Announcements CS Ice Cream Social 9/5 3:30-4:30, ECCR 265 includes poster session, student group presentations

Concept Learning Examples  Word meanings  Edible foods  Abstract structures (e.g., irony) glorch not glorch not glorch

Supervised Approach To Concept Learning Both positive and negative examples provided Typical models (both in ML and Cog Sci) circa 2000 required both positive and negative examples

Contrast With Human Learning Abiliites Learning from positive examples only Learning from a small number of examples  E.g., word meanings  E.g., learning appropriate social behavior  E.g., instruction on some skill What would it mean to learn from a small number of positive examples? + + +

Tenenbaum (1999) Two dimensional continuous feature space Concepts defined by axis-parallel rectangles e.g., feature dimensions  cholesterol level  insulin level e.g., concept healthy

Learning Problem Given a set of given a set of n examples, X = {x 1, x 2, x 3, …, x n }, which are instances of the concept… Will some unknown example Y also be an instance of the concept? Problem of generalization

Hypothesis (Model) Space H: all rectangles on the plane, parameterized by (l 1, l 2, s 1, s 2 ) h: one particular hypothesis Note: |H| = ∞ Consider all hypotheses in parallel In contrast to non-Bayesian approach of maintaining only the best hypothesis at any point in time.

Prediction Via Model Averaging Will some unknown input y be in the concept given examples X = {x 1, x 2, x 3, …, x n }? Q: y is a positive example of the concept (T,F)  P(Q | X) = ⌠ h p(Q & h | X) dh  P(Q & h | X) = p(Q | h, X) p(h | X)  P(Q | h, X) = P(Q | h) = 1 if y is in h  p(h | X) ~ P(X | h) p(h) priorlikelihood Chain rule Marginalization Conditional independence and deterministic concepts Bayes rule

Priors and Likelihood Functions Priors, p(h)  Location invariant  Uninformative prior (prior depends only on area of rectangle)  Expected size prior Likelihood function, p(X|h)  X = set of n examples  Size principle x

Expected size prior

Generalization Gradients MIN: smallest hypothesis consistent with data weak Bayes: instead of using size principle, assumes examples are produced by process independent of the true class Dark line = 50% prob.

Experimental Design Subjects shown n dots on screen that are “randomly chosen examples from some rectangle of healthy levels”  n drawn from {2, 3, 4, 6, 10, 50} Dots varied in horizontal and vertical range  r drawn from {.25,.5, 1, 2, 4, 8} units in a 24 unit window Task  draw the ‘true’ rectangle around the dots

Experimental Results

Number Game Experimenter picks integer arithmetic concept C  E.g., prime number  E.g., number between 10 and 20  E.g., multiple of 5 Experimenter presents positive examples drawn at random from C, say, in range [1, 100] Participant asked whether some new test case belongs in C

Empirical Predictive Distributions

Hypothesis Space  Even numbers  Odd numbers  Squares  Multiples of n  Ends in n  Powers of n  All numbers  Intervals [n, m] for n>0, m<101  Powers of 2, plus 37  Powers of 2, except for 32

Observation = 16 Likelihood function  Size principle Prior  Intuition

Observation = Likelihood function  Size principle Prior  Intuition

Posterior Distribution After Observing 16

Model Vs. Human Data MODEL HUMAN DATA

Summary of Tenenbaum (1999) Method  Pick prior distribution (includes hypothesis space)  Pick likelihood function (size principle)  leads to predictions for generalization as a function of r (range) and n (number of examples) Claims people generalize optimally given assumptions about priors and likelihood Bayesian approach provides best description of how people generalize on rectangle task. Explains how people can learn from a small number of examples, and only positive examples.

Important Ideas in Bayesian Models Generative models  Likelihood function Consideration of multiple models in parallel  Potentially infinite model space Inference  prediction via model averaging  role of priors diminishes with amount of evidence Learning  trade off between model simplicity and fit to data  Bayesian Occam’s Razor

Ockham's Razor If two hypotheses are equally consistent with the data, prefer the simpler one. Simplicity can accommodate fewer observations smoother fewer parameters restricts predictions more (“sharper” predictions) Examples 1 st vs. 4 th order polynomial small rectangle vs. large rectangle in Tenenbaum model medieval philosopher and monk tool for cutting (metaphorical)

Motivating Ockham's Razor Aesthetic considerations A theory with mathematical beauty is more likely to be right (or believed) than an ugly one, given that both fit the same data. Past empirical success of the principle Coherent inference, as embodied by Bayesian reasoning, automatically incorporates Ockham's razor Two theories H 1 and H 2 PRIORS LIKELIHOODS

Ockham's Razor with Priors Jeffreys (1939) probabililty text more complex hypotheses should have lower priors Requires a numerical rule for assessing complexity e.g., number of free parameters e.g., Vapnik-Chervonenkis (VC) dimension

Subjective vs. Objective Priors subjective or informative prior specific, definite information about a random variable objective or uninformative prior vague, general information Philosophical arguments for certain priors as uninformative Maximum entropy / least committment e.g., interval [a b]: uniform e.g., interval [0, ∞) with mean 1/λ: exponential distribution e.g., mean μ and std deviation σ: Gaussian  Independence of measurement scale e.g., Jeffrey’s prior 1/(θ(1-θ)) for θ in [0,1] expresses same belief whether we talk about θ or logθ

Ockham’s Razor Via Likelihoods Coin flipping example H 1 : coin has two heads H 2 : coin has a head and a tail Consider 5 flips producing HHHHH H 1 could produce only this sequence H 2 could produce HHHHH, but also HHHHT, HHHTH,... TTTTT P(HHHHH | H 1 ) = 1, P(HHHHH | H 2 ) = 1/32 H 2 pays the price of having a lower likelihood via the fact it can accommodate a greater range of observations H 1 is more readily rejected by observations

Simple and Complex Hypotheses H2H2 H1H1

Bayes Factor BIC is approximation to Bayes factor A.k.a. likelihood ratio

Hypothesis Classes Varying In Complexity E.g., 1 st, 2 nd, and 3 d order polynomials Hypothesis class is parameterized by w v

Rissanen (1976) Minimum Description Length Prefer models that can communicate the data in the smallest number of bits. The preferred hypothesis H for explaining data D minimizes: (1) length of the description of the hypothesis (2) length of the description of the data with the help of the chosen theory L: length

MDL & Bayes L: some measure of length (complexity) MDL: prefer hypothesis that min. L(H) + L(D|H) Bayes rule implies MDL principle P(H|D) = P(D|H)P(H) / P(D) –log P(H|D) = –log P(D|H) – log P(H) + log P(D) = L(D|H) + L(H) + const

Relativity Example Explain deviation in Mercury's orbit at perihelion with respect to prevailing theory E: Einstein's theoryα = true deviation F: fudged Newtonian theorya = observed deviation

Relativity Example (Continued) Subjective Ockham's razor result depends on one's belief about P(α|F) Objective Ockham's razor for Mercury example, RHS is Applies to generic situation