# gjm comments on New Philosophical Work on Solomonoff Induction

• Al­most all hy­pothe­ses have high com­plex­ity. There­fore most high-com­plex­ity hy­pothe­ses must have low prob­a­bil­ity.

(To put it differ­ently: let p(n) be the to­tal prob­a­bil­ity of all hy­pothe­ses with com­plex­ity n, where I as­sume we’ve defined com­plex­ity in some way that makes it always a pos­i­tive in­te­ger. Then the sum of the p(n) con­verges, which im­plies that the p(n) tend to 0. So for large n the to­tal prob­a­bil­ity of all hy­pothe­ses of com­plex­ity n must be small, never mind the prob­a­bil­ity of any par­tic­u­lar one.)

Note: all this tells you only about what hap­pens in the limit. It’s all con­sis­tent with there be­ing some par­tic­u­lar high-com­plex­ity hy­pothe­ses with high prob­a­bil­ity.

• But why should the prob­a­bil­ity for lower-com­plex­ity hy­pothe­ses be any lower?

• But why should the prob­a­bil­ity for lower-com­plex­ity hy­pothe­ses be any lower?

It shouldn’t, it should be higher.

If you just meant ”… be any higher?” then the an­swer is that if the prob­a­bil­ities of the higher-com­plex­ity hy­pothe­ses tend to zero, then for any par­tic­u­lar low-com­plex­ity hy­poth­e­sis H all but finitely many of the higher-com­plex­ity hy­pothe­ses have lower prob­a­bil­ity. (That’s just part of what “tend­ing to zero” means.)