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B-Course searches for the single most probable model. While this goal is easy to understand, it can also be criticized. Searching for the single model also has some unwanted consequences.
The answer is tautological: "because it is the most probable model". The quest for the most probable model is easy to comprehend, since it is plausible to think that world is "one and only" and since our model should be model of the world it also should be "one and only". Philosophically maybe, but our incompleteness in modeling makes the story more difficult.
Because of the restrictions in our set of models we consider and because of the finite data sample, it may well happen, that many different models look almost equally probable. Why should we then pick the single most probable one? Maybe if we had gathered one more data vector some other model would appear to be the most probable. This is totally legitimate question. In the future, B-Course is likely to search for many different probable models rather than just the single most probable one. The user may also be offered the possibility to enquire the probability of any model she/he wants to.
When there are many models that have approximately the same probability than the most probable model, those other models should also be consulted when we make predictions (generalizations) for the data we have not seen (i.e., when we want to say something about cases that were not in our data sample). To be exact, all the possible models should be used when making predictions and the contribution of each model should be proportional to model's probability. If we pick just one model, our predictions (generalizations) are worse than if we use many models. Only if we are lucky and the most probable model has very much higher probability than any other model, other models can be discarded and the predictions can be done with the most probable model. This is natural, since in this case we are almost sure that our model is the right one.
Using just a single model also does not allow us to estimate our certainty about the predictions that are based on the model. This is due to the facts that picking one model equals pretending that we have found the true model of the world, and when the truth is known, there is no uncertainty left. However, pretending that we have the correct model does not make our model correct and our certainty is but an illusion. Of course, if it so happens, that the best model has the probability one (or very near one), we are safe.
B-Course, version 2.0.0 |