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When we are comparing two models against some data, will we obtain the same (set of) posterior odds for the models both when we use the Bayes' factor and when we use the discriminant rule?

If not, which one is considered more "accurate" and why is the other being used?

I have found that the Bayes' factor is described also as:

$B_{i,j} = \frac{\pi_{data}(D|M_i)}{\pi_{data}(D|M_j)} = \frac{L_{max}^{(i)}}{L_{max}^{(j)}} \frac{W_i}{W_j}$

where $W_i$ is the Ochham's factor: $W_i = \int_{V_{\theta_{i}}} \pi_{prior}(\theta_i|M_i) \frac{L_{(i)}(\theta_i)}{L_{max}^{(i)}} d\theta_j$

Is $\frac{L_{max}^{(i)}}{L_{max}^{(j)}} \frac{W_i}{W_j}$ anyhow relevant to the ratio of 'allocations' at each model ($\frac{allocations-to-model_i}{allocations-to-model_j}$), made by the Bayes' Discriminant rule?

Edit-: After the response of conjugateprior, I would like to add this:

I am wondering whether a discriminant analysis (such as ML discriminant rule i.e. Bayesian discr. for priors 1/2 and 1/2) is an acceptable way of doing model selection when we are comparing two models from which we inferred posterior information about some parameter/hyperparameter via Bayes' theorem. I.e. although we assume that the data are generated by one model, if you do discriminant analysis, won't the percentage of the mixture reveal the model that the Bayesians' describe as 'more likely'?

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By 'Bayes Discriminant Rule' do you mean simply Bayes theorem (as here)? If you do then I think you are asking is:

Is model comparison using Bayes Factors is just like data point allocation in a mixture model on the basis of posterior probabilities of generation by each component, thinking of the Bayes Factor as a component likelihood.

The anwer to that question seems to be (mostly) no.

First, model comparison assumes that all the data was generated from one model or another, or that one model is more likely to have generated all the data than the other model. Either way one does not get to assign model responsibility per data point. Second, the mixture context estimates component mixing parameters whereas Bayes Factor comparisons tend to give models a priori equal prior probabilities of being the generating model. The similarity is that Bayes theorem is applied.

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  • $\begingroup$ Thank you, I can see the difference now. Now, I am wondering whether a discriminant analysis (such as ML discriminant rule i.e. Bayesian discr. for priors 1/2 and 1/2) is an acceptable way of doing model selection when we are comparing two models from which we inferred posterior information about some parameter/hyperparameter via Bayes' theorem. I.e. although we assume that the data are generated by one model, if you do discriminant analysis, won't the percentage of the mixture reveal the model that the Bayesians' describe as 'more likely'? $\endgroup$ – Low Yield Bond Aug 25 '14 at 0:02

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