Subjective probability

What does it mean to say that a flipped coin has a 50% prob­a­bil­ity of land­ing heads?

There are mul­ti­ple ways to an­swer this ques­tion, de­pend­ing on what you mean by “prob­a­bil­ity”. This page dis­cusses “sub­jec­tive prob­a­bil­ities,” which are a tool for quan­tify­ing your per­sonal un­cer­tainty about the world.

Imag­ine flip­ping a coin and slap­ping it against your wrist. It’s already landed ei­ther heads or tails. The fact that you don’t know whether it’s heads or tails is a fact about you, not a fact about the coin. Ig­no­rance is in the mind, not in the world.

So your mind is rep­re­sent­ing the coin, and you’re un­sure about which way the coin came up. Those prob­a­bil­ities, rep­re­sented in your brain, are your sub­jec­tive prob­a­bil­ities. Prob­a­bil­ity the­ory is con­cerned with the for­mal­iza­tion, study, and ma­nipu­la­tion of sub­jec­tive prob­a­bil­ities.

If prob­a­bil­ities are sim­ply sub­jec­tive men­tal states, what does it mean to say that prob­a­bil­ities are “good,” “cor­rect,” “ac­cu­rate,” or “true”? The sub­jec­tivist an­swer, roughly, is that a prob­a­bil­ity dis­tri­bu­tion be­comes more ac­cu­rate as it puts more of its prob­a­bil­ity mass on the true pos­si­bil­ity within the set of all pos­si­bil­ities it con­sid­ers. For more on this see Cor­re­spon­dence vi­su­al­iza­tions for differ­ent in­ter­pre­ta­tions of “prob­a­bil­ity”.

Sub­jec­tive prob­a­bil­ities, given even a small grain of truth, will be­come more ac­cu­rate as they in­ter­act with re­al­ity and ex­e­cute Bayesian up­dates. Your sub­jec­tive be­lief about “Is it cloudy to­day?” is ma­te­ri­ally rep­re­sented in your brain, and be­comes more ac­cu­rate as you look up at the sky and causally in­ter­act with it: light from the clouds in the sky comes down, en­ters your retina, is trans­duced to nerve im­pulses, pro­cessed by your vi­sual cor­tex, and then your sub­jec­tive be­lief about whether it’s cloudy be­comes more ac­cu­rate.

‘Sub­jec­tive prob­a­bil­ity’ des­ig­nates our view of prob­a­bil­ity as an epistemic state, some­thing in­her­ently in the mind, since re­al­ity it­self is not un­cer­tain. It doesn’t mean ‘ar­bi­trary prob­a­bil­ity’ or ‘prob­a­bil­ity that some­body just made up with no con­nec­tion to re­al­ity’. Your be­lief that it’s cloudy out­side (or sunny) is a be­lief, but not an ar­bi­trary or made-up be­lief. The same can be true about your state­ment that you think it’s 90% likely to be sunny out­side, be­cause it was sunny this morn­ing and it’s sum­mer, even though you’re cur­rently in an in­te­rior room and you haven’t checked the weather. The out­doors it­self is not wa­ver­ing be­tween sunny and cloudy; but your guess that it’s 9 times more likely to be sunny than cloudy is not un­grounded.

Sev­eral co­her­ence the­o­rems sug­gest that clas­si­cal prob­a­bil­ities are a uniquely good way of quan­tify­ing the rel­a­tive cred­i­bil­ity we at­tach to our guesses; e.g. that even if it’s prob­a­bly sunny, it’s still more likely for it to be cloudy out­side than for the Moon to be made of green cheese. This in turn says that while the prob­a­bil­ities them­selves may ex­ist in our minds, the laws that gov­ern the ma­nipu­la­tion and up­dat­ing of these prob­a­bil­ities are as solid as any other math­e­mat­i­cal fact.

For an ex­am­ple of a solid law gov­ern­ing sub­jec­tive prob­a­bil­ity, see Ar­bital’s guide to Bayes’ rule.

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Parents:

  • Bayesian reasoning

    A prob­a­bil­ity-the­ory-based view of the world; a co­her­ent way of chang­ing prob­a­bil­is­tic be­liefs based on ev­i­dence.

  • Interpretations of "probability"

    What does it mean to say that a fair coin has a 50% prob­a­bil­ity of com­ing up heads?