Waterfall diagram

Water­fall di­a­grams, like fre­quency di­a­grams, provide a way of vi­su­al­iz­ing Bayes’ Rule. Re­call that Bayes’ rule (in the odds form) says that the pos­te­rior odds be­tween any two hy­pothe­ses is equal to the prior odds times the rel­a­tive like­li­hoods.

For ex­am­ple, in the Dise­a­sitis prob­lem, a pa­tient is 20% likely to be sick (and 80% likely to be healthy) a pri­ori, and they take a test that is 3x more likely to come back pos­i­tive if they are sick. The odds of a pa­tient be­ing sick given a pos­i­tive test are thus \((1 : 4) \times (3 : 1) = (3 : 4).\)

Us­ing wa­ter­fall di­a­grams, we can vi­su­al­ize the prior odds as two sep­a­rate streams of wa­ter at the top of a wa­ter­fall, and the rel­a­tive like­li­hoods as the pro­por­tion of wa­ter from each stream that makes it to the shared pool at the bot­tom. The pos­te­rior odds can then be vi­su­al­ized as the pro­por­tion of wa­ter in the shared pool that came from each differ­ent prior stream.

Waterfall diagram

See Water­fall di­a­grams and rel­a­tive odds for a walk­through of the di­a­gram.

Water­fall di­a­grams make it clear that, when calcu­lat­ing the pos­te­rior odds, what mat­ters is the rel­a­tive pro­por­tion be­tween how much each gal­lon of red wa­ter vs each gal­lon of blue wa­ter makes it into the shared pool. If 45% of red wa­ter and 15% of blue wa­ter made it to the bot­tom, that would give the same rel­a­tive pro­por­tion of red and blue wa­ter in the shared pool at the bot­tom as 90% and 30%.

Same relative proportions

Thus, it is only the rel­a­tive like­li­hoods (and not the ab­solute like­li­hoods) that mat­ter when calcu­lat­ing pos­te­rior odds.

Similarly, chang­ing the wa­ter flows at the top of the wa­ter­fall from (20 gal­lons/​sec red wa­ter : 80 gal­lons/​sec blue wa­ter) to (40 gal­lons/​sec red wa­ter : 160 gal­lons/​sec blue wa­ter) would dou­ble the to­tal wa­ter at bot­tom, but not change the rel­a­tive pro­por­tions of blue and red wa­ter. So only the rel­a­tive prior odds mat­ter to the rel­a­tive pos­te­rior odds.

Children:

Parents:

  • Bayes' rule

    Bayes’ rule is the core the­o­rem of prob­a­bil­ity the­ory say­ing how to re­vise our be­liefs when we make a new ob­ser­va­tion.