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Bayesian Reasoning. Thomas Bayes (1702-1761). Pierre-Simon Laplace (1749-1827). A/Prof Geraint Lewis A/Prof Peter Tuthill. “Probability theory is nothing but common sense, reduced to calculation.”. Laplace. Are you a Bayesian or Frequentist?. 4.
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Bayesian Reasoning Thomas Bayes (1702-1761) Pierre-Simon Laplace (1749-1827) A/Prof Geraint Lewis A/Prof Peter Tuthill “Probability theory is nothing but common sense, reduced to calculation.” Laplace
Are you a Bayesian or Frequentist? 4 “There are 3 kinds of lies: Lies, Damned Lies, and Statistics” ...and Bayesian Statistics Benjamin Disraeli Frequentists Fig 1. A Frequentist Statistician Fig 2. Bayesian Statistics Conference
What is Inference? If A is true then B is true(Major Premise) A = A,B(in Boolean notation) Deductive Inference (Logic) Aristotle 4th Century B.C. AB A is true (Minor Premise) thereforeB is true (conclusion) } T → T STRONG SYLLOGISMS B is False (Minor Premise) thereforeA is False (conclusion) F ← F Inductive Inference (Plausible Reasoning) B is true (Minor Premise) thereforeA is more plausible } t ← T WEAK SYLLOGISMS A is false (Minor Premise) thereforeB is less plausible F → f
What is Inference? Deductive Logic: Effects or outcomes Cause Inductive Logic: Effects or observations Possible Causes
What is a Probability? Frequentists Bayesians P(A|B) = Real number measure of the plausibility of proposition A, given (conditional upon) the truth of proposition B P(A) = long run relative frequency of A occurring in identical repeats of an observation “A” is restricted to propositions about random variables “A” can be any logical proposition All probabilities are conditional; we must be explicit what our assumptions B are (no such thing as an absolute probability!)
Probability depends on our state of Knowledge Monte Hall A B C ?
Probability depends on our state of Knowledge 7 Red 5 Blue ? 1st draw 2nd draw 5/12 Blue 7/12 Red
The Desiderata of Bayesian Probability Theory • Degrees of plausibility are represented by real numbers (higher degree of belief represented by a larger number) • With extra evidence supporting a proposition, the plausibility should increase monotonically up to a limit (certainty). • Consistency. Multiple ways to arrive at a conclusion must all produce the same answer (see book for additional details)
Logic and Probability • In the certainty limit, where probabilities go to zero (falsehood) or one (truth), then the sum and product rules reduce to formal Boolean deductive logic (strong syllogisms). • Bayesian Probability is therefore an extension of formal logic into intermediate states of knowledge. • Bayesian inference gives a measure of our state of knowledge about nature, not a measure of nature itself.
The two rules underlyingprobability theory P(A|B) + P(A|B) = 1 SUM RULE: P(A,B|C) = P(A|C) P(B|A,C) PRODUCT RULE: = P(B|C) P(A|B,C) Blue, Left Blue Eyes Right Handed Left Handed All Kangaroos Brown Eyes
Bayes’ Theorem Posterior P(Hi|I) P(D|Hi I) P(Hi|D,I) = Bayes Theorem: P(D|I) Hi=proposition asserting truth of a hypothesis of interest I =proposition representing prior information D =proposition representing the data P(D|Hi I) =Likelihood: probability of obtaining the data given that the hypothesis is true P(Hi|I) =Prior: probability of hypothesis before new data P(D|I) =Normalization factor (prob all hypothesis i sum to 1)
Example: The Gambler’s coin problem P(H|I) P(D|HI) P(H|D,I) = P(D|I) Normalization factor – Ignore this for now as only need relative merit Prior – what do we know about the coin? Assume H=pdf(head) is uniformly distributed 0-1 Likelihood –if we assume the data D gives R heads in N tosses: P(D|HI) HR (1-H)N-R The full distribution, assuming independence of throws, is the Binomial Distribution. We omit terms not containing H, and use a proportionality.
Data Example: A fair coin? H H T T
