Bayesian Model Averaging: How to use in this example? Let ${M_1, M_2}$ denote two competing forecasting models.
With Bayesian model averaging we can get
$p(y_{T+h}|y_{1:T}) = \sum_{j=1}^2p(y_{T+h}|y_{1:T},M_j)*p(M_j|y_{1:T})$
$1:T$ represents the training set and
$h$ the h-ahead forecast of a out-of-sample set $N$
My problem is now to compute the j-th posterior model probalitites (PMP):
$p(M_j|y_{1:T}) = \frac{p(y_{1:T}|M_j)*p(M_j)}{\sum_{l=1}^2p(y_{1:T}|M_l)*p(M_l)}$
Assuming equal prior weights the function reduces to 
$p(M_j|y_{1:T}) = \frac{p(y_{1:T}|M_j)}{\sum_{l=1}^2p(y_{1:T}|M_l)}$
My problem is now, that I dont know how to compute $p(y_{1:T}|M_j)$.
I have the densities/histograms of the training-data (realized data) as well as from both models for the training-data.
Is this enough to compute the above marginal likelihood for each model? Are there any useful approximations?
Can I use maybe use BIC for the weights?
 A: I would frame the problem this way:
$p(M_j|y_{1:T}) = \frac{\big(\int{p(y_{1:T}|\theta,M_j)p(\theta|M_j)d\theta}\big)p(M_j)}{\sum_{l=1}^2\big(\int{p(y_{1:T}|\theta,M_l)p(\theta|M_l)d\theta} \big) p(M_l)}$
where $\theta=(\theta_1,\dots,\theta_d)$ is the vector of random parameters on which your models depend. Actually, $d$ may be different for different models,  but since you have only two models, I'll keep the notation simple. If you have $d_1\neq d_2$, then in the following just consider $d=\max{(d_1,d_2)}$. From the above formula, it's obvious that you need two "ingredients":


*

*you need prior probabilities for the models. You assumed that $p(M_1)=p(M_2)=0.5$, so that's already taken care of, but it's good to remember that your results are conditional on this assumption.

*you need the marginal likelihoods for each model, or equivalently the Bayes factors. That is to say, for each model $M_j$ you need to integrate the likelihood $p(y_{1:T}|\theta,M_j)$ with respect to $p(\theta|M_j)$, which is the prior distribution of $\theta$ for model $M_j$. Now, in a discrete case, where $\theta$ can assume only one possible value under model $M_j$, this step is easy. For example, this is often the case for the model derived under the null, in Bayesian hypothesis testing. But in the continuous case, this becomes a bit more complicated. If $p(y_{1:T}|\theta,M_j)$ and $p(\theta|M_j)$ are a conjugate pair, then this is easy, thus I assume they aren't. I can think of 3 options:


*

*if $T\gg d$, then the likelihood will be extremely peaked around the MLE of $\theta$. In this case, the Laplace approximation should work well. Another large sample approximation is the one based on the BIC. However, I don't know a lot about it: I only used it for linear regression models. Since you're talking about forecasts, I guess you're doing time series modeling. I know that BIC is used also for time series modeling. I would at least cross-check with another method. 

*since you have an explicit expression for $p(\theta|M_j)$, you can sample from it. This means that you can compute $\int{p(y_{1:T}|\theta,M_j)p(\theta|M_j)d\theta}$ by Monte Carlo integration ( unless of course you chose a Cauchy prior). Since likelihood functions are nasty beasts, usually this will converge slowly, because most of the MC samples $\theta_i$ will correspond to very small likelihood values. Importance sampling will improve upon that. 

*if $d$ is small (say, less than 9) numerical integration may work very well. Sparse grid Gaussian quadrature or adaptive Gaussian quadrature will do the trick. If $d$ is very small (for example, less than 4 or 5), even tensor grid Gaussian quadrature could work.


A: After stressful days I think I have found a idea here:
http://www.immagic.com/eLibrary/ARCHIVES/GENERAL/WIKIPEDI/W120607B.pdf
Therefore the BIC of a specific model $i$ can be approximated by
$BIC = n*ln(s_e^2) + k*ln(n)$
with
$n$ : number of observations
$s_e^2$ : error variance
$k$: the number of free parameters
now this BIC can be used to derivate the Posterior Model Probabilities (weights) for the case that the models have uniform distributed priori weights
