I have two models $M_1$ and $M_2$ that I am using to try and compare to observed data $D$. $M_1$ is an $n_1$-dimensional model, and $M_2$ is an $n_2$-dimensional problem. The Bayes factor $K$ to compare the models can be calculated using:

$K = P(D|M_1)/P(D|M_2) $

assuming no prior preference for either model. The numerator and denominator can be written as

$P(D|M_i) = \int P(D|\mathbf{w},M_i) P(\textbf{w}|M_i) d\mathbf{w}$

where $\mathbf{w}$ is the parameter vector, so the integral is over parameter space.

Now say that due to e.g. computational constraints, one can only compute $M_1$ and $M_2$ for a finite number of random samples of the parameter vector $\mathbf{w}$, where the number of samples is given by $s_1$ and $s_2$. Would it be acceptable to then say that the integral above becomes a summation over the random samples, and assuming the random samples are uniformly distributed through parameter space, $P(\textbf{w}|M_i)$ becomes $1/s_i$, so that:

$P(D|M_i) = \sum_{j=1}^{s_i} P(D|\mathbf{w_j},M_i) / s_i$

and so what is being compared in the Bayes factor $K$ is the ratio of the average probability over all the samples for each model?


2 Answers 2


Yes, you can do that. However, your I'd like to play with your formulas a little bit.

If the model is determined by the parameters, than $P(D|M_i)=\int P(D|w,M_i)P(M_i|w)*P(w)dw$ should be more appropriate. Since I guess the model is determined by the parameters in a deterministic (instead of stochastic) way, the formula can be abbreviated to $P(D|M_i)=\int P(D|w,M_i)*P(w)dw$.

Given this and only a finite uniform sample over the parameter space, your approximation is indeed correct.


Model evidence $P(D|M_i)$ can be viewed as an expectation of $P(D|w, M_i)$ with respect to distribution $P(w|M_i)$. You can then use Monte-Carlo methods to estimate it with required precision.

Other suitable options include using Laplace Approximation and then finding closed-form solution for evidence (as they do in RVM).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.