Say I have a black box generating data samples, and I want to estimate the parameters of the black box from the samples.
The black box works like this: it has a parameter m (a real number), and to generate a value v, it first generates v0 according to a normal distribution (with mean m and variance 1), and if v0 is positive it returns v0, if not it returns 0.
So my data samples will be a bunch of zeroes and positive real numbers.
So my question is, from a sample, how do I estimate m?
And what kind of mathematical tools do I use to reason about this?
To me, this looks like a straightforward case of bayesian probability, where I would use p(samples|m) to get p(m|samples) and have some prior on the distribution of m.
So uncle Bayes would say: $p(m|series) = p(series|m) * p(m) / p(series)$
Since the samples are independant, $p(samples|m) = \prod p(sample|m)$
...but some of those $p(sample|m)$ are "discrete probabilities" (when the value is 0), and some are continuous probabilities! Can I muliply them just like that?
(Same goes for calculating $p(samples)$)
Can someone help me clear the confusion?