1.11 Machine learning lecture 12  (Page 8/10)

So let me draw a picture that would explain this and I think – Many of my friends and I often don’t remember is less than or great than or whatever, and the way that many of us remember the sign of that in equality is by drawing the following picture. For this example, let’s say, X is equal to 1 with a probability of one-half and X is equal to 6 worth probability 1 whole. So I’ll illustrate this inequality with an example. So let’s see. So X is 1 with probability one-half and X is 6 with probably with half and so the expected value of X is 3.5. It would be in the middle here. So that’s the expected value of X. The horizontal axis here is the X axis. And so F of the expected value of X, you can read of as this point here. So this is F of the expected value of X. Where as in contrast, let’s see. If X is equal to 1 then here’s F of 1 and if X equaled a 6 then here’s F of 6 and the expected value of F of X, it turns out, is now averaging on the vertical axis. We’re 50 percent chance you get F of 1 with 50 percent chance you get F of 6 and so these expected value of F of X is the average of F of 1 and F of 6, which is going to be the value in the middle here. And so in this example you see that the expected value of F of X is greater than or equal to F of the expected value of X. Okay.

And it turns out further that if F double prime of X makes [inaudible]than Z row, if this happens, we say F is strictly convex then the inequality holds an equality or in other words, E of F of X equals F of EX, if and only if, X is a constant with probability 1. Well, another way of writing this is X equals EX. Okay. So in other words, if F is a strictly convex function, then the only way for this inequality to hold its equality is if the random variable X always takes on the same value. Okay. Any questions about this? Yeah.

Student:

[Inaudible]

Instructor (Andrew Ng) :Say that again?

Student: What is the strict [inaudible]?

Instructor (Andrew Ng) :I still couldn’t hear that. What is –

Student: What is the strictly convex [inaudible]?

Instructor (Andrew Ng) :Oh, I see. If double prime of X is strictly greater than 0 that’s my definition for strictly convex. If the second derivative of X is strictly greater than 0 then that’s what it means for F to be strictly convex.

Student:

[Inaudible]

Instructor (Andrew Ng) :I see. Sure. So for example, this is an example of a convex function that’s not strictly convexed because there’s part of this function is a straight line and so F double prime would be zero in this portion. Let’s see. Yeah. It’s just a less formal way of saying strictly convexed just means that you can’t have a convex function within a straight line portion and then [inaudible]. Speaking very informally, think of this as meaning that there aren’t any straight line portions. Okay. So here’s the derivation for the general version of EM. The problem was face is as follows. We have some model for the joint distribution of X of Z, but we observe only X, and our goal is to maximize the law of likelihood of the parameters of model.