Pascal's Muggle: Infinitesimal Priors and Strong Evidence

43 08 May 2013 12:43AM

Short form:  Pascal's Muggle

tl;dr:  If you assign superexponentially infinitesimal probability to claims of large impacts, then apparently you should ignore the possibility of a large impact even after seeing huge amounts of evidence.  If a poorly-dressed street person offers to save 10(10^100) lives (googolplex lives) for \$5 using their Matrix Lord powers, and you claim to assign this scenario less than 10-(10^100) probability, then apparently you should continue to believe absolutely that their offer is bogus even after they snap their fingers and cause a giant silhouette of themselves to appear in the sky.  For the same reason, any evidence you encounter showing that the human species could create a sufficiently large number of descendants - no matter how normal the corresponding laws of physics appear to be, or how well-designed the experiments which told you about them - must be rejected out of hand.  There is a possible reply to this objection using Robin Hanson's anthropic adjustment against the probability of large impacts, and in this case you will treat a Pascal's Mugger as having decision-theoretic importance exactly proportional to the Bayesian strength of evidence they present you, without quantitative dependence on the number of lives they claim to save.  This however corresponds to an odd mental state which some, such as myself, would find unsatisfactory.  In the end, however, I cannot see any better candidate for a prior than having a leverage penalty plus a complexity penalty on the prior probability of scenarios.

In late 2007 I coined the term "Pascal's Mugging" to describe a problem which seemed to me to arise when combining conventional decision theory and conventional epistemology in the obvious way.  On conventional epistemology, the prior probability of hypotheses diminishes exponentially with their complexity; if it would take 20 bits to specify a hypothesis, then its prior probability receives a 2-20 penalty factor and it will require evidence with a likelihood ratio of 1,048,576:1 - evidence which we are 1048576 times more likely to see if the theory is true, than if it is false - to make us assign it around 50-50 credibility.  (This isn't as hard as it sounds.  Flip a coin 20 times and note down the exact sequence of heads and tails.  You now believe in a state of affairs you would have assigned a million-to-one probability beforehand - namely, that the coin would produce the exact sequence HTHHHHTHTTH... or whatever - after experiencing sensory data which are more than a million times more probable if that fact is true than if it is false.)  The problem is that although this kind of prior probability penalty may seem very strict at first, it's easy to construct physical scenarios that grow in size vastly faster than they grow in complexity.

I originally illustrated this using Pascal's Mugger:  A poorly dressed street person says "I'm actually a Matrix Lord running this world as a computer simulation, along with many others - the universe above this one has laws of physics which allow me easy access to vast amounts of computing power.  Just for fun, I'll make you an offer - you give me five dollars, and I'll use my Matrix Lord powers to save 3↑↑↑↑3 people inside my simulations from dying and let them live long and happy lives" where ↑ is Knuth's up-arrow notation.  This was originally posted in 2007, when I was a bit more naive about what kind of mathematical notation you can throw into a random blog post without creating a stumbling block.  (E.g.:  On several occasions now, I've seen someone on the Internet approximate the number of dust specks from this scenario as being a "billion", since any incomprehensibly large number equals a billion.)  Let's try an easier (and way smaller) number instead, and suppose that Pascal's Mugger offers to save a googolplex lives, where a googol is 10100 (a 1 followed by a hundred zeroes) and a googolplex is 10 to the googol power, so 1010100 or 1010,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 lives saved if you pay Pascal's Mugger five dollars, if the offer is honest.