Bayesian models as a tool for revealing inductive biases Tom Griffiths University of California, Berkeley

Inductive problems blicket toma dax wug blicket wug S  X Y X  {blicket,dax} Y  {toma, wug} Learning languages from utterances Learning functions from (x,y) pairs Learning categories from instances of their members

Revealing inductive biases Many problems in cognitive science can be formulated as problems of induction –learning languages, concepts, and causal relations Such problems are not solvable without bias (e.g., Goodman, 1955; Kearns & Vazirani, 1994; Vapnik, 1995) What biases guide human inductive inferences? How can computational models be used to investigate human inductive biases?

Models and inductive biases Transparent

Reverend Thomas Bayes Bayesian models

Bayes’ theorem Posterior probability LikelihoodPrior probability Sum over space of hypotheses h: hypothesis d: data

Three advantages of Bayesian models Transparent identification of inductive biases through hypothesis space, prior, and likelihood Opportunity to explore a range of biases expressed in terms that are natural to the problem at hand Rational statistical inference provides an upper bound on human inferences from data

Two examples Causal induction from small samples (Josh Tenenbaum, David Sobel, Alison Gopnik) Statistical learning and word segmentation (Sharon Goldwater, Mark Johnson)

Two examples Causal induction from small samples (Josh Tenenbaum, David Sobel, Alison Gopnik) Statistical learning and word segmentation (Sharon Goldwater, Mark Johnson)

Blicket detector (Dave Sobel, Alison Gopnik, and colleagues) See this? It’s a blicket machine. Blickets make it go. Let’s put this one on the machine. Oooh, it’s a blicket!

–Two objects: A and B –Trial 1: A B on detector – detector active –Trial 2: B on detector – detector inactive –4-year-olds judge whether each object is a blicket A: a blicket (100% say yes) B: almost certainly not a blicket (16% say yes) “One cause” (Gopnik, Sobel, Schulz, & Glymour, 2001) AB Trial B Trial AB A Trial

Hypotheses: causal models Defines probability distribution over variables (for both observation, and intervention) E BA E BA E BA E BA (Pearl, 2000; Spirtes, Glymour, & Scheines, 1993)

Prior and likelihood: causal theory Prior probability an object is a blicket is q –defines a distribution over causal models Detectors have a deterministic “activation law” –always activate if a blicket is on the detector –never activate otherwise (Tenenbaum & Griffiths, 2003; Griffiths, 2005)

Prior and likelihood: causal theory P(E=1 | A=0, B=0): P(E=0 | A=0, B=0): P(E=1 | A=1, B=0): P(E=0 | A=1, B=0): P(E=1 | A=0, B=1): P(E=0 | A=0, B=1): P(E=1 | A=1, B=1): P(E=0 | A=1, B=1): E BA E BA E BA E BA P(h 00 ) = (1 – q) 2 P(h 10 ) = q(1 – q)P(h 01 ) = (1 – q) qP(h 11 ) = q 2

Modeling “one cause” P(E=1 | A=0, B=0): P(E=0 | A=0, B=0): P(E=1 | A=1, B=0): P(E=0 | A=1, B=0): P(E=1 | A=0, B=1): P(E=0 | A=0, B=1): P(E=1 | A=1, B=1): P(E=0 | A=1, B=1): E BA E BA E BA E BA P(h 00 ) = (1 – q) 2 P(h 10 ) = q(1 – q)P(h 01 ) = (1 – q) qP(h 11 ) = q 2

Modeling “one cause” P(E=1 | A=0, B=0): P(E=0 | A=0, B=0): P(E=1 | A=1, B=0): P(E=0 | A=1, B=0): P(E=1 | A=0, B=1): P(E=0 | A=0, B=1): P(E=1 | A=1, B=1): P(E=0 | A=1, B=1): E BA E BA E BA P(h 10 ) = q(1 – q)P(h 01 ) = (1 – q) qP(h 11 ) = q 2

Modeling “one cause” P(E=1 | A=0, B=0): P(E=0 | A=0, B=0): P(E=1 | A=1, B=0): P(E=0 | A=1, B=0): P(E=1 | A=0, B=1): P(E=0 | A=0, B=1): P(E=1 | A=1, B=1): P(E=0 | A=1, B=1): E BA P(h 10 ) = q(1 – q) A is definitely a blicket B is definitely not a blicket

–Two objects: A and B –Trial 1: A B on detector – detector active –Trial 2: B on detector – detector inactive –4-year-olds judge whether each object is a blicket A: a blicket (100% say yes) B: almost certainly not a blicket (16% say yes) “One cause” (Gopnik, Sobel, Schulz, & Glymour, 2001) AB Trial B Trial AB A Trial

Building on this analysis Transparent

Other physical systems From stick-ball machines… …to lemur colonies (Kushnir, Schulz, Gopnik, & Danks, 2003) (Griffiths, Baraff, & Tenenbaum, 2004) (Griffiths & Tenenbaum, 2007)

Two examples Causal induction from small samples (Josh Tenenbaum, David Sobel, Alison Gopnik) Statistical learning and word segmentation (Sharon Goldwater, Mark Johnson)

Bayesian segmentation In the domain of segmentation, we have: –Data: unsegmented corpus (transcriptions). –Hypotheses: sequences of word tokens. Optimal solution is the segmentation with highest prior probability = 1 if concatenating words forms corpus, = 0 otherwise. Encodes assumptions about the structure of language

Brent (1999) Describes a Bayesian unigram model for segmentation. –Prior favors solutions with fewer words, shorter words. Problems with Brent’s system: –Learning algorithm is approximate (non-optimal). –Difficult to extend to incorporate bigram info.

A new unigram model (Dirichlet process) Assume word w i is generated as follows: 1. Is w i a novel lexical item? Fewer word types = Higher probability

A new unigram model (Dirichlet process) Assume word w i is generated as follows: 2. If novel, generate phonemic form x 1 …x m : If not, choose lexical identity of w i from previously occurring words: Shorter words = Higher probability Power law = Higher probability

Unigram model: simulations Same corpus as Brent (Bernstein-Ratner, 1987) : –9790 utterances of phonemically transcribed child-directed speech (19-23 months). –Average utterance length: 3.4 words. –Average word length: 2.9 phonemes. Example input: yuwanttusiD6bUk lUkD*z6b7wIThIzh&t &nd6dOgi yuwanttulUk&tDIs...

Example results

What happened? Model assumes (falsely) that words have the same probability regardless of context. Positing amalgams allows the model to capture word-to-word dependencies. P( D&t ) =.024 P( D&t | WAts ) =.46 P( D&t | tu ) =.0019

What about other unigram models? Brent’s learning algorithm is insufficient to identify the optimal segmentation. –Our solution has higher probability under his model than his own solution does. –On randomly permuted corpus, our system achieves 96% accuracy; Brent gets 81%. Formal analysis shows undersegmentation is the optimal solution for any (reasonable) unigram model.

Bigram model (hierachical Dirichlet process) Assume word w i is generated as follows: 1.Is (w i-1,w i ) a novel bigram? 2.If novel, generate w i using unigram model (almost). If not, choose lexical identity of w i from words previously occurring after w i-1.

Example results

Conclusions Both adults and children are sensitive to the nature of mechanisms in using covariation Both adults and children can use covariation to make inferences about the nature of mechanisms Bayesian inference provides a formal framework for understanding how statistics and knowledge interact in making these inferences –how theories constrain hypotheses, and are learned

A probabilistic mechanism? Children in Gopnik et al. (2001) who said that B was a blicket had seen evidence that the detector was probabilistic –one block activated detector 5/6 times Replace the deterministic “activation law”… –activate with p = 1-  if a blicket is on the detector –never activate otherwise

Deterministic vs. probabilistic Probability of being a blicket One cause Deterministic Probabilistic mechanism knowledge affects intepretation of contingency data

At end of the test phase, adults judge the probability that each object is a blicket AB Trial B Trial BA I. Familiarization phase: Establish nature of mechanism II. Test phase: one cause Manipulating mechanisms same block

Manipulating mechanisms Expose to different kinds of mechanism –deterministic: detector always activates –probabilistic: detector activates with p = 1-  Test with “one cause” trials Model makes two qualitative predictions: –people will infer nature of mechanism –evaluation of B as a blicket will increase with the probabilistic mechanism (Griffiths & Sobel, submitted)

Probability of being a blicket One cause Bayes People Deterministic Probabilistic Manipulating mechanisms (n = 12 undergraduates per condition)

Probability of being a blicket One cause One control Three control Deterministic Probabilistic Bayes People Manipulating mechanisms (n = 12 undergraduates per condition)

At end of the test phase, adults judge the probability that each object is a blicket AB Trial B Trial BA I. Familiarization phase: Establish nature of mechanism II. Test phase: one cause Acquiring mechanism knowledge same block

Learning causal theories Apply Bayes’ rule as before: Sum over causal structures (h j ) to get P(d|T)

Results with children Tested 24 four-year-olds (mean age 54 months) Instead of rating, yes or no response Significant difference in one cause B responses –deterministic: 8% say yes –probabilistic: 79% say yes No significant difference in one control trials –deterministic: 4% say yes –probabilistic: 21% say yes (Griffiths & Sobel, submitted)

–Two objects: A and B –Trial 1: A B on detector – detector active –Trial 2: A on detector – detector active –4-year-olds judge whether each object is a blicket A: a blicket (100% of judgments) B: probably not a blicket (66% of judgments) “Backwards blocking” (Sobel, Tenenbaum & Gopnik, 2004) AB Trial A Trial AB

After each trial, adults judge the probability that each object is a blicket. AB Trial A Trial BA I. Pre-training phase: Establish baserate for blickets (q) II. Backwards blocking phase: Manipulating plausibility

AB Trial A Trial Initial

Comparison to previous results Proposed boundaries are more accurate than Brent’s, but fewer proposals are made. Result: word tokens are less accurate. Boundary Precision Boundary Recall Brent GGJ Token F-score Brent.68 GGJ.54 Precision: #correct / #found = [= hits / (hits + false alarms)] Recall: #found / #true = [= hits / (hits + misses)] F-score: an average of precision and recall.

Quantitative evaluation Compared to unigram model, more boundaries are proposed, with no loss in accuracy: Accuracy is higher than previous models: Boundary Precision Boundary Recall GGJ (unigram) GGJ (bigram) Token F-scoreType F-score Brent (unigram) GGJ (bigram).77.63

Two examples Causal induction from small samples (Josh Tenenbaum, David Sobel, Alison Gopnik) Statistical learning and word segmentation (Sharon Goldwater, Mark Johnson)