### Do you see the Matrix? Derivation of Linear chi^2 minimization

This blog post is primarily for my Ay117 students. However, if you've ever wondered where chi-squared minimization comes from, here's my derivation.
 Figure 1: Either a scene from The Matrix or the hallway in your astronomy building.
Yesterday in class we reviewed the concept of "chi-squared minimization," starting with Bayes' Theorem
$P(\{a\} | {d}) \propto P(\{d\} | {a}) P(\{a\})$
In other words, if we wish to assess the probability of a hypothesis that is expressed in terms of the parameters $\{a\}$ conditioned on our data $\{d\}$, we first calculate how likely we were to get our data under the hypothesis (first term on the right), and multiply this "likelihood" by our prior notion that a given set of parameters is representative of the truth.

Supposing that we have data that are independent from one another, and normally distributed, then our likelihood term can be written
$P(\{d\} | {a}) = \prod_i \frac{1}{\sqrt{2\pi \sigma^2}} \exp{\left[\frac{1}{2}\left( \frac{y_i - f(x_i)}{\sigma_i}\right)^2\right]}$
As for the priors, we'll make the fast and loose assumption that they are constant ($P(a_0) = P(a_1) = ... = {\rm const}$). It is computationally advantageous to compute the log-likelihood
$l = \ln{P(\{d\} | {a})} = C - \frac{1}{2} \sum_{i=0}^{N-1} \left[ \frac{y_i - f(x_i)}{\sigma_i}\right]^2 = C - \frac{1}{2} \chi^2$
Since our goal is to find the parameters that maximize the likelihood, this is equivalent to maximizing the log-likelihood, which is in turn equivalent to minimizing that $\chi^2$ thingy.

For the specific problem of fitting an Mth-degree polynomial, $f(x_i) = \sum_{j=0}^{M-1}a_j x_i^j$, and this results in a linear system of equations that can be solved for the best-fitting parameters.

In class, I got my notation all scrambled, and I neglected the measurement uncertainties $\sigma_i$. My bad! Here's what should have appeared on the board (worked out this morning over breakfast, so be sure to check my work!).

To be clear, the "weights" are $w_i = 1/\sigma_i^2$. Zooming in on the key part:

The first problem of the next Class Activity will be to write a function that takes abscissa and ordinate values, and the associated uncertainties, and computes the best-fitting coefficients for a polynomial of arbitrary dimension $M$.

### An annual note to all the (NSF) haters

It's that time of year again: students have recently been notified about whether they received the prestigious NSF Graduate Student Research Fellowship. Known in the STEM community as "The NSF," the fellowship provides a student with three years of graduate school tuition and stipend, with the latter typically 5-10% above the standard institutional support for first- and second-year students. It's a sweet deal, and a real accellerant for young students to get their research career humming along smoothly because they don't need to restrict themselves to only advisors who have funding: the students fund themselves!
This is also the time of year that many a white dude executes what I call the "academic soccer flop." It looks kinda like this:

It typically sounds like this: "Congrats! Of course it's easier for you to win the NSF because you're, you know, the right demographic." Or worse: "She only won because she's Hispanic."…

### On the Height of J.J. Barea

Dallas Mavericks point guard J.J. Barea standing between two very tall people (from: Picassa user photoasisphoto).

Congrats to the Dallas Mavericks, who beat the Miami Heat tonight in game six to win the NBA championship.

Okay, with that out of the way, just how tall is the busy-footed Maverick point guard J.J. Barea? He's listed as 6-foot on NBA.com, but no one, not even the sports casters, believes that he can possibly be that tall. He looks like a super-fast Hobbit out there. But could that just be relative scaling, with him standing next to a bunch of extremely tall people? People on Yahoo! Answers think so---I know because I've been Google searching "J.J. Barea Height" for the past 15 minutes.

So I decided to find a photo and settle the issue once and for all.