Bayesian reasoning

Ac­cord­ing to its ad­vo­cates, Bayesian rea­son­ing is a way of see­ing the world, and our be­liefs about the world, in the light of prob­a­bil­ity the­ory, in par­tic­u­lar Bayes’s The­o­rem or Bayes’s Rule. This prob­a­bil­ity-the­o­retic way of see­ing the world can ap­ply to sci­en­tific is­sues, to tasks in ma­chine learn­ing, and to ev­ery­day life.

To start learn­ing, visit Ar­bital’s Guide to Bayes’s Rule.

After that, con­sider vis­it­ing the Bayesian up­date page.


  • Bayesian update

    Bayesian up­dat­ing: the ideal way to change prob­a­bil­is­tic be­liefs based on ev­i­dence.

  • Bayes' rule

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

  • Prior probability

    What we be­lieved be­fore see­ing the ev­i­dence.

  • Interest in mathematical foundations in Bayesianism

    “Want” this req­ui­site if you pre­fer to see ex­tra in­for­ma­tion about the math­e­mat­i­cal foun­da­tions in Bayesi­anism.

  • Posterior probability

    What we be­lieve, af­ter see­ing the ev­i­dence and do­ing a Bayesian up­date.

  • Humans doing Bayes

    The hu­man use of Bayesian rea­son­ing in ev­ery­day life

  • Ignorance prior

    Key equa­tions for quan­ti­ta­tive Bayesian prob­lems, de­scribing ex­actly the right shape for what we be­lieved be­fore ob­ser­va­tion.

  • Strictly confused

    A hy­poth­e­sis is strictly con­fused by the raw data, if the hy­poth­e­sis did much worse in pre­dict­ing it than the hy­poth­e­sis it­self ex­pected.

  • Finishing your Bayesian path on Arbital

    The page that comes at the end of read­ing the Ar­bital Guide to Bayes’ rule

  • Prior

    A state of prior knowl­edge, be­fore see­ing in­for­ma­tion on a new prob­lem. Po­ten­tially com­pli­cated.

  • Empirical probabilities are not exactly 0 or 1

    “Cromwell’s Rule” says that prob­a­bil­ities of ex­actly 0 or 1 should never be ap­plied to em­piri­cal propo­si­tions—there’s always some prob­a­bil­ity, how­ever tiny, of be­ing mis­taken.

  • Subjective probability

    Prob­a­bil­ity is in the mind, not in the en­vi­ron­ment. If you don’t know whether a coin came up heads or tails, that’s a fact about you, not a fact about the coin.

  • Likelihood


  • Probability theory

    The logic of sci­ence; co­her­ence re­la­tions on quan­ti­ta­tive de­grees of be­lief.