Saturday, August 6, 2016

Bayes 23: enumerated events revisited

Another idea that I've glimpsed from the book on boosting is the handling of the enumerated events, previously described in the part 19. The part 6 of my notes on boosting describes how the decision stumps can be treated as only "half-stumps": actionable if the answer is "yes" and non-actionable if the answer is "no" (or vice versa). This is actually the same thing as I complained of before as the mistreatment of the Bayesian formula where the negative answer is treated the same as the lack of answer. But taken in the right context, it makes sense.

If we take a complementary pair of such half-symptoms (asking the same question, and one of them equaling the negative answers with "don't know", another one equaling the positive answer with "don't know"), their combined effect on the probability of the hypotheses will be exactly the same as of one full symptom. In the weight-based model, the weights of the hypotheses after the complementary pair will be only half of those after one full symptom but they will all be scaled proportionally, making no difference. Alternatively, if we equal the negative answers not with "don't know" but with "irrelevant", even the weights will stay the same.

The interesting thing is that these half-symptoms can be straightforwardly extended to the multiple-choice questions. Each choice can be equaled with one half-symptom. So if the answer to this choice is "yes" then it takes effect, if "no" then it gets skipped. In the end exactly one choice takes effect. Or potentially the answer can also be "none of the above" and then this symptom will be simply skipped. It should also be relatively straightforward to accommodate the answers like "it's probably one of these two", taking both answers at half-weight. I didn't work through the exact formulas yet but I think it shouldn't be difficult.

The approach of taking the answer at a partial weight also provides a possible answer to "should we treat this problem as model-specific or generic?": it allows to mix both together, taking say the model-specific approach at the weight 0.99 and the generic at 0.01. Then if the model-specific approach finds a matching hypothesis, great, if not then the answer found with the generic approach will outweigh it. This weight of the generic approach should be higher than the confidence cap of the "can't happen" answer: the generic weight of 0.01 would probably work decently well together with the capping probability of 0.001.

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