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Information theory, MDL and human cognition

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### Information theory, MDL and human cognition

Nick Chater

Department of Psychology

University College London

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 Maximize: Maximize: Just what Bayes recommends Equivalence generalizes to parametric M(θ) to ‘full’ Bayes; and in other ways

- Minimize:
- -log2Pr(D, M)

- Pr(D, M) = Pr(M|D)Pr(D)

- Pr(M|D)

- (if choosing a single model)

- 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)

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

Overview

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

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 model

- 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? model

- 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, modelλ-calculus

272, combinators

Compact Universal Turing machinesDue to Jon Tromp, 2007

Not much room to hide, here!

A key result: model

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 complexityPrediction by simplicity model

- 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 model(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… model

- 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.

Overview model

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

A problem of model selection? modelOr: 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 model

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… model

- Convert parameters to constants
- y(x)=a125x125+a124x124+…+a0
- 126 parameters

- y(x)=.003x125 + .02x124+…+3x – 24.3
- 0 parameters

- y(x)=a125x125+a124x124+…+a0
- 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 model

- ML Gaussian is a delta function on one point

An impressive fit!

A related problem for Bayes? model

- The mixture of delta functions model (!)

A related problem for Bayes? model

- The mixture of delta functions model (!)

An even more impressive fit!

No! model

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 dataThe models get a spectacular boost; but is even more spectacularly unlikely…

- So we need to take care with priors! model
- 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…)

Overview model

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

There are quite a few… model

Here: model

- Perceptual organization
- Language Acquisition
- Similarity and generalization

Long tradition of simplicity in perception model(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

Under modelgeneral 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 paradoxWithout 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 grammars

Measures of learning performance

Learning method

Positive evidence only; computability

Statistical

Simplicity

An “ideal” learning set-up (cf ideal observers)Overgeneralization Theorem grammars(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 grammars

- 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 grammars

- 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 LawA derivation function of psychological “distance”

Shepard’s generalization measure

for “typical” items

Assuming items roughly the same complexity function of psychological “distance”

The universal law

The asymmetry of similarity function of psychological “distance”

- What thing is this like?

- And what is this like? function of psychological “distance”

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 guessing game…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) guessing game…

A simple array

A complex array guessing game…

Summary guessing game…

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

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