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Hey there, I know this is somewhat a basic question, but got to have it right. Given a joint distribution p(x,y) and a set of parameters w, We have a posterior of w given by p(w|x,y)=p(w)*p(y|x,w)/p(y|x) P(y|x) is defined as the marginal likelihood, that is, we marginilized w out, by doing the integral of p(x,y,w) over w. My question is, what is the actual meaning of this likelihood, I have been looking at it from a gaussian process perspective, but the wikipedia page has as an explanation that it helps us decide which models fits the data the best. Other than this, does it have a more profound meaning. Thanks |
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int dw P(y|x,w)P(w) = P(y|x) is, as Philemon wrote, the posterior distribution of y given x, according to your model. If your model is a gaussian process, this will be a normal distribution that tells you the expected value of y according to your model and the confidence the model has in this model. SO it basically helps us to select which Covariance Matrix works best for the same data?
(Jan 05 '11 at 05:28)
Leon Palafox
Yes. If you look at P(y|x) for the training/validation set you can choose a covariance function K (not matrix, as the space is infinite-dimensional) that maximizes P(y|x,K)P(K)
(Jan 05 '11 at 07:09)
Alexandre Passos ♦
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I'm not 100% sure what kind of answer you are looking for. As the wikipedia page mentions, when you integrate w out p(y|x) is only conditioned on the type of model you used and no specific parametrization of it. You generally also use this quantity to make predictions in a Bayesian setting about y for new values of x. A more correct notation would be p(y|x,M) in which M identifies your model architecture. |
