I got the idea for the following problem from Tom Campbell-Ricketts, author of the Maximum Entropy blog at http://maximum-entropy-blog.blogspot.com. And he got the idea from E.T. Jaynes, author of the classic Probability Theory: The Logic of Science:
Suppose that a radioactive source emits particles toward a Geiger counter at an average rate of r particles per second, but the counter only registers a fraction, f, of the particles that hit it. If f is 10% and the counter registers 15 particles in a one second interval, what is the posterior distribution of n, the actual number of particles that hit the counter, and r, the average rate particles are emitted?
To get started on a problem like this, think about the chain of causation that starts with the parameters of the system and ends with the observed data:
The source emits particles at an average rate, r.
During any given second, the source emits n particles toward the counter.
Out of those n particles, some number, k, get counted.
The probability that an atom decays is the same at any point in time, so radioactive decay is well modeled by a Poisson process. Given r, the distribution of n is Poisson distribution with parameter r.
And if we assume that the probability of detection for each particle is independent of the others, the distribution of k is the binomial distribution with parameters n and f.
Given the parameters of the system, we can find the distribution of the data. ...