CS340: Bayesian concept learning. Kevin Murphy Based on Josh Tenenbaum s PhD thesis (MIT BCS 1999)

Transcription

1 CS340: Bayesian concept learning Kevin Murpy Based on Jos Tenenbaum s PD tesis (MIT BCS 1999)

2 Concept learning (binary classification) from positive and negative examples

3 Concept learning from positive only examples How far out sould te rectangle go? No negative examples to act as an upper bound.

4 Human learning vs macine learning/ statistics Most ML metods for learning "concepts" suc as "dog" require a large number of positive and negative examples But people can learn from small numbers of positive only examples (look at te doggy!) Tis is called "one sot learning"

5 Everyday inductive leaps How can we learn so muc about... Meanings of words Properties of natural kinds Future outcomes of a dynamic process Hidden causal properties of an object Causes of a person s action (beliefs, goals) Causal laws governing a domain... from suc limited data?

6 Te Callenge How do we generalize successfully from very limited data? Just one or a few examples Often only positive examples Pilosopy: Induction called a problem, a riddle, a paradox, a scandal, or a myt. Macine learning and statistics: Focus on generalization from many examples, bot positive and negative.

7 Te solution: Bayesian inference Bayes rule: P ( H D) = P( H ) P( D P( D) H ) Various compelling (teoretical and experimental) arguments tat one sould represent one s beliefs using probability and update tem using Bayes rule

8 Bayesian belief updating Posterior probability p( d) = Likeliood H p( d ) p( ) p( d ) p( ) Prior probability Likeliood Posterior Prior Bayesian inference = Inverse probability teory

9 Derivation of Bayes rule By te defn of conditional prob p(a=a B=b)= p(a=a,b=b) p(b=b) By cain rule p(a=a,b=b)=p(b=b A=a)p(A=a) By rule of total probability p(b=b)= a p(b=b,a=a ) Hence we get Bayes rule ifp(b=b)>0 p(a=a B=b)= p(b=b A=a)p(A=a) a p(b=b A=a)p(A=a)

10 Bayesian inference: key ingredients Hypotesis space H Prior p() Likeliood p(d ) Algoritm for computing posterior p( D) = H p d p p d p d p ) ( ) ( ) ( ) ( ) (

11 Two examples Te number game inferring abstract patterns from sequences of integers Te ealty levels game inferring rectangles from points in R 2

12 Te number game Learning task: Observe one or more examples (numbers) Judge weter oter numbers are yes or no.

13 Te number game Examples of yes numbers Hypoteses multiples of 10 even numbers??? multiples of 10 even numbers numbers near 60

14 Human performance Diffuse similarity Rule: multiples of 10 Focused similarity: numbers near 50-60

15 Human performance Diffuse similarity Rule: multiples of 10 Focused similarity: numbers near Some penomena to explain: People can generalize from just positive examples. Generalization can appear eiter graded (uncertain) or all-or-none (confident).

16 H: Hypotesis space of possible concepts: X = {x 1,..., x n }: n examples of a concept C. Evaluate ypoteses given data using Bayes rule: p() [ prior ]: domain knowledge, pre-existing biases p(x ) [ likeliood ]: statistical information in examples. p( X) [ posterior ]: degree of belief tat is te true extension of C. = H p X p p X p X p ) ( ) ( ) ( ) ( ) ( Bayesian model

17 Hypotesis space Matematical properties (~50): odd, even, square, cube, prime, multiples of small integers powers of small integers same first (or last) digit Magnitude intervals (~5000): all intervals of integers wit endpoints between 1 and 100 Hypotesis can be defined by its extension ={x:(x)=1, x=1,2,...,100}

18 Likeliood p(x ) Size principle: Smaller ypoteses receive greater likeliood, and exponentially more so as n increases. p n 1 ( ) = if x, K, 1 X size( ) = 0 if any x i x n Follows from assumption of randomly sampled examples (strong sampling). Captures te intuition of a representative sample.

19 Example of likeliood X={20,40,60} H1 = multiples of 10 = {10,20,,100} H2 = even numbers = {2,4,,100} H3 = odd numbers = {1,3,,99} P(X H1) = 1/10 * 1/10 * 1/10 p(x H2) = 1/50 * 1/50 * 1/50 P(X H3) = 0

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21 Likeliood function 100 x 1 =1x 2 =1 Since p( x ) is a distribution over vectors of lengt n, we require tat, for all, x It is easy to see tis is true, e.g., for =even numbers, n=2 100 p(x 1,x 2 )= x 1 =1x 2 =1 p(x 1 )p(x 2 )= x 1 evenx 2 even p(x )=1 If x is fixed, we do not require p(x )=1 Hence we are free to multiply te likeliood by any constant independent of =1

22 Illustrating te size principle

23 Illustrating te size principle Data sligtly more of a coincidence under 1

24 Illustrating te size principle Data muc more of a coincidence under 1