# Chapter 14. A Hierarchical Model

# The Geiger counter problem

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

rparticles per second, but the counter only registers a fraction,f, of the particles that hit it. Iffis 10% and the counter registers 15 particles in a one second interval, what is the posterior distribution ofn, the actual number of particles that hit the counter, andr, 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. ...

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