EM algorithm special case I'm considering a collection of Bayes Factors, $\mbox{BF}(j)$, $j=1, ..., J$, so that the overall evidence against $H_0$ is represented by the overall Bayes Factor
 $$\frac{P(x|H_0 \mbox{ false})}{P(x | H_0 \mbox{ true})} = \sum_{j=1}^{J} w_j \mbox{BF}(j)\,,$$ 
where $\sum_{j=1}^Jw_j = 1$.
I now have access to a restricted set data from which I would like to estimate the optimal weights $w_j$: I can to compute the Bayes Factors $\mbox{BF}_i{(j)}$ for each observation $i=1, ..., n$ and for each $j=1, ..., J$, and I can form:
$$\prod_{i=1}^n \sum_{j=1}^{J} w_j \mbox{BF}_i(j)\,.$$
So the problem consists in finding weights that maximize this expression. I was wondering if we could consider this as a likelihood and compute a sort of EM algorithm, similar to that used for Gaussian mixtures.
However, as far as I understand, since I already have the $\mbox{BF}_i(j)$ computed, I wouldn't need to optimize the means and variances for each mixture(?)
...and in this case, I don't see how to apply the EM algorithm anymore.
I would be very grateful if somebody could help me on this.
Is the EM approach a good option? If yes, how exactly should I apply it? If no, is there a simple other strategies that guaranties $0 \leq w_j \leq 1$ $\forall j$ and $\sum_{j=1}^Jw_j = 1$ ?
Many thanks in advance!
 A: If you treat the function to maximise$$\prod_{i=1}^n \sum_{j=1}^{J} w_j \mbox{BF}_i(j)$$as a pseudo-likelihood, this likelihood is a formal mixture where only the weights $w_j$ are unknown. Hence you can apply EM for that purpose (of finding a local optimum). The algorithm is as follows:


*

*Start with an arbitrary value of $\mathbf{w}^{(0)}=(w_1,\ldots,w_J)$

*Given the current value of the parameter $\mathbf{w}^{(t)}$, derive the expectations (E) of the latent variables $z_{ij}$ as$$\mathbb{E}_{\mathbf{w}^{(t)}}[Z_{ij}|x_i]=\dfrac{w_j^{(t)}BF_i(j)}{\sum_{k=1}^J w_j^{(t)}BF_i(j)}$$

*Update the weights from $\mathbf{w}^{(t)}$ to $\mathbf{w}^{(t)}$ as
$$w_j^{(t+1)}=\dfrac{\sum_{i=1}^n \mathbb{E}_{\mathbf{w}^{(t)}}[Z_{ij}|x_i]}{\sum_{k=1}^J \sum_{i=1}^n \mathbb{E}_{\mathbf{w}^{(t)}}[Z_{ik}|x_i]}$$

*Increase $t$ to $t+1$ and get back to Step 2. unless $\mathbf{w}^{(t)}=\mathbf{w}^{(t)}$


The justification for this representation comes from writing the mixture$$ \sum_{j=1}^{J} w_j \mbox{BF}_i(j)$$as the marginal of$$w_j^{z_{ij}} \mbox{BF}_i(j)^{z_{ij}}$$where $\mathbf{z_i}=(z_{i1},\ldots,z_{iJ})$ is a multinomial $\mathcal{M}(1;\mathbf{w})$, i.e. with only one component equal to one and all others equal to zero.
