Sunday, March 23, 2008

Bayes, Laplace, Bayesian and neo-Bayesians

More evidence that Laplace was the father (or perhaps the dedicated single mother) of Bayesian inference can be found here.

Contains this little jem: ...the term Bayesian "was first used in print by R.A. Fisher in the 1950 introduction to his 1930 paper on fiducial inference entitled Inverse Probability [where in he seeks to] 'distinguish [his result] from the Bayesian probability a posteriori.'"

I also enjoyed the many references to Stigler's Law which states that: "no scientific discovery is named after its original discoverer." The author also notes that " is worth noting that Stigler proposed [this Law] in the spirit of a self-proving theorem," but then neglects to mention from whom it was stolen...

Also contains an interesting aside on objective vs subjective probability. I particularly, like the comment indicating that many philosophers disliked the notion of subjective probability, but ultimately the sheer utility of the concept won the day. Anyway, the definition of objective probability likens it to a frequentist situation where pr(H) = 1/2 every time i toss a fair coin. This theory can be tested by throwing many coin tosses and showing that the average number of heads converges to 1/2. So its objective in the sense that the number 1/2 refers to a quantity which can actually be observed.

On the other hand the typical exemplar of a subjective probability is a statement of belief about what the weather will be like tomorrow. The distinction being that since tomorrow only comes once there is no way to verify that, on this particular day, there is a 50% chance of rain that is similar to the method used for the coin toss.

However, it seems to me that either this example or this distinction is rather quite poorly thought out. The repeatability and inter-toss independence of a coin is itself an assumption which should be subject to verification. This is no different than questioning the reasonability of comparing the clouds (or temp/pressure/etc) of today with those of yesterday. True we've got alot more evidence about coins, but lets face it, even after a billion coin tosses, we can neither conclude for certain that we have a fair one, or even that we have been tossing the "same" coin all this time.

This lead me to think that this issue pertains to the debate, brought to my attention by JJ, concerning whether or not a rational doxastic state (fancy word for belief) could have probability one. To my mind the resolution of that issue was that only conditional probabilities could take on probability 1. Such conditionals are syllogisms. In the context of this discussion, I would suggest that conditional probabilities which represent statements of model assumptions are objective. While probabilistic statements concerning empirical quantities are subjective. This is because additional evidence or even the consideration of additional models can augment degrees of beliefs regarding empirical quantities, while it is true by assumption that if we have a fair coin it will come up 50% heads if we toss it an infinite number of times. This is true regardless of how many times we actually toss it.

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