Part II Graphical models Challenges of probabilistic models

Part II: Graphical models

Challenges of probabilistic models • Specifying well-defined probabilistic models with many variables is hard (for modelers) • Representing probability distributions over those variables is hard (for computers/learners) • Computing quantities using those distributions is hard (for computers/learners)

Representing structured distributions Four random variables: X 1 X 2 X 3 X 4 coin toss produces heads pencil levitates friend has psychic powers friend has two-headed coin Domain {0, 1}

Joint distribution • Requires 15 numbers to specify probability of all values x 1, x 2, x 3, x 4 – N binary variables, 2 N-1 numbers • Similar cost when computing conditional probabilities 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111

How can we use fewer numbers? Four random variables: X 1 X 2 X 3 X 4 coin toss produces heads Domain {0, 1}

Statistical independence • Two random variables X 1 and X 2 are independent if P(x 1|x 2) = P(x 1) – e. g. coinflips: P(x 1=H|x 2=H) = P(x 1=H) = 0. 5 • Independence makes it easier to represent and work with probability distributions • We can exploit the product rule: If x 1, x 2, x 3, and x 4 are all independent…

Expressing independence • Statistical independence is the key to efficient probabilistic representation and computation • This has led to the development of languages for indicating dependencies among variables • Some of the most popular languages are based on “graphical models”

Part II: Graphical models • Introduction to graphical models – representation and inference • Causal graphical models – causality – learning about causal relationships • Graphical models and cognitive science – uses of graphical models – an example: causal induction

Part II: Graphical models • Introduction to graphical models – representation and inference • Causal graphical models – causality – learning about causal relationships • Graphical models and cognitive science – uses of graphical models – an example: causal induction

Graphical models • Express the probabilistic dependency structure among a set of variables (Pearl, 1988) • Consist of – a set of nodes, corresponding to variables – a set of edges, indicating dependency – a set of functions defined on the graph that specify a probability distribution

Undirected graphical models • Consist of X 1 X 3 X 4 – a set of nodes X 2 X 5 – a set of edges – a potential for each clique, multiplied together to yield the distribution over variables • Examples – statistical physics: Ising model, spinglasses – early neural networks (e. g. Boltzmann machines)

Directed graphical models • Consist of X 1 X 3 X 4 – a set of nodes X 2 X 5 – a set of edges – a conditional probability distribution for each node, conditioned on its parents, multiplied together to yield the distribution over variables • Constrained to directed acyclic graphs (DAGs) • Called Bayesian networks or Bayes nets

Bayesian networks and Bayes • Two different problems – Bayesian statistics is a method of inference – Bayesian networks are a form of representation • There is no necessary connection – many users of Bayesian networks rely upon frequentist statistical methods – many Bayesian inferences cannot be easily represented using Bayesian networks

Properties of Bayesian networks • Efficient representation and inference – exploiting dependency structure makes it easier to represent and compute with probabilities • Explaining away – pattern of probabilistic reasoning characteristic of Bayesian networks, especially early use in AI

Properties of Bayesian networks • Efficient representation and inference – exploiting dependency structure makes it easier to represent and compute with probabilities • Explaining away – pattern of probabilistic reasoning characteristic of Bayesian networks, especially early use in AI

Efficient representation and inference Four random variables: X 1 X 2 X 3 X 4 coin toss produces heads pencil levitates friend has psychic powers friend has two-headed coin P(x 4) X 4 P(x 1|x 3, x 4) X 3 P(x 3) X 1 X 2 P(x 2|x 3)

The Markov assumption Every node is conditionally independent of its nondescendants, given its parents where Pa(Xi) is the set of parents of Xi (via the product rule)

Efficient representation and inference Four random variables: X 1 X 2 X 3 X 4 coin toss produces heads pencil levitates friend has psychic powers friend has two-headed coin 1 P(x 4) X 4 4 total = 7 (vs 15) P(x 1|x 3, x 4) X 3 P(x 3) 1 X 2 P(x 2|x 3) 2 P(x 1, x 2, x 3, x 4) = P(x 1|x 3, x 4)P(x 2|x 3)P(x 4)

Reading a Bayesian network • The structure of a Bayes net can be read as the generative process behind a distribution • Gives the joint probability distribution over variables obtained by sampling each variable conditioned on its parents

Reading a Bayesian network Four random variables: X 1 X 2 X 3 X 4 coin toss produces heads pencil levitates friend has psychic powers friend has two-headed coin P(x 4) X 4 P(x 1|x 3, x 4) X 3 P(x 3) X 1 X 2 P(x 2|x 3) P(x 1, x 2, x 3, x 4) = P(x 1|x 3, x 4)P(x 2|x 3)P(x 4)

Reading a Bayesian network • The structure of a Bayes net can be read as the generative process behind a distribution • Gives the joint probability distribution over variables obtained by sampling each variable conditioned on its parents • Simple rules for determining whether two variables are dependent or independent • Independence makes inference more efficient

Computing with Bayes nets P(x 4) X 4 P(x 1|x 3, x 4) X 3 P(x 3) X 1 X 2 P(x 2|x 3) P(x 1, x 2, x 3, x 4) = P(x 1|x 3, x 4)P(x 2|x 3)P(x 4)

Computing with Bayes nets sum over 8 values P(x 4) X 4 P(x 1|x 3, x 4) X 3 P(x 3) X 1 X 2 P(x 2|x 3) P(x 1, x 2, x 3, x 4) = P(x 1|x 3, x 4)P(x 2|x 3)P(x 4)

Computing with Bayes nets P(x 4) X 4 P(x 1|x 3, x 4) X 3 P(x 3) X 1 X 2 P(x 2|x 3) P(x 1, x 2, x 3, x 4) = P(x 1|x 3, x 4)P(x 2|x 3)P(x 4)

Computing with Bayes nets sum over 4 values P(x 4) X 4 P(x 1|x 3, x 4) X 3 P(x 3) X 1 X 2 P(x 2|x 3) P(x 1, x 2, x 3, x 4) = P(x 1|x 3, x 4)P(x 2|x 3)P(x 4)

Computing with Bayes nets • Inference algorithms for Bayesian networks exploit dependency structure • Message-passing algorithms – “belief propagation” passes simple messages between nodes, exact for tree-structured networks • More general inference algorithms – exact: “junction-tree” – approximate: Monte Carlo schemes (see Part IV)

Properties of Bayesian networks • Efficient representation and inference – exploiting dependency structure makes it easier to represent and compute with probabilities • Explaining away – pattern of probabilistic reasoning characteristic of Bayesian networks, especially early use in AI

Explaining away Rain Sprinkler Grass Wet Assume grass will be wet if and only if it rained last night, or if the sprinklers were left on:

Explaining away Rain Sprinkler Grass Wet Compute probability it rained last night, given that the grass is wet:

Explaining away Rain Sprinkler Grass Wet Compute probability it rained last night, given that the grass is wet:

Explaining away Rain Sprinkler Grass Wet Compute probability it rained last night, given that the grass is wet:

Explaining away Rain Sprinkler Grass Wet Compute probability it rained last night, given that the grass is wet:

Explaining away Rain Sprinkler Grass Wet Compute probability it rained last night, given that the grass is wet: Between 1 and P(s)

Explaining away Rain Sprinkler Grass Wet Compute probability it rained last night, given that the grass is wet and sprinklers were left on: Both terms = 1

Explaining away Rain Sprinkler Grass Wet Compute probability it rained last night, given that the grass is wet and sprinklers were left on:

Explaining away Rain Sprinkler Grass Wet “Discounting” to prior probability.

Contrast w/ production system Rain Sprinkler Grass Wet • Formulate IF-THEN rules: – IF Rain THEN Wet – IF Wet THEN Rain IF Wet AND NOT Sprinkler THEN Rain • Rules do not distinguish directions of inference • Requires combinatorial explosion of rules

Contrast w/ spreading activation Rain Sprinkler Grass Wet • Excitatory links: Rain Wet, Sprinkler Wet • Observing rain, Wet becomes more active. • Observing grass wet, Rain and Sprinkler become more active • Observing grass wet and sprinkler, Rain cannot become less active. No explaining away!

Contrast w/ spreading activation Rain Sprinkler Grass Wet • Excitatory links: Rain Wet, Sprinkler Wet • Inhibitory link: Rain Sprinkler • Observing grass wet, Rain and Sprinkler become more active • Observing grass wet and sprinkler, Rain becomes less active: explaining away

Contrast w/ spreading activation Rain Sprinkler Burst pipe Grass Wet • Each new variable requires more inhibitory connections • Not modular – whether a connection exists depends on what others exist – big holism problem – combinatorial explosion

Contrast w/ spreading activation (Mc. Clelland & Rumelhart, 1981)

Graphical models • Capture dependency structure in distributions • Provide an efficient means of representing and reasoning with probabilities • Allow kinds of inference that are problematic for other representations: explaining away – hard to capture in a production system – more natural than with spreading activation

Part II: Graphical models • Introduction to graphical models – representation and inference • Causal graphical models – causality – learning about causal relationships • Graphical models and cognitive science – uses of graphical models – an example: causal induction

Causal graphical models • Graphical models represent statistical dependencies among variables (ie. correlations) – can answer questions about observations • Causal graphical models represent causal dependencies among variables (Pearl, 2000) – express underlying causal structure – can answer questions about both observations and interventions (actions upon a variable)

Bayesian networks Nodes: variables Links: dependency Four random variables: X 1 X 2 X 3 X 4 Each node has a conditional probability distribution coin toss produces heads pencil levitates friend has psychic powers friend has two-headed coin Data: observations of x 1, . . . , x 4 P(x 4) X 4 P(x 1|x 3, x 4) X 3 P(x 3) X 1 X 2 P(x 2|x 3)

Causal Bayesian networks Nodes: variables Links: causality Four random variables: X 1 X 2 X 3 X 4 Each node has a conditional probability distribution coin toss produces heads pencil levitates friend has psychic powers friend has two-headed coin Data: observations of and interventions on x 1, . . . , x 4 P(x 4) X 4 P(x 1|x 3, x 4) X 3 P(x 3) X 1 X 2 P(x 2|x 3)

Interventions Four random variables: Cut all incoming links for the node that we intervene on X 1 X 2 X 3 X 4 Compute probabilities with “mutilated” Bayes net coin toss produces heads pencil levitates friend has psychic powers friend has two-headed coin hold down pencil P(x 4) X 4 X 3 P(x 3) X P(x 1|x 3, x 4) X 1 X 2 P(x 2|x 3)

Learning causal graphical models C B E • Strength: how strong is a relationship? • Structure: does a relationship exist?

Causal structure vs. causal strength C B E • Strength: how strong is a relationship?

Causal structure vs. causal strength B C B w 0 w 1 w 0 E C E • Strength: how strong is a relationship? – requires defining nature of relationship

Parameterization • Structures: h 1 = C B h 0 = E E • Parameterization: C B 0 1 0 0 1 1 Generic h 1: P(E = 1 | C, B) p 00 p 10 p 01 p 11 C B h 0: P(E = 1| C, B) p 0 p 1

Parameterization • Structures: h 1 = B C w 0 w 1 h 0 = C B w 0 E E w 0, w 1: strength parameters for B, C • Parameterization: C B 0 1 0 0 1 1 Linear h 1: P(E = 1 | C, B) 0 w 1 w 0 w 1+ w 0 h 0: P(E = 1| C, B) 0 0 w 0

Parameterization • Structures: h 1 = B C w 0 w 1 h 0 = C B w 0 E E w 0, w 1: strength parameters for B, C • Parameterization: C B 0 1 0 0 1 1 “Noisy-OR” h 1: P(E = 1 | C, B) 0 w 1 w 0 w 1+ w 0 – w 1 w 0 h 0: P(E = 1| C, B) 0 0 w 0

Parameter estimation • Maximum likelihood estimation: maximize i P(bi, ci, ei; w 0, w 1) • Bayesian methods: as in Part I

Causal structure vs. causal strength C B E • Structure: does a relationship exist?

Approaches to structure learning • Constraint-based: C B – dependency from statistical tests (eg. 2) – deduce structure from dependencies E (Pearl, 2000; Spirtes et al. , 1993)

Approaches to structure learning • Constraint-based: C B – dependency from statistical tests (eg. 2) – deduce structure from dependencies E (Pearl, 2000; Spirtes et al. , 1993)

Approaches to structure learning • Constraint-based: C B – dependency from statistical tests (eg. 2) – deduce structure from dependencies E (Pearl, 2000; Spirtes et al. , 1993)

Approaches to structure learning • Constraint-based: C B – dependency from statistical tests (eg. 2) – deduce structure from dependencies E (Pearl, 2000; Spirtes et al. , 1993) Attempts to reduce inductive problem to deductive problem

Approaches to structure learning • Constraint-based: C B – dependency from statistical tests (eg. 2) – deduce structure from dependencies E (Pearl, 2000; Spirtes et al. , 1993) • Bayesian: – compute posterior probability of structures, given observed data P(h|data) P(data|h) P(h) C B E E P(h 1|data) P(h 0|data) (Heckerman, 1998; Friedman, 1999)

Bayesian Occam’s Razor P(d | h ) h 0 (no relationship) h 1 (relationship) All possible data sets d For any model h,

Causal graphical models • Extend graphical models to deal with interventions as well as observations • Respecting the direction of causality results in efficient representation and inference • Two steps in learning causal models – strength: parameter estimation – structure: structure learning

Part II: Graphical models • Introduction to graphical models – representation and inference • Causal graphical models – causality – learning about causal relationships • Graphical models and cognitive science – uses of graphical models – an example: causal induction

Uses of graphical models • Understanding existing cognitive models – e. g. , neural network models • Representation and reasoning – a way to address holism in induction (c. f. Fodor) • Defining generative models – mixture models, language models (see Part IV) • Modeling human causal reasoning

Human causal reasoning • How do people reason about interventions? (Gopnik, Glymour, Sobel, Schulz, Kushnir & Danks, 2004; Lagnado & Sloman, 2004; Sloman & Lagnado, 2005; Steyvers, Tenenbaum, Wagenmakers & Blum, 2003) • How do people learn about causal relationships? – parameter estimation (Shanks, 1995; Cheng, 1997) – constraint-based models (Glymour, 2001) – Bayesian structure learning (Steyvers et al. , 2003; Griffiths & Tenenbaum, 2005)

Causation from contingencies C present C absent (c+) (c-) E present (e+) a c E absent (e-) b d “Does C cause E? ” (rate on a scale from 0 to 100)

Two models of causal judgment • Delta-P (Jenkins & Ward, 1965): • Power PC (Cheng, 1997): Power

Buehner and Cheng (1997) P 0. 00 People P Power 0. 25 0. 50 0. 75 1. 00

Buehner and Cheng (1997) People P Power Constant P, changing judgments

Buehner and Cheng (1997) People P Power Constant causal power, changing judgments

Buehner and Cheng (1997) People P Power P = 0, changing judgments

Causal structure vs. causal strength B C B w 0 w 1 w 0 E C E • Strength: how strong is a relationship? • Structure: does a relationship exist?

Causal strength • Assume structure: B C w 0 w 1 E • P and causal power are maximum likelihood estimates of the strength parameter w 1, under different parameterizations for P(E|B, C): – linear P, Noisy-OR causal power

Causal structure • Hypotheses: h 1 = C B E h 0 = C B E • Bayesian causal inference: support = P(d|h 1) P(d|h 0) likelihood ratio (Bayes factor) gives evidence in favor of h 1

Buehner and Cheng (1997) People P (r = 0. 89) Power (r = 0. 88) Support (r = 0. 97)

The importance of parameterization • Noisy-OR incorporates mechanism assumptions: – generativity: causes increase probability of effects – each cause is sufficient to produce the effect – causes act via independent mechanisms (Cheng, 1997) • Consider other models: – statistical dependence: 2 test – generic parameterization (cf. Anderson, 1990)

People Support (Noisy-OR) 2 Support (generic)

Generativity is essential P(e+|c+) P(e+|c-) 100 50 0 8/8 6/8 4/8 2/8 0/8 Support • Predictions result from “ceiling effect” – ceiling effects only matter if you believe a cause increases the probability of an effect

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!

“Backwards blocking” (Sobel, Tenenbaum & Gopnik, 2004) A B AB Trial A Trial – 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% say yes) • B: probably not a blicket (34% say yes)

A Possible hypotheses A B A E A A E B A A A B A A A B E B E B E E B E B E A E B A B E

Bayesian inference • Evaluating causal models in light of data: • Inferring a particular causal relation:

Bayesian inference With a uniform prior on hypotheses, and the generic parameterization Probability of being a blicket A B 0. 32 0. 34

Modeling backwards blocking Assume… • Links can only exist from blocks to detectors • Blocks are blickets with prior probability q • Blickets always activate detectors, but detectors never activate on their own – deterministic Noisy-OR, with wi = 1 and w 0 = 0�

Modeling backwards blocking P(h 00) = (1 – q)2 A P(E=1 | A=0, B=0): P(E=1 | A=1, B=0): P(E=1 | A=0, B=1): P(E=1 | A=1, B=1): B P(h 01) = (1 – q) q A B P(h 10) = q(1 – q) A B P(h 11) = q 2 A B E E 0 0 0 1 1 0 1 0 1 1 1

Modeling backwards blocking P(h 00) = (1 – q)2 A P(E=1 | A=1, B=1): B P(h 01) = (1 – q) q A B P(h 10) = q(1 – q) A B P(h 11) = q 2 A B E E 0 1 1 1

Modeling backwards blocking P(h 01) = (1 – q) q A B P(h 10) = q(1 – q) A B P(h 11) = q 2 A B E E E P(E=1 | A=1, B=0): 0 1 1 P(E=1 | A=1, B=1): 1 1 1

Manipulating prior probability (Tenenbaum, Sobel, Griffiths, & Gopnik, submitted) Initial AB Trial A Trial

Summary • Graphical models provide solutions to many of the challenges of probabilistic models – defining structured distributions – representing distributions on many variables – efficiently computing probabilities • Causal graphical models provide tools for defining rational models of human causal reasoning and learning