Bayesianism : a lightning introduction Ronojoy Adhikari The Institute of Mathematical Sciences

Message from a non-Bayesian

Message from a non-Bayesian

Message from a non-Bayesian “Now this is the peculiarity of scientific method, that when once it has become a habit of mind, that mind converts all facts whatsoever into science. The field of science is unlimited; its solid contents are endless, every group of natural phenomena, every phase of social life, every stage of past or present development is material for science. The unity of all science consists alone in its method, not in its material. The man who classifies facts of any kind whatever, who sees their mutual relation and describes their sequence, is applying the scientific method and is a man of science. The facts may belong to the past history of mankind, to the social statistics of our great cities, to the atmosphere of the most distant stars, to the digestive organs of a worm, or to the life of a scarcely visible bacillus. It is not the facts themselves which form science, but the method in which they are dealt with.”

Message from a non-Bayesian “Now this is the peculiarity of scientific method, that when once it has become a habit of mind, that mind converts all facts whatsoever into science. The field of science is unlimited; its solid contents are endless, every group of natural phenomena, every phase of social life, every stage of past or present development is material for science. The unity of all science consists alone in its method, not in its material. The man who classifies facts of any kind whatever, who sees their mutual relation and describes their sequence, is applying the scientific method and is a man of science. The facts may belong to the past history of mankind, to the social statistics of our great cities, to the atmosphere of the most distant stars, to the digestive organs of a worm, or to the life of a scarcely visible bacillus. It is not the facts themselves which form science, but the method in which they are dealt with.”

Message from a non-Bayesian “Now this is the peculiarity of scientific method, that when once it has become a habit of mind, that mind converts all facts whatsoever into science. The field of science is unlimited; its solid contents are endless, every group of natural phenomena, every phase of social life, every stage of past or present development is material for science. The unity of all science consists alone in its method, not in its material. The man who classifies facts of any kind whatever, who sees their mutual relation and describes their sequence, is applying the scientific method and is a man of science. The facts may belong to the past history of mankind, to the social statistics of our great cities, to the atmosphere of the most distant stars, to the digestive organs of a worm, or to the life of a scarcely visible bacillus. It is not the facts themselves which form science, but the method in which they are dealt with.”

Scientific inference and Bayesian probability Cause Possible Causes Effects or Outcomes Effects or Observations Deductive logic Inductive logic P(H|D) = P(D|H)P(H)/P(D) posterior = likelihood x prior / evidence Mathematical derivation. after D. Sivia in Data Analysis : A Bayesian Tutorial

Bayesian analysis through two examples Coin tossing Brownian motion

Bernoulli trials

Bernoulli trials A coin is thrown 6 times with the outcome D = {hhhhtt}

Bernoulli trials What is the probability of the proposition H = The coin is fair A coin is thrown 6 times with the outcome D = {hhhhtt}

Bernoulli trials What is the probability of the proposition H = The coin is fair A coin is thrown 6 times with the outcome D = {hhhhtt} H = The coin is fair ¯ H = The coin is not fair Possible “causes” Effects or Observations D = {hhhhtt}

Bernoulli trials P(D|H) = Probability of D given H = Probability of outcome given a fair coin P(h|H) = P(t|H) = 1 2 P(D|H) = P(hhhhtt|H) = 1 2 · 1 2 · 1 2 · 1 2 · 1 2 · 1 2 P(h| ¯ H) = , P(t| ¯ H) = 1 P(D| ¯ H) = P(hhhhtt| ¯ H) = · · · · (1 ) · (1 )

Bernoulli trials P(D| ) = P(hhhhtt| ) = · · · · (1 ) · (1 ) P( |D) = P( |hhhhtt) = Probability of the hypothesis ✓ given the data H ! = 1 2 ¯ H ! 6= 1 2 P( |D) P(D| ) know this want this

Bernoulli trials P( |D) = P(D| )P( ) P(D)

Bernoulli trials 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 P( |D) = P(D| )P( ) P(D) Posterior distribution with uniform prior