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I have two beta-distributions:

$H_1 = Beta(\alpha_1, \beta_1) $

and

$H_2 = Beta(\alpha_2, \beta_2) $

(parameters are known), and I'd like to estimate whether a new sample $D$ rather comes from $H_1$ or $H_2$.

It seems that Bayes factor is the solution

$K = \frac{\int \Pr(\theta_1|H_1)\Pr(D|\theta_1,H_1)\,d\theta_1} {\int \Pr(\theta_2|H_2)\Pr(D|\theta_2,H_2)\,d\theta_2}$

but I don't really see how to compute it. How do I get and integrate $Pr(\theta)$ and $Pr(D|\theta)$?

Also, how does it generalize to Dirichlets?

Thanks, JP.

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  • $\begingroup$ What is the likelihood of your data? I'm assuming Binomial? $\endgroup$
    – user44764
    May 15 '14 at 14:22
  • $\begingroup$ @Matthew Well, actually my real data is multinomial, that's why I ask for the generalization, but I'd like first to understand the binomial/beta-distribution case. $\endgroup$
    – user45607
    May 15 '14 at 14:27
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You said in the comments that this is for a Binomial likelihood, which you want to extend to the Multinomial-Dirichlet situation.

To evaluate $K$ you're looking to evaluate the likelihood of your data given each model. In the Beta-Binomial case of $N$ trials where $$ p \sim \text{Beta}(\alpha, \beta) \\ y \sim \text{Binomial}(N, p), $$ then the data likelihood is $$ \begin{eqnarray*} \pi(y \mid \alpha, \beta) &=& \int_0^1 \text{Binomial}(y \mid N, p) \text{Beta}(p \mid \alpha, \beta) \,dp. \\ &=& \int_0^1 {N \choose y} p^y (1-p)^{N-y} \times \frac{1}{\text{B}(\alpha,\beta)} p^{\alpha-1} (1-p)^{\beta-1} \,dp \\ &=& {N \choose y} \frac{1}{\text{B}(\alpha,\beta)} \int_0^1 p^{\alpha - 1 + y} (1-p)^{\beta - 1 + N -y} \,dp \\ &=& {N \choose y} \frac{\text{B}(\alpha+y,\beta + N - y)}{\text{B}(\alpha,\beta)} \end{eqnarray*} $$

where $\text{B}$ is the beta function; you should be able to compute it with some software package or even some online tool. This probability distribution is called the Beta-Binomial distribution. In this case, $$ K = \frac{\text{B}(\alpha_1 +y,\beta_1 + N - y) \text{B}(\alpha_2,\beta_2)}{ \text{B}(\alpha_2 +y,\beta_2 + N - y) \text{B}(\alpha_1,\beta_1)} $$

In the multiple category case there is the Dirichlet-Multinomial distribution which gives $K$ in a similar form with Gamma functions.

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    $\begingroup$ Thanks a lot! This answer exactly covers what I missed from my readings so far, that's why I mark it as best answer. But new readers should definitely read catchmeifyoutry's answer as well, which is perfectly complementary. $\endgroup$
    – user45607
    May 15 '14 at 15:58
  • $\begingroup$ @Matthew: Shouldn't the data likelihood as you computed it be the numerator of your result? The normalizing constant of the posterior beta distribution is B(a+y,b+N-y). In my opinion, it is not a ratio of beta functions $\endgroup$
    – beginneR
    Oct 30 '14 at 17:24
  • $\begingroup$ @beginneR The p.m.f. for the Beta-Binomial distribution is as I derived. If you instead mean that I have incorrectly determined the model likelihood, then observe from the law of total probability that $P(D\mid M) = \int_\mathbb{R} P(D \mid \theta) p(\theta \mid M) d\theta$, where $p(\theta \mid M)$ is just the prior probability of the parameter $\theta$ under the model $M$. $\endgroup$
    – user44764
    Oct 30 '14 at 20:51
  • $\begingroup$ Maybe I am missing something here. Is it correct that you derived the denominator of Bayes Law for binomial data, also called the prior predictive distribution? I thought that the BF is a ratio of the denominator terms of two models (p(D|M1)/p(D|M2)). Or, put differently, the BF is the ratio of two normalizing constants for the posterior distributions? $\endgroup$
    – beginneR
    Nov 1 '14 at 12:34
  • $\begingroup$ @beginneR The models being discussed here are probability distributions on the parameters; that is, $M_1 \equiv p \sim \text{Beta}(\alpha_1, \beta_1)$ and $M_2 \equiv p \sim \text{Beta}(\alpha_2, \beta_2)$. So in order to obtain $P(M_i \mid D)$ it is necessary to integrate over the parameter $p$ as shown $\endgroup$
    – user44764
    Nov 3 '14 at 3:44
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I'll answer for the more general case of a Dirichlet prior and Multinomial likelihood, which is what you are interested in anyway.

The integral has a closed form solution, which can be found in this Wikipedia article on the Drichlet-Multinomial.

$$p(X|\alpha_0) = DirMult(X| \alpha_0) = \int_\theta Cat(X|\theta) Dir(\theta|\alpha_0) d \theta$$ $$= \frac{\Gamma(\sum_k \alpha_{0k})}{\Gamma(\sum_k n_k + \sum_k \alpha_{0k})} \prod_k \frac{\Gamma(n_k + \alpha_{0k})}{\Gamma(\alpha_{0k})}$$

where X are the observations, which we can summarize into occurrence counts $n_k$ ($n_i$ is the number of times event $i$ occurred in observations $X$), $a0$ is the vector of parameters of the Dirichlet prior. $\Gamma$ is the Gamma function.

An important but subtitle thing to note is that this solution actually assumes a Categorical likelihood instead of Multinomial, as the name suggests. That means that for a proper multinomial, you should include the multinomial coefficient.

Derivation

It can be good to study how to obtain this result when confronted with other conjugate pairs

  • From the Dirichlet distribution we know that

$$\int_\theta \left [ \frac{\Gamma(\sum_k \alpha_{0k})}{\prod_k \Gamma(\alpha_{0k})} \prod_k \theta_k^{(\alpha_{0k} - 1)} \right ] d\theta = 1$$ thus $$\int_{\theta} \left [ \prod_k \theta_k^{(\alpha_k - 1)} \right ] d\theta = \frac{\prod_k \Gamma(\alpha_k)} {\Gamma(\sum_k \alpha_k)}$$

  • Therefore,

$$p(X|\alpha_0) = \int_\theta Cat(X|\theta) Dir(\theta|\alpha_0) d \theta$$

$$= \int_\theta \left [ \prod_k \theta_k^{n_k} \times \frac{\Gamma(\sum_k \alpha_{0k})}{\prod_k \Gamma(\alpha_{0k})} \prod_k \theta_k^{(\alpha_{0k} - 1)} \right ] d\theta$$

$$= \frac{\Gamma(\sum_k \alpha_{0k})}{\prod_k \Gamma(\alpha_{0k})} \int_\theta \left [ \prod_k \theta_k^{(n_k + \alpha_{0k} - 1)} \right ] d\theta$$

$$= \frac{\Gamma(\sum_k \alpha_{0k})}{\prod_k \Gamma(\alpha_{0k})} \times \frac{\prod_k \Gamma(\alpha_{0k} + n_k)} {\Gamma(\sum_k \alpha_{0k} + \sum_k n_k)}$$

$$= \frac{\Gamma(\sum_k \alpha_{0k})}{\Gamma(\sum_k n_k + \sum_k \alpha_{0k})} \prod_k \frac{\Gamma(n_k + \alpha_{0k})}{\Gamma(\alpha_{0k})}$$

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