CS 188: Artificial Intelligence. Probability recap

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1 CS 188: Artificial Intelligence Bayes Nets Representation and Independence Pieter Abbeel UC Berkeley Many slides over this course adapted from Dan Klein, Stuart Russell, Andrew Moore Conditional probability Product rule Chain rule Probability recap, independent iff: and are conditionally independent given Z iff: 2 1

2 Probabilistic Models Models describe how (a portion of) the world works Models are always simplifications May not account for every variable May not account for all interactions between variables All models are wrong; but some are useful. George E. P. Box What do we do with probabilistic models? We (or our agents) need to reason about unknown variables, given evidence Example: explanation (diagnostic reasoning) Example: prediction (causal reasoning) Example: value of information 3 Bayes Nets: Big Picture Two problems with using full joint distribution tables as our probabilistic models: Unless there are only a few variables, the joint is WA too big to represent explicitly. For n variables with domain size d, joint table has d n entries --- exponential in n. Hard to learn (estimate) anything empirically about more than a few variables at a time Bayes nets: a technique for describing complex joint distributions (models) using simple, local distributions (conditional probabilities) More properly called graphical models We describe how variables locally interact Local interactions chain together to give global, indirect interactions 4 2

3 Bayes Nets Representation Informal first introduction of Bayes nets through causality intuition More formal introduction of Bayes nets Conditional Independences Probabilistic Inference Learning Bayes Nets from Data 5 Graphical Model Notation Nodes: variables (with domains) Can be assigned (observed) or unassigned (unobserved) Arcs: interactions Similar to CSP constraints Indicate direct influence between variables Formally: encode conditional independence (more later) For now: imagine that arrows mean direct causation (in general, they don t!) 6 3

4 Example: Coin Flips N independent coin flips 1 2 n No interactions between variables: absolute independence 7 Example: Traffic Variables: R: It rains T: There is traffic R Model 1: independence T Model 2: rain causes traffic Why is an agent using model 2 better? 8 4

5 Example: Traffic II Let s build a causal graphical model Variables T: Traffic R: It rains L: Low pressure D: Roof drips B: Ballgame C: Cavity 9 Example: Alarm Network Variables B: Burglary A: Alarm goes off M: Mary calls J: John calls E: Earthquake! 10 5

6 Bayes Net Semantics Let s formalize the semantics of a Bayes net A set of nodes, one per variable A directed, acyclic graph A conditional distribution for each node A collection of distributions over, one for each combination of parents values A 1 A n CPT: conditional probability table Description of a noisy causal process A Bayes net = Topology (graph) + Local Conditional Probabilities 11 Probabilities in BNs Bayes nets implicitly encode joint distributions As a product of local conditional distributions To see what probability a BN gives to a full assignment, multiply all the relevant conditionals together: Example: This lets us reconstruct any entry of the full joint Not every BN can represent every joint distribution The topology enforces certain conditional independencies 12 6

7 Example: Coin Flips 1 2 n h 0.5 t 0.5 h 0.5 t 0.5 h 0.5 t 0.5 Only distributions whose variables are absolutely independent can be represented by a Bayes net with no arcs. 13 Example: Traffic R +r 1/4 r 3/4 T +r +t 3/4 t 1/4 r +t 1/2 t 1/2 14 7

8 Example: Alarm Network B P(B) +b Burglary Earthqk E P(E) +e b e John calls A J P(J A) +a +j 0.9 +a j 0.1 a +j 0.05 a j 0.95 Alarm Mary calls A M P(M A) +a +m 0.7 +a m 0.3 a +m 0.01 a m 0.99 B E A P(A B,E) +b +e +a b +e a b e +a b e a 0.06 b +e +a 0.29 b +e a 0.71 b e +a b e a Example Bayes Net: Insurance 16 8

9 Example Bayes Net: Car 17 Build your own Bayes nets!

10 Size of a Bayes Net How big is a joint distribution over N Boolean variables? 2 N How big is an N-node net if nodes have up to k parents? O(N * 2 k+1 ) Both give you the power to calculate BNs: Huge space savings! Also easier to elicit local CPTs Also turns out to be faster to answer queries (coming) 21 Bayes Nets Representation Informal first introduction of Bayes nets through causality intuition More formal introduction of Bayes nets Conditional Independences Probabilistic Inference Learning Bayes Nets from Data 22 10

11 Representing Joint Probability Distributions Table representation: number of parameters: d n -1 Chain rule representation: number of parameters: (d-1) + d(d-1) + d 2 (d-1)+ +d n-1 (d-1) = d n -1 Size of CPT = (number of different joint instantiations of the preceding variables) times (number of values current variable can take on minus 1) Both can represent any distribution over the n random variables. Makes sense same number of parameters needs to be stored. Chain rule applies to all orderings of the variables, so for a given distribution we can represent it in n! = n factorial = n(n-1)(n-2) 2.1 different ways with the chain rule 23 Chain Rule à Bayes net Chain rule representation: applies to ALL distributions Pick any ordering of variables, rename accordingly as x 1, x 2,, x n Exponential in n number of parameters: (d-1) + d(d-1) + d 2 (d-1)+ +d n-1 (d-1) = d n -1 Bayes net representation: makes assumptions Pick any ordering of variables, rename accordingly as x 1, x 2,, x n Pick any directed acyclic graph consistent with the ordering Assume following conditional independencies: P (x i x 1 x i 1 )=P (x i parents( i )) à Joint: number of parameters: (maximum number of parents = K) Linear in n 24 Note: no causality assumption made anywhere. 11

12 Causality? When Bayes nets reflect the true causal patterns: Often simpler (nodes have fewer parents) Often easier to think about Often easier to elicit from experts BNs need not actually be causal Sometimes no causal net exists over the domain E.g. consider the variables Traffic and Drips End up with arrows that reflect correlation, not causation What do the arrows really mean? Topology may happen to encode causal structure Topology only guaranteed to encode conditional independence 25 Example: Traffic Basic traffic net Let s multiply out the joint R T r 1/4 r 3/4 r t 3/4 t 1/4 r t 1/2 t 1/2 r t 3/16 r t 1/16 r t 6/16 r t 6/

13 Example: Reverse Traffic Reverse causality? T t 9/16 t 7/16 r t 3/16 r t 1/16 r t 6/16 R t r 1/3 r 2/3 r t 6/16 t r 1/7 r 6/7 27 Example: Coins Extra arcs don t prevent representing independence, just allow non-independence h 0.5 t 0.5 h 0.5 t 0.5 h 0.5 t 0.5 h h 0.5 t h 0.5 Adding unneeded arcs isn t wrong, it s just inefficient h t 0.5 t t

14 Bayes Nets Representation Informal first introduction of Bayes nets through causality intuition More formal introduction of Bayes nets Conditional Independences Probabilistic Inference Learning Bayes Nets from Data 29 Bayes Nets: Assumptions To go from chain rule to Bayes net representation, we made the following assumption about the distribution: P (x i x 1 x i 1 )=P (x i parents( i )) Turns out that probability distributions that satisfy the above ( chain-ruleà Bayes net ) conditional independence assumptions often can be guaranteed to have many more conditional independences These guaranteed additional conditional independences can be read off directly from the graph Important for modeling: understand assumptions made when choosing a Bayes net graph 30 14

15 Example Z W Conditional independence assumptions directly from simplifications in chain rule: Additional implied conditional independence assumptions? 31 Independence in a BN Given a Bayes net graph Important question: Are two nodes guaranteed to be independent given certain evidence? Equivalent question: Are two nodes independent given the evidence in all distributions that can be encoded with the Bayes net graph? Before proceeding: How about opposite question: Are two nodes guaranteed to be dependent given certain evidence? No! For any BN graph you can choose all CPT s such that all variables are independent by having P( Pa() = pa) not depend on the value of the parents. Simple way of doing so: pick all entries in all CPTs equal to 0.5 (assuming binary variables) 15

16 Independence in a BN Given a Bayes net graph Are two nodes guaranteed to be independent given certain evidence? If no, can prove with a counter example I.e., pick a distribution that can be encoded with the BN graph, i.e., pick a set of CPT s, and show that the independence assumption is violated If yes, For now we are able to prove using algebra (tedious in general) Next we will study an efficient graph-based method to prove yes: D-separation D-separation: Outline Study independence properties for triples Any complex example can be analyzed by considering relevant triples 34 16

17 Causal Chains This configuration is a causal chain Z : Low pressure : Rain Z: Traffic Is it guaranteed that is independent of Z? No! One example set of CPTs for which is not independent of Z is sufficient to show this independence is not guaranteed. Example: P(y x) = 1 if y=x, 0 otherwise P(z y) = 1 if z=y, 0 otherwise Then we have P(z x) = 1 if z=x, 0 otherwise hence and Z are not independent in this example 35 Causal Chains This configuration is a causal chain Z : Low pressure : Rain Z: Traffic Is it guaranteed that is independent of Z given? es! Evidence along the chain blocks the influence 36 17

18 Common Cause Another basic configuration: two effects of the same cause Is it guaranteed that and Z are independent? No! Counterexample: Choose P( )=1 if x=y, 0 otherwise, Choose P(z y) = 1 if z=y, 0 otherwise. Then P(x z)=1 if x=z and 0 otherwise, hence and Z are not independent in this example and hence it is not guaranteed that if a distribution can be encoded with the Bayes net structure on the right that and Z are independent in that distribution Z : Project due : Piazza busy Z: Lab full 37 Common Cause Another basic configuration: two effects of the same cause Is it guaranteed that and Z are independent given? Z es! Observing the cause blocks influence between effects. : Project due : Piazza busy Z: Lab full 38 18

19 Common Effect Last configuration: two causes of one effect (v-structures) Are and Z independent? es: the ballgame and the rain cause traffic, but they are not correlated Still need to prove they must be (try it!) Are and Z independent given? No: seeing traffic puts the rain and the ballgame in competition as explanation? This is backwards from the other cases Observing an effect activates influence between possible causes. Z : Raining Z: Ballgame : Traffic 39 Reachability (D-Separation) Question: Are and conditionally independent given evidence vars {Z}? es, if and separated by Z Consider all (undirected) paths from to No active paths = independence! A path is active if each triple is active: Causal chain A B C where B is unobserved (either direction) Common cause A B C where B is unobserved Common effect (aka v-structure) A B C where B or one of its descendents is observed All it takes to block a path is a single inactive segment Active Triples Inactive Triples 19

20 D-Separation Given query i? j { k1,..., kn } Shade all evidence nodes For all (undirected!) paths between and Check whether path is active If active return: not guaranteed that i j { k1,..., kn } (If reaching this point all paths have been checked and shown inactive) Return: guaranteed tat i j { k1,..., kn } 41 Example es R B T T 42 20

21 Example L es es R B D T es T 43 Example Variables: R: Raining T: Traffic R D: Roof drips S: I m sad Questions: T S D es 44 21

22 All Conditional Independences Given a Bayes net structure, can run d- separation to build a complete list of conditional independences that are guaranteed to be true, all of the form i j { k1,..., kn } 45 Possible to have same full list of conditional independencies for different BN graphs? es! Examples: If two Bayes Net graphs have the same full list of conditional independencies then they are able to encode the same set of distributions

23 Topology Limits Distributions Given some graph topology G, only certain joint distributions can be encoded The graph structure guarantees certain (conditional) independences (There might be more independence) Adding arcs increases the set of distributions, but has several costs Z Z Z Z Z Z Z Full conditioning can encode any distribution Z Z Z 48 Bayes Nets Representation Summary Bayes nets compactly encode joint distributions Guaranteed independencies of distributions can be deduced from BN graph structure D-separation gives precise conditional independence guarantees from graph alone A Bayes net s joint distribution may have further (conditional) independence that is not detectable until you inspect its specific distribution 49 23

24 Representation Bayes Nets Conditional Independences Probabilistic Inference Enumeration (exact, exponential complexity) Variable elimination (exact, worst-case exponential complexity, often better) Probabilistic inference is NP-complete Sampling (approximate) Learning Bayes Nets from Data 53 24

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