Bayesian inference Marcel Lthi Graphics and Vision Research Group - PowerPoint PPT Presentation

> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Bayesian inference Marcel Lüthi Graphics and Vision Research Group Department of Mathematics and Computer Science University of Basel

University of Basel > DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Probabilities: What are they? Four possible interpretations: 1. Long-term frequencies • Relative frequency of an event over time 2. Physical tendencies (propensities) • Arguments about a physical situation (causes of relative frequencies) 3. Degree of belief (Bayesian probabilities) • Subjective beliefs about events/hypothesis/facts 4. Logic • Degree of logical support for a particular hypothesis

University of Basel > DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Degree of belief: An Example • Dentist example: Does the patient have a cavity? 𝑄 cavity = 0.1 𝑄 cavity toothache) = 0.8 𝑄 cavity toothache, gum problems) = 0.4 Bu But t the the patien tient t eith either r has as a a cavi vity or or does oes not ot • There is no 80% cavity! • Having a cavity should not depend on whether the patient has a toothache or gum problems These statements do not contradict each other, they summarize the dentist’s knowledge about the patient 3 AIMA: Russell & Norvig, Artificial Intelligence. A Modern Approach, 3 rd edition,

University of Basel > DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Uncertainty: Bayesian Probability • Bayesian probabilities rely on a subjective perspective: • Probabilities express our current knowledge . • Can change when we learn or see more • More data -> more certain about our result. Subjectivity : There is no single, real underlying distribution. A probability distribution expresses our knowledge – It is different in different situations and for different observers since they have different knowledge. • Subjective != Arbitrary • Given belief, conclusions follow by laws of probability calculus 4

University of Basel > DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Belief Updates Model Ob Observ rvatio ion Pos osterior Face distribution Concrete points Face distribution Possibly uncertain consistent with observation Prior belief More knowledge Posterior belief Consistency: Laws of probability calculus!

University of Basel > DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Two important rules Probabilistic model: joint distribution of points 𝑄 𝑦 1 , 𝑦 2 Marginal Conditional Distribution of certain points only Distribution of points conditioned on known values of others 𝑄 𝑦 1 |𝑦 2 = 𝑄 𝑦 1 , 𝑦 2 𝑄 𝑦 1 = ෍ 𝑄(𝑦 1 , 𝑦 2 ) 𝑄 𝑦 2 𝑦 2 Product rule: 𝑄 𝑦 1 , 𝑦 2 = 𝑞 𝑦 1 𝑦 2 𝑞(𝑦 2 )

University of Basel > DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Certain Observation • Observations are known values X • Distribution of 𝑌 after observing y 1 𝑧 1 , … , 𝑧 𝑂 : 𝑄 𝑌|𝑧 1 … 𝑧 𝑂 y i • Conditional probability 𝑄 𝑌|𝑧 1 … 𝑧 𝑂 = 𝑄 𝑌, 𝑧, … , 𝑧 𝑂 y N 𝑄 𝑧 1 , … , 𝑧 𝑂

University of Basel > DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Towards Bayesian Inference • Update belief about 𝑌 by observing 𝑧 1 , … , 𝑧 𝑂 𝑄 𝑌 → 𝑄 𝑌 𝑧 1 , … , 𝑧 𝑂 • Factorize joint distribution 𝑄 𝑌, 𝑧 1 , … , 𝑧 𝑂 = 𝑄 𝑧 1 , … , 𝑧 𝑂 |𝑌 𝑄 𝑌 • Rewrite conditional distribution 𝑄 𝑌|𝑧 1 , … , 𝑧 𝑂 = 𝑄 𝑌, 𝑧 1 , … , 𝑧 𝑂 = 𝑄 𝑧 1 , … , 𝑧 𝑂 |𝑌 𝑄 𝑌 𝑄 𝑧 1 , … , 𝑧 𝑂 𝑄 𝑧 1 , … , 𝑧 𝑂 More generally: distribution of model points 𝑌 given data 𝑍 : 𝑄 𝑌|𝑍 = 𝑄 𝑌, 𝑍 = 𝑄 𝑍|𝑌 𝑄 𝑌 𝑄 𝑍 𝑄 𝑍

University of Basel > DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Uncertain Observation • Observations with uncertainty X Model needs to describe how observations are distributed y 1 + 𝜁 with joint distribution 𝑄 𝑌, 𝑍 • Still conditional probability y i + 𝜁 But joint distribution is more complex • Joint distribution factorized y N + 𝜁 𝑄 𝑌, 𝑍 = 𝑄 𝑍|𝑌 𝑄 𝑌 • Likelihood 𝑄 𝑍|𝑌 • Prior 𝑄 𝑌

University of Basel > DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Likelihood Join Joint Lik Likelih ihood Prio rior 𝑄 𝑌, 𝑍 = 𝑄 𝑍|𝑌 𝑄 𝑌 • Likelihood x prior: factorization is more flexible than full joint • Prior: distribution of core model without observation • Likelihood: describes how observations are distributed

University of Basel > DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Bayesian Inference • Conditional/Bayes rule: method to update beliefs Likelih Lik ihood Prio rior 𝑄 𝑌|𝑍 = 𝑄 𝑍|𝑌 𝑄 𝑌 Pos osterior 𝑄 𝑍 Mar argin inal l Lik Likelih ihood • Each observation updates our belief (changes knowledge!) 𝑄 𝑌 → 𝑄 𝑌 𝑍 → 𝑄 𝑌 𝑍, 𝑎 → 𝑄 𝑌 𝑍, 𝑎, 𝑋 → ⋯ • Bayesian Inference: How beliefs evolve with observation • Recursive: Posterior becomes prior of next inference step

University of Basel > DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Marginalization • Models contain irrelevant/hidden variables e.g. points on chin when nose is queried • Marginalize over hidden variables ( Z ) 𝑄 𝑍, 𝑎|𝑌 𝑄 𝑌 𝑄 𝑌 𝑍 = ෍ 𝑄 𝑌, 𝑎 𝑍 = ෍ 𝑄 𝑍, 𝑎 𝐼 𝐼

University of Basel > DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE General Bayesian Inference • Observation of additional variables • Common case, e.g. image intensities, surrogate measures (size, sex, …) • Coupled to core model via likelihood factorization • General Bayesian inference case: • Distribution of data 𝑍 • Parameters 𝜄 𝑄 𝜄|𝑍 = 𝑄 𝑍|𝜄 𝑄 𝜄 𝑄 𝑍|𝜄 𝑄 𝜄 = 𝑄 𝑍 ∫ 𝑄 𝑍|𝜄 𝑄 𝜄 𝑒𝜄 𝑄 𝜄|𝑍 ∝ 𝑄 𝑍|𝜄 𝑄 𝜄

University of Basel > DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Summary: Bayesian Inference • Belief : formal expression of an observer’s knowledge • Subjective state of knowledge about the world • Beliefs are expressed as probability distributions • Formally not arbitrary: Consistency requires laws of probability • Observations change knowledge and thus beliefs • Bayesian inference formally updates prior beliefs to posteriors • Conditional Probability • Integration of observation via likelihood x prior factorization 𝑄 𝜄|𝑍 = 𝑄 𝑍|𝜄 𝑄 𝜄 𝑄 𝑍 18