Someone asked me "why did you say in the Gaussian processes tutorial that Gaussian processes are ill-conditioned?" Here is an answer in the form of a tiny exercise.

Take the simplest case where the function is just a constant.

y(x) = Y for all x. (Y is Gaussian distributed and not known a priori)

Make a GP with the appropriate covariance function.

Now

(1) think about the inference of the GP given data {x,t}. (Not very difficult is it?)

(2) see what happens when you use the standard GP matrix inversion formalism to solve the problem. What is the well-conditioned-ness of the matrix you must invert?

## Saturday, January 31, 2009

## Friday, January 9, 2009

### Soaring, Cryptography and Nuclear Weapons

Please go to nuclearrisk.org and read renowned cryptographer Martin Hellman's article Soaring, Cryptography and Nuclear Weapons.

If you feel the urge to pass the message on, pass it on.

## Saturday, January 3, 2009

### Why traders make bad investments

I heard this on Radio 4 - Paul Wilmott's explanation of why traders will rationally put their bank's money into suboptimal risky investments.

(Of course this isn't "the" reason why bad investments happen. Rather, this cartoon presents a simple world in which we can see why traditional incentives for employees lead to bad outcomes.)

Imagine Joe is a trader at a fancy investment bank where all the many other traders are lemmings who unanimously invest over the next 6 months in deal A, which has a 50% chance of making a big return, and a 50% chance of not. Joe is a wise trader and has identified deal B, which has a 75% chance of making a big return for his bank. Deals A and B are independent random variables. What should Joe do with the bank's money?

Well, we need to specify Joe's utility function, which is dominated by his bonus. The rule at his bank is that Joe gets a huge bonus if the bank makes a big return, AND Joe's chosen investment made a big return. Otherwise, no bonus.

If Joe invests in deal A (like all the other lemmings alongside whom he works) then he has a 50% chance of getting a huge bonus.

If Joe instead invests his relatively small part of the bank's money in deal B, then he will get a bonus only if _both_ deal A and deal B come out as successes - because the bank will get a big profit only if deal A succeeds. So Joe has a 37.5% chance of getting a big bonus.

So Joe's rational decision is to invest the bank's money in deal A, the inferior investment.

(Of course this isn't "the" reason why bad investments happen. Rather, this cartoon presents a simple world in which we can see why traditional incentives for employees lead to bad outcomes.)

Imagine Joe is a trader at a fancy investment bank where all the many other traders are lemmings who unanimously invest over the next 6 months in deal A, which has a 50% chance of making a big return, and a 50% chance of not. Joe is a wise trader and has identified deal B, which has a 75% chance of making a big return for his bank. Deals A and B are independent random variables. What should Joe do with the bank's money?

Well, we need to specify Joe's utility function, which is dominated by his bonus. The rule at his bank is that Joe gets a huge bonus if the bank makes a big return, AND Joe's chosen investment made a big return. Otherwise, no bonus.

If Joe invests in deal A (like all the other lemmings alongside whom he works) then he has a 50% chance of getting a huge bonus.

If Joe instead invests his relatively small part of the bank's money in deal B, then he will get a bonus only if _both_ deal A and deal B come out as successes - because the bank will get a big profit only if deal A succeeds. So Joe has a 37.5% chance of getting a big bonus.

So Joe's rational decision is to invest the bank's money in deal A, the inferior investment.

Labels:
decision theory,
investment,
probabilities,
risk

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