Exercise 3.17 - Marginal likelihood for beta-binomial under uniform prior

Answers

The marginal likelihood is given by:

p (N_{1} | N) = \int_{0}^{1} p (N_{1}, 𝜃 | N) d 𝜃 = \int_{0}^{1} p (N_{1} | 𝜃, N) \cdot p (𝜃) d 𝜃 .

Plug in:

p (N_{1} | 𝜃, N) = Bin (N_{1} | 𝜃, N),

p (𝜃) = Beta (𝜃 | 1, 1) .

Thus:

\begin{align} p (N_{1} | N) = & \int_{0}^{1} (\binom{N}{N_{1}}) \cdot 𝜃^{N_{1}} \cdot {(1 - 𝜃)}^{N - N_{1}} d 𝜃 \\ = & (\binom{N}{N_{1}}) \cdot B (N_{1} + 1, N - N_{1} + 1) \\ = & \frac{N!}{N_{1}! \cdot (N - N_{1})!} \cdot \frac{N_{1}! \cdot (N - N_{1})!}{(N + 1)!} \\ = & \frac{1}{N + 1} . \end{align}

Where $B$ is the regulizer for a Beta distribution:

B (a, b) = \frac{Γ (a) \cdot Γ (b)}{Γ (a + b)} .

The physics behind this setting is that if no prior information is introduced then all $N + 1$ possibilities are equally likely to appear.

solour_lfq

2021-03-24 13:42