
September 20th (Rochester, NY) — House on the Hill
While studying the Theory of Signal Detection, particularly, determination of optimum criterion, we come to this equation:
Bopt = (p(N)/p(Y))[(value{n|N} - cost{y|N})/(value{y|Y} - cost{n|Y})]
where p(N) and p(Y) are the probabilities that a signal is not present (N) and that a signal is present (Y), the rest of the formula involves the value and cost added and subtracted given a decision. For instance, value{n|N} gives the value of saying ‘no’ when there is no signal present, while cost{n|Y} gives the cost of saying ‘no’ when there is a signal present. Cost values are negative, so you only get zero when cost and value are zero. Got it? Good.
Pascal’s gambit (Wikipedia entry) posits that ‘God is present’ (Y) or ‘God is not present’ (N). He goes on to say that proving the existence of God is near impossible therefore we should take the probability that God is present to be ridiculously low, i.e. p(Y) approaches zero, which, ignoring the rest of the formula for now, makes the optimum criterion, Bopt, approach infinity, which, in turn, suggests you should not say ‘yes’ when asked if God is present.
Now when you consider the rest of the formula, Bopt tips back towards zero. Pascal suggests that the value gained by saying God is present, value{y|Y}, is infinite, i.e. the heavenly life, while the cost of saying God is not present when God is present, cost{n|Y}, is an infinite loss - condemned to hell. Since both of these are in the denominator, one over infinity is zero, so the Bopt goes to zero. Pascal goes on to suggest that no value or cost is associated with saying God is present when God is not present, i.e. value{n|N} and cost{y|N} are both zero, he concludes that the safest best is to say yes God is present.
Aside from the problems cited on that Wiki entry, my own problem is this, how does one interpret an indeterminate form like zero divided by infinity? I’ve thought about recasting the problem into an indeterminate form that will allow repeated application of L’Hopital’s rule, but then what is changing, what do I take the derivative with respect to? My conclusion is this: the problem is poorly posed, because the value and cost functions are actually Heaviside functions and their parity is not determined by the question of whether God is present but whether there is an afterlife. Interestingly enough, the derivative of a Heaviside is a delta function at the time of death! So, for optimal gain, live like a heathen at convert or recant at your deathbed!