lew@ihuxr.UUCP (12/01/83)
Bayes' Theorem gives a very simple answer when applied to survival functions. If S(t) is the probability of still being alive at time t, the probability, S'(t'), of being alive at time t0 + t', given that we were alive at time t0, is just S(t0 + t')/S(t0) . If S(t) is an exponential decay, representing a constant probability per unit time of being killed, S'(t') = S(t'). This means that if we feel certain for some reason that there is a certain probability per unit time of being killed (call it p), this certainty is rationally maintainable in the face of any run of good luck, however long. This is equivalent to maintaining a 50/50 expectation of heads/tails after a run of heads. If we are uncertain as to the exact value of p, we can give our estimation of the likelihood of various values in the form of a distribution function, f(p). This means that we think the probability that p has value between p' and p' + dp' is given by f(p') * dp'. The survival function is then given by: S(t) = int 0 to inf of f(p) * exp( -p*t) * dp This is just the weighted average of the survival functions for all p. Note that the Bayesian result ( S'(t') = S(t'+t0)/S(t0) ) just redefines our distribution function: S'(t') = int 0 to inf of f'(p) * exp( -p*t) * dp f'(p) = f(p) * exp( -p*t0 ) / S(t0) This is just another form of Bayes' Theorem. Note that the model based on a distribution over p is equivalent to simply specifying the appropriate survival function. A nice choice of f(p) is: f(p) = (1/p0) * exp( - p/p0 ) This gives S(t) = 1/(1+p0*t). The reader can verify that S'(t') = S(t'+t0)/S(t0) = 1/(1+p1*t') ; p1 = p0/(1+p0*t0) ... so that our survival function retains the same form but with a time constant which is optimistically revised the longer we survive. Lew Mammel, Jr. ihnp4!ihuxr!lew P.S. This choice of f(p) gives something very close to Herb Norton's answer. If we start with p0 = 1/yr, after 35 years we have p1 = 1/36 yrs. I would emphasize, though that this isn't the only answer. The answer always depends on our a priori expectations.