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Old 06-07-2012, 10:48 AM
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yaser yaser is offline
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Location: Pasadena, California, USA
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Default Re: Questions on the Bayesian Prior

Originally Posted by sakumar View Post
Say in gradient descent (or perceptron) we start with an initial guess for w. Then we proceed to modify w as we process the training data till we get a satisfactory w.

I believe our initial w is not the same as the "Bayesian prior". Why is that? Is it because we are guessing an actual number instead of imposing a probability distribution on the values for w? How would gradient descent change if we modeled w as a Bayesian? In perceptron, when we change w based on a data point that contradicts w's predictions, that is not the same as "conditioning on data"? Why?
You are correct. The initial {\bf w}(0) is not the same as a prior. It does affect the outcome by preferring one local minimum over another, but that is often mitigated by picking the best of several runs.

If we had a prior, and we used gradient descent to pick the hypothesis with the maximum posterior, it would be working on an augmented error of sorts, part of which comes from the prior and plays the role of a regularizer.

Also, somewhat related, I think: Prof Abu-Mostafa explained in the last lecture the second condition where it would be ok to assume a Bayesian Prior. The way I understood it was that if we had enough data points that successive updates would eventually dilute our (perhaps poor estimate of the) original prior, then we're OK? Is my understanding correct? So is that similar to saying our Eout will be low if M is large enough?
Indeed, dilution of the role of the prior is the effect we get when we have sufficient data, but it is not the same as saying that E_{\rm out} is low, as the latter depends on other factors such as noise.
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