Week 4: Fishing for Information

Hi all!  Hope everything is going well and that you guys don't catch this stupid sickness that's been going around.
I have to admit, I was all proud of myself for coming up with a punny blog title, and then I saw that everyone else also came up with punny titles this week.  I suppose that's what I get for waiting until Sunday to post.

This week, I will be talking about Fisher Information (one of the main research components of my onsite adviser, Dr. Frieden) and its applications in econophysics.
Fisher information is a statistical tool developed by Ronald Fisher in the 1920s.  He was one of the chief inventors of modern information theory, which of course plays a large role in many different sectors of our lives.  (Computers science, physics, chemistry, and economics, to name a few.)

The basic idea behind Fisher information is to tell us how easy it is to learn about a probability distribution by sampling from it.  Say we have a probability density p that depends on some parameter (traditionally theta; I'll use t), so the density function is p(t).

(If you're interested in the mathematical equation for Fisher information, tell me in the comments and I will try to approximate it with this awful HTML formatting.  I'm focusing more on conceptual understanding here.)

The important result, though, is that the variance in any estimate of the parameter t is equal to 1/I.  
Therefore, the more Fisher information "I" that we have, the better our estimate of t, so it becomes possible to make more precise estimates from data.

(For systems of multiple parameters, this still works, but you have to use something called the Cramer-Rao inequality and the Fisher information matrix, so it's a bit uglier.)

Dr. Frieden's research is a bit of a deviation from standard Fisher Information; he claims that when  that when we observe a system, we do in fact measure some information "I."  However, due to a variety of factors, this value we observe can never be the exact value; rather, the exact, platonic value of the information is given by the letter J.

In any given system, we would want to minimize "I-J"; that is, make the perceived value of the system as close to the real value of the system as possible.  In other words, we are optimizing the information flow by minimizing the Fisher information.

Now, we can begin to apply this theory (called "Extreme physical information, or EPI, because I-J is a minimum/extremum) to financial markets.  After all, finance is completely dependent on information; the intrinsic value of a stock is believed to incorporate all available information about its value.  Stock traders fight each other for clues; in fact, this dependence on information is what makes insider trading such a big deal.  (And why people still attempt it, even though it's illegal.)

To summarize the applications in Fisher information, the probability density function is considered to be the probability of price fluctuation of some given stock, and can be generalized to include the entire market.  The trade price is considered to be the "measurement" in the EPI process.  It turns out that we can construct equilibrium distributions (including yield curves, which measure the volatility of the value of a stock) as well as more dynamic constructions.

There are also some interesting parallels; for one thing, the derived expression for economic valuation has the same general solution as that for stationary quantum mechanics.  (Both obey the Schrodinger equation.)  Also, interest rate dynamics are shown to be analogous to the Fokker-Plank expression for diffusion processes.

Ultimately, Fisher information is a statistical tool that is generally applied in the natural sciences; however, it can also be applied in economics, resulting in similar expressions to other physical properties.  Therefore, it is an interesting example of the range of tools available to those pursuing econophysics.

This coming week, I'm planning to start the grand scavenger hunt for supplies for my experimental work.  I'll be sure and keep you all updated about that, but until then, enjoy your week, and feel free to comment.