Sample spaces are too large

When we are rea­son­ing about an un­cer­tain world us­ing prob­a­bil­ity the­ory, we have a sam­ple space of pos­si­ble ways the world could be, and we as­sign prob­a­bil­ities to out­comes in the sam­ple space with a prob­a­bil­ity dis­tri­bu­tion. Un­for­tu­nately, of­ten the sam­ple space (which is sup­posed to con­tain ev­ery pos­si­ble way the world might turn out) is very large and pos­si­bly in­finite.

For ex­am­ple, if we are think­ing about a box with some par­ti­cles in it, our sam­ple space is the enor­mous set of all pos­si­ble ar­range­ments of the par­ti­cles. To rea­son use­fully about the box, we’d have to do com­pu­ta­tions us­ing a gi­gan­tic table spec­i­fy­ing a sep­a­rate num­ber for ev­ery sin­gle dis­tinct ar­range­ment of par­ti­cles. Two ar­range­ments that are pretty much the same, as far as we care, take up just as much of our com­pu­ta­tional re­sources as two im­por­tantly differ­ent ar­range­ments. So, even though we could in prin­ci­ple rea­son con­sis­tently un­der un­cer­tainty just us­ing a prob­a­bil­ity dis­tri­bu­tion over a sam­ple space, we can­not do so eas­ily, be­cause we are try­ing to keep track of too much.

examples

Parents:

  • Sample space

    The set of pos­si­ble things that could hap­pen in a part of the world that you are un­cer­tain about.