> The probability of observing the data you observed

Yes, in discrete cases. In continuous cases, you have to work with a probability density. I think this is one of the hurdles people encounter when they're first exposed to Bayesian stats. The probabilities, technically speaking, are zero.

The important insight in Bayesian work is that it's often not the probabilities themselves that matter but the ratios thereof, since from those alone you can compute posteriors.

Indeed, but it's a bit tedious to always say, "probabilities (in the discrete case) or probability densities (in the continuous case)."

In general, you have sums in the discrete case and integrals in the continuous case, but most formulas are otherwise the same.

That's quite true. Also, one could argue that continuous probabilities in practice are discrete probabilities due to finite resolution--we just don't care to specify what the resolution is.

I once had a TA job for an undergrad stats course. This was the "non calculus" course for the psych majors. I had also taken the "math" version of the same course, where we spent two semesters and proved everything. I honestly never came up with a satisfactory layman's explanation why continuous distributions are necessary, or what "continuous" is. I knew that we used calculus to derive the formulas that they were faced with memorizing, but that would have been irrelevant to them.

The best explanation I can think of today is: Use the one that makes the math easier or more readable.

Many measurements are continuous.

What is the age of a rock? That's not a discrete quantity: it could be anything.

Plus or minus what?

I know the importance of continuous sets in the study of statistics as a branch of math. But I don't know of any measurements that can't be represented by integer multiples of a unit for all practical purposes. And the students in the non-calculus stats course can't grasp what continuity is anyway.

> Plus or minus what?

That's what the probability density specifies.

> But I don't know of any measurements that can't be represented by integer multiples of a unit for all practical purposes.

You can always discretize any real number, but why would you? I don't see how integers are easier to deal with than real numbers. Calculus can be viewed as the limit in which you discretize numbers infinitely finely. Once you know how things work in that limit, it's generally easier to use calculus than to work with discretized quantities. One example: summations are often more difficult than integrals, and one way of approximating sums is to turn them into integrals.

From my perspective, calculus is a basic part of mathematics that everyone should be expected to learn in school. In the US, calculus is often viewed as some sort of intimidating subject that only extremely clever people can grasp, and then only in late high-school or in university, but in East Asia, it's taught to children as a matter of course.

I thought likelihood doesn’t necessarily sum to one so it’s not a probability.

It will always sum (or integrate) to one with respect to the data. For example, given likelihood p(x1, x2, ..., x_N|params), summing (or integrating) over all possible values of x_1 ... x_N will indeed yield 1.