equation models in graduate school, I asked where the parameters in our model came from. My question caused a similar level of awkwardness as a child asking where babies come from. Differential equation parameters come from analyzing data, and so there may be a significant amount of uncertainty regarding their values.
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