Predicting tomorrow's trade count, Poisson-Gamma gives Negative Binomial

Question

You have modeled daily trade counts as 	ext{Poisson}(\lambda) and, after observing data, hold a posterior \lambda \sim 	ext{Gamma}(a, b) (shape a, rate b), say a = 8, b = 4. Derive the predictive distribution of tomorrow's count, and compare its spread to a plain Poisson.

Accepted Answer

You are uncertain about \lambda, so the honest forecast averages the Poisson over the whole Gamma posterior rather than plugging in a single \lambda. The posterior predictive for a new count X is p(x) = \int0^\infty \underbrace{\frac{e^{-\lambda}\lambda^{x}}{x!}}{	ext{Poisson}} \cdot \underbrace{\frac{b^{a}}{\Gamma(a)}\lambda^{a-1}e^{-b\lambda}}{	ext{Gamma}}\, d\lambda. Pull out constants and use \int0^\infty \lambda^{x+a-1}e^{-(b+1)\lambda}\,d\lambda = \frac{\Gamma(x+a)}{(b+1)^{x+a}}: p(x) = \frac{\Gamma(x+a)}{x!\,\Gamma(a)}\left(\frac{b}{b+1}ight)^{a}\left(\frac{1}{b+1}ight)^{x}. This…

Predicting tomorrow's trade count, Poisson-Gamma gives Negative Binomial

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