Occam’s Razor and Unfalsifiable Claims
“Let’s flip a coin to decide who is right. Heads I win, tails you lose.”
That is the classic example of an unfalsifiable claim, claims which are consistent with all possible evidence, meaning they make no predictions. Many, including myself, have criticized such claims before as any theory which makes no predictions is incapable of being correct because if all data can be said to count as evidence for it, all data can be said to count against it. However I recently came across a wonderful, but complex, explanation of this principle and Occam’s razor in Bayesian terms by Tom Campbell-Ricketts and here I’d like to expound on part of his work here in simpler terms.
Occam’s razor is a kind of common dictum which tells us a simple theory is preferable to a complicated one. While there have been many attempts to explain why we should accept this statement because it has proven so useful, there’s a simple explanation to do so according to Bayes’ theorem. Whenever you add additional parameters to a theory the prior probability of that theory goes down. This is because the prior probability of any given parameter within a theory is always less than 1. Therefore multiplying parameters together decreases the prior probability of the theory as a whole.
For practical purposes that means before even examining the data to be explained, a theory with few flexible parameters is preferable to one with many flexible parameters, the essence of Occam’s razor. Taking this principle to its logical extension means the elimination of theories with infinite parameters, as all unfalsifiable claims are because theories which are consistent with all possible data sets have infinite parameters and the prior probability of any given parameter is less than 1. Therefore theories which are consistent with all possible data sets are forced to combine (multiply) the prior probability of all parameters, therefore the prior probability of these claims is (effectively) 0. If the prior probability is 0, then the posterior probability is necessarily 0.
QED.
This is a powerful explanation of what’s wrong with unfalsifiable claims however there was something else in Campbell-Ricketts’ post which I hadn’t considered before which has equally strong implications for some faulty beliefs. More to come soon.
