Information theory, MDL and human cognition

Information theory, MDL and human cognition Nick Chater Department of Psychology University College London n.chater@ucl.ac.uk

Overview • Bayes and MDL: An overview • Universality and induction • Some puzzles about model fitting • Cognitive science applications

Bayes and MDL: A simplified story • Shannon’s coding theorem. • For distribution Pr(A), optimal code will assign -log2Pr(A) to code event A • MDL model selection: • choose, M, that yields the shortest code for D, i.e., minimize: • -log2Pr(D, M)

A simple equivalence • Minimize: • -log2Pr(D, M) • Maximize: • Pr(D, M) = Pr(M|D)Pr(D) • Maximize: • Pr(M|D) • Just what Bayes recommends • (if choosing a single model) • Equivalence generalizes to parametric M(θ) to ‘full’ Bayes; and in other ways • Chater, 1996, for application to the simplicity and likelihood principles in perceptual organization

Codes or priors? Which comes first? 1. The philosophical issue • Bayesian viewpoint: Probabilities as basic • calculus for degrees of beliefs (probability theory) • decision theory (probabilities meet action) • brain as a probabilistic calculation machine (whether belief propagation, dynamic programming…)

Simplicity/MDL viewpoint: Codes as basic • Rissanen: data is all there is; distributions are a fiction • Code structure is primary; code interpretation is secondary • Probabilities defined over events; but “events” are cognitive constructs • Leeuwenberg & Boselie (1988)

MDL viewpoint: Take codes as basic… …when we know most about representation, e.g., grammars Codes or priors? Which comes first? 2. The practical issue • Bayesian viewpoint: • Take probabilities as basic… • …when we know most about probability, e.g, image statistics:

Bayesian viewpoint (e.g., Geisler et al, 2001) Good continuation—most lines continue in the same direction in real images

In, e.g., linguistics, representations are given by theory And we can roughly assess the complexity of grammars (by length) Not so clear how directly to set a prior over all grammars (though can define a generative process in simple cases…) Simplicity/MDL viewpoint(e.g., Goldsmith, 2001) S  NP VP VP  V NP VP  V NP PP NP  Det Noun NP  NP PP “Binding contraints” Gzip as a handy approximation!?!

Simplicity/MDL and Bayes are closely related • Lets now explore the simplicity perspective

Suppose we want a prior so neutral that it never rules out a model Possible, if limit to computable models Mixture of all (computable) priors, with weights, i, that decline fairly fast: Then, this multiplicatively dominates all priors though neutral priors will mean slow learning m(x) are “universal” priors The most neutral possible prior…

The most neutral possible coding language • Universal programming languages (Java, matlab, UTMs, etc) • K(x) = length of shortest program in Java, matlab, UTM, that generates x (K is uncomputable) • Invariance theorem • any languages L1, L2, c, • x |KL1(x)-KL2(x)| ≤c • Mathematically justifies talk of K(x), not KJava(x) , KMatlab(x),…

So does this mean that choice of language doesn’t matter? • Not quite! • c can be large • And, for any L1, c0, L2, x such that • |KL1(x)-KL2(x)| ≥c0 • The problem of the one-instruction code for the entire data set… But Kolmogorov complexity can be made concrete…

210 bits, λ-calculus 272, combinators Compact Universal Turing machines Due to Jon Tromp, 2007 Not much room to hide, here!

A key result: K(x) = -log2m(x) o(1) Where m is a universal prior Analogous to the Shannon’s source coding theorem And for any computable q, K(x) ≤ -log2q(x) o(1) For typical x drawn from q(x) Any data, x, that is likely for any sensible probability distribution has low K(x) Neutral priors and Kolmogorov complexity

Prediction by simplicity • Find shortest ‘program/explanation’ for current ‘corpus’ (binary string) • Predict using that program • Strictly, use ‘weighted sum’ of explanations, weighted by brevity

Prediction is possible (Solomonoff, 1978)Summed error has finite bound • sj is summed squared error between prediction and true probability on item j • So prediction converges [faster than 1/nlog(n)], for corpus size n • Computability assumptions only (no stationarity needed)

Summary so far… • Simplicity/MDL - close and deep connections with Bayes • Defines universal prior (i.e., based on simplicity) • Can be made “concrete” • General prediction results • A convenient “dual” framework to Bayes, when codes are easier than probabilities Li, M. & Vitanyi, P. (1997) (2nd Edition). Introduction to Kolmogorov complexity theory and its applications. Berlin: Springer.

A problem of model selection? Or: why simplicity won’t go away • Where do priors come from? • Well, priors can be given by hyper-priors • And hyper-priors by hyper-hyper-priors • But it can’t go on forever! • And we need priors over models we’re only just thought of • And, in some contexts, over models we haven’t yet thought of (!) • Code length in our representation language is a fixed basis • Nb. Building probabilistic models = augmenting our language with new coding schemes

Bayesian model selection prefers y(x)=a2x2+a1x+a0 Not y(x)=a125x125+a124x124+…+a0 The hidden role of simplicity… But who says how many parameters a function has got??

A trick… • Convert parameters to constants • y(x)=a125x125+a124x124+…+a0 • 126 parameters • y(x)=.003x125 + .02x124+…+3x – 24.3 • 0 parameters • And hence is favoured by Bayesian (and all other) model selection criteria All the virtues of theft over honest toil

Zoubin’s problem for ML • ML Gaussian is a delta function on one point

ML: Gaussian is a delta functions on one point An impressive fit!

A related problem for Bayes? • The mixture of delta functions model (!)

A related problem for Bayes? • The mixture of delta functions model (!) An even more impressive fit!

No! Sense of moral outrage Model must be fitted post-hoc It would be different if I’d thought of it before the data arrived (cf empirical Bayes) Yes! ! But order of data acquisition has no role in Bayes Confirmation is just the same, whenever I thought of the model Should the “cheating” model get a huge boost from this data The models get a spectacular boost; but is even more spectacularly unlikely…

So we need to take care with priors! • y = x • High prior; compact to state • y=.003x125 + .02x124+…+3x-24 • Low prior; not compact to state • With a different representation language, could have the opposite bias • But we start from where we are; our actual representations We can discoverthat things are simpler than we though (i.e., simplicity is not quite so subjective…)

There are quite a few…

Here: • Perceptual organization • Language Acquisition • Similarity and generalization

Long tradition of simplicity in perception (Mach, Koffka, Leeuwenberg); e.g., Gestalt laws (x) (x,v) + (x,v) (x) + (x,v) + (x) (x) + + (x,v) (x) (x,v) + (x) (x,v) (v) Grouped 6 + 1 vectors Ungrouped 6 x 2 vectors

Undergeneral grammars predict that good sentences are not allowed just wait til one turns up Overgeneral grammars predict that bad sentences are actually ok Need negative evidence---say a bad sentence, and get corrected And language acquisition: where it helps resolve an apparent learnability paradox

Without negative evidence can never eliminate overgeneral grammars “Mere” non-occurrence of sentences is not enough… …because almost all acceptable sentences also never occur Backed-up by formal results (Gold, 1967; though Feldman, Horning et al) Argument for innateness? The logical problem of language acquisition(e.g., Hornstein & Lightfoot, 1981; Pinker, 1979)

Linguistic environment Measures of learning performance Learning method Positive evidence only; computability Statistical Simplicity An “ideal” learning set-up (cf ideal observers)

Overgeneralization Theorem (Chater & Vitányi) • Suppose learner has probability j of erroneously guessing an ungrammatical jth word • Intuitive explanation: • overgeneralization underloads probabilities of grammatical sentences; • Small probabilities implies longer code lengths

Absence as implicit negative evidence • Overgeneral grammars predict missing sentences • And their absence is a clue that the grammar is wrong Method can be “scaled-down” to consider learnability of specific linguistic constructions

Similarity and categorization • Cognitive dissimilarity: representational “distortion” required to get from x to y • DU(x,y) = K(y|x) • Not symmetrical • K(y|x) > K(x|y) when • K(y) > K(x) • Deletion is easy…

Generalization (strictly confusability) is an exponention function of psychological “distance” Shepard’s (1987) Universal Law

A derivation Shepard’s generalization measure for “typical” items

Assuming items roughly the same complexity The universal law

The asymmetry of similarity • What thing is this like?

And what is this like?

A heuristic measure of amount of information: Shannon’s guessing game… 1. Pony? 2. Cow? 3. Dog? … 345. Pegasus√ 1. Pony? 2. Cow? 3. Dog? … 345. Pegasus√ 345!

Asymmetry of codelengthsasymmetry of similarity • Horse: guess #345 gets Pegasus. log2Pr(#345) is very small. • Pegasus: guess #2 gets Horse. log2(Pr(#2)) is very small. • So Pegasus is more like Horse, than Horse is like Pegasus • Many other examples of asymmetry, and many measures (search times, memory confusions…), which seem to fit this pattern

Treisman & Souther (1985) A simple array

A complex array

Summary • MDL/Kolmogorov complexity close relation with Bayes • Basis for a “universal” prior • Variety of applications to cognitive science

Information theory, MDL and human cognition