Brownian thought space

Cognitive science, mostly, but more a sometimes structured random walk about things.

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Chronically curious モ..

Sunday, April 06, 2008

Statistical inference

Imagine that a statistical reasoner was trying to build an appropriate model of when I go to the gym. What would such a model entail? In a very general sense, it would be some kind of prediction about the (say hourly) probability that I go to the gym. So, given some data, for each of the 24 hours in a day the model would make some prediction about how likely I am to go to the gym. Observing the data (so far), it might make certain generalizations. (1) The probability of going to the gym is zero at 00(hrs), is some value >0 between 0800 and 1900, and then drops down to zero after 1900. Further, if the model had access to my (woefully inadequate) Google calendar, it would know that the probability that I went to the gym during the hours that my Google cal said I was supposed to be at a meeting was close to zero. Add more observable, contextual variables. The probability that I go to the gym rises in the hours following my eating a protein bar. It is also higher when my car is parked at the campus. The availability of a fresh change of clothes increases the probability , while my drinking a coke drives it lower. Eventually, I'm quite certain that these factors taken together would come up with a model of my gym-going behavior with a pretty good level of accuracy. You could consult this model and know, at any given hour, how likely it is that I am actually at the gym, and you won't be terribly wrong. However! if you thought that the model was a causal model, you would have got the whole causal structure inverted: as far as I'm concerned, I go to the gym when it darn well pleases me. Of course I cannot go when I'm fast asleep, and I do tend to sleep mostly between 1900 and 0800 the next day. Of course I'd rather sit in meetings than go gymming. But these are not the reasons I go to the gym; these are the restrictions that permit or disallow my going to the gym. This is just like the Buddhist idea of substantial causes and cooperative conditions. The first is the actual cause; the second are factors that make the effect possible. In my reading, the failure of Behaviorism is in identifying the cooperative conditions as substantial causes. I think that this also the pitfall that's right next to every statistical learning view of acquisition in cognition. As Chomsky, Pinker and others have belabored, the point is that you need to posit mental causes for observed behaviors. Observed behaviors in themselves don't constitute a causal theory. However, what the causal theory itself should look like is a tough problem. The reason is that we can postulate as many or as few mental variables, but we won't necessarily know which is the right set. It seems that some sort of Occam's razor over introspectively (intuitively) hypothesized causal models would be the best bet. It's in this light that I'm viewing the research of Josh Tenenbaum, for example. And here's where the Bayesian approach might be most useful- I think of it as the best implementation (so far) of Occam's razor.

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