Sometimes you can derive a probability distributions from a list of properties it must have. For example, there are several properties that lead inevitably to the normal distribution or the Poisson distribution.
Although such derivations are attractive, they don’t apply that often, and they’re suspect when they do apply. There’s often some effect that keeps the prerequisite conditions from being satisfied in practice, so the derivation doesn’t lead to the right result.
The Poisson may be the best example of this. It’s easy to argue that certain count data have a Poisson distribution, and yet empirically the Poisson doesn’t fit so well because, for example, you have a mixture of two populations with different rates rather than one homogeneous population. (Averages of Poisson distributions have a Poisson distribution. Mixtures of Poisson distributions don’t.)
The best scenario is when a theoretical derivation agrees with empirical analysis. Theory suggests the distribution should be X, and our analysis confirms that. Hurray! The theoretical and empirical strengthen each other’s claims.
Theoretical derivations can be useful even when they disagree with empirical analysis. The theoretical distribution forms a sort of baseline, and you can focus on how the data deviate from that baseline.
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