## Log Likelihood

A problem from yesterday: Suppose you observe n iid normal variables from the normal density, X ∼ N(µ, σ 2 ), where σ 2 is known. (A) Find the maximum likelihood estimator of the mean µ.

This is the normal density function. The trick is you take the log to make it easier.

L is for likelihood, where L(θ; x) = f(x; θ).  L(θ|X) = n ∏ i=1 Pr(Xi = xi |θ).

So, what you do, I THINK, is you take the 1st equation, P(x), and substitute it into that second equation and then take the LOG to MAKE IT EASIER, then take the first derivative to maximize, the second derivative to make sure it’s a max, not a min. After that, it’s okay to cry.

Any time someone suggests that taking a log of a function makes it easier, then I know I’ve wandered into the wrong role playing group. Dungeons and Dragons to the left, Rulers of the Math Geekdom to your right, and People Who Just Want to Take a Nap straight ahead.

As crazy complicated as all this is, it’s an apt metaphor for my life. Simple questions like: “Are you coming home this weekend?” require 2 rounds of computations and end with an answer like: “Probably not.”

Where do you think you’ll work when you finish your degree?  (long pause as I do the mental math shuffle). I don’t know, but I hope it’s somewhere warm.

Let’s go to lunch.dinner/tailgate sometime…. The Magic Abacus says unlikely.

All of the uncertainty is excruciating, but it’s a really good life lesson. Control of our lives is an illusion, one we grip with desperation (lucky socks, pen, underwear anyone?). Maybe this practice of letting go is good for me.