Inference in Bayesian Networks CE417: Introduction to Artificial - PowerPoint PPT Presentation

Inference in Bayesian Networks CE417: Introduction to Artificial Intelligence Sharif University of Technology Spring 2018 Soleymani Slides are based on Klein and Abdeel, CS188, UC Berkeley.

Bayes ’ Nets  Representation  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 2

Recap: Bayes ’ Net Representation  A directed, acyclic graph, one node per random variable  A conditional probability table (CPT) for each node A collection of distributions over X, one for each combination of  parents ’ values  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: 3

Example: Alarm Network E P(E) B P(B) B urgla E arth +e 0.002 +b 0.001 ry qk -e 0.998 -b 0.999 A lar m B E A P(A|B,E) J oh M a +b +e +a 0.95 n ry +b +e -a 0.05 call call +b -e +a 0.94 s s A J P(J|A) A M P(M|A) +b -e -a 0.06 +a +j 0.9 +a +m 0.7 -b +e +a 0.29 +a -j 0.1 +a -m 0.3 -b +e -a 0.71 -a +j 0.05 -a +m 0.01 -b -e +a 0.001 -a -j 0.95 -a -m 0.99 -b -e -a 0.999 [Demo: BN Appl 4

Video of Demo BN Applet 5

Example: Alarm Network B P(B) E P(E) B E +b 0.001 +e 0.002 -b 0.999 -e 0.998 A A J P(J|A) A M P(M|A) B E A P(A|B,E) +a +j 0.9 +a +m 0.7 +b +e +a 0.95 +a -j 0.1 +a -m 0.3 J M +b +e -a 0.05 -a +j 0.05 -a +m 0.01 +b -e +a 0.94 -a -j 0.95 -a -m 0.99 +b -e -a 0.06 -b +e +a 0.29 -b +e -a 0.71 -b -e +a 0.001 -b -e -a 0.999 6

Example: Alarm Network B P(B) E P(E) B E +b 0.001 +e 0.002 -b 0.999 -e 0.998 A A J P(J|A) A M P(M|A) B E A P(A|B,E) +a +j 0.9 +a +m 0.7 +b +e +a 0.95 +a -j 0.1 +a -m 0.3 J M +b +e -a 0.05 -a +j 0.05 -a +m 0.01 +b -e +a 0.94 -a -j 0.95 -a -m 0.99 +b -e -a 0.06 -b +e +a 0.29 -b +e -a 0.71 -b -e +a 0.001 -b -e -a 0.999 7

Bayes ’ Nets Representation  Conditional Independences  Probabilistic Inference  Enumeration (exact, exponential complexity)  Variable elimination (exact, worst-case exponential  complexity, often better)  Inference is NP-complete Sampling (approximate)  Learning Bayes ’ Nets from Data  8

Inference  Examples:  Inference: calculating some useful quantity from a joint  Posterior probability probability distribution  Most likely explanation: 9

Inference by Enumeration * Works fine with General case:   We want: multiple query  Evidence variables: variables, too Query* variable:  All variables  Hidden variables:    Step 2: Sum out H to get joint Step 3: Normalize Step 1: Select the entries consistent of Query and evidence with the evidence 10

Inference by Enumeration in Bayes ’ Net  Given unlimited time, inference in BNs is easy B E  Reminder of inference by enumeration by example: A J M 11

Burglary example: full joint probability 𝐵 𝐹 𝑄 𝑘, ¬𝑛, 𝑐, 𝐵, 𝐹 𝑄 𝑐 𝑘, ¬𝑛 = 𝑄 𝑘, ¬𝑛, 𝑐 = 𝑄 𝑘, ¬𝑛 𝐶 𝐵 𝐹 𝑄 𝑘, ¬𝑛, 𝑐, 𝐵, 𝐹 𝐵 𝐹 𝑄 𝑘 𝐵 𝑄 ¬𝑛 𝐵 𝑄 𝐵 𝑐, 𝐹 𝑄 𝑐 𝑄(𝐹) = 𝐶 𝐵 𝐹 𝑄 𝑘 𝐵 𝑄 ¬𝑛 𝐵 𝑄 𝐵 𝐶, 𝐹 𝑄 𝐶 𝑄(𝐹) Short-hands 𝑘: 𝐾𝑝ℎ𝑜𝐷𝑏𝑚𝑚𝑡 = 𝑈𝑠𝑣𝑓 ¬𝑐: 𝐶𝑣𝑠𝑕𝑚𝑏𝑠𝑧 = 𝐺𝑏𝑚𝑡𝑓 … 12

Inference by Enumeration? 13

Factor Zoo 14

Factor Zoo I  Joint distribution: P(X,Y) T W P Entries P(x,y) for all x, y  hot sun 0.4 Sums to 1 hot rain 0.1  cold sun 0.2 cold rain 0.3  Selected joint: P(x,Y) A slice of the joint distribution  T W P Entries P(x,y) for fixed x, all y  cold sun 0.2 Sums to P(x)  cold rain 0.3  Number of capitals = dimensionality of the table 15

Factor Zoo II  Single conditional: P(Y | x) Entries P(y | x) for fixed x, all y  T W P Sums to 1  cold sun 0.4 cold rain 0.6  Family of conditionals: P(X |Y) Multiple conditionals  T W P  Entries P(x | y) for all x, y hot sun 0.8  Sums to |Y| hot rain 0.2 cold sun 0.4 cold rain 0.6 16

Factor Zoo III  Specified family: P( y | X ) Entries P(y | x) for fixed y,  but for all x  Sums to … who knows! T W P hot rain 0.2 cold rain 0.6 17

Factor Zoo Summary  In general, when we write P(Y 1 … Y N | X 1 … X M )  It is a “ factor, ” a multi-dimensional array  Its values are P(y 1 … y N | x 1 … x M )  Any assigned (=lower-case) X or Y is a dimension missing (selected) from the array 18

Example: Traffic Domain  RandomVariables +r 0.1  R: Raining -r 0.9 R  T:Traffic +r +t 0.8  L: Late for class! T +r -t 0.2 -r +t 0.1 -r -t 0.9 L +t +l 0.3 +t -l 0.7 -t +l 0.1 -t -l 0.9 19

Inference by Enumeration: Procedural Outline  Track objects called factors  Initial factors are local CPTs (one per node) +r 0.1 +r +t 0.8 +t +l 0.3 -r 0.9 +r -t 0.2 +t -l 0.7 -r +t 0.1 -t +l 0.1 -r -t 0.9 -t -l 0.9  Any known values are selected  E.g. if we know , the initial factors are +r 0.1 +r +t 0.8 +t +l 0.3 -r 0.9 +r -t 0.2 -t +l 0.1 -r +t 0.1 -r -t 0.9  Procedure: Join all factors, then eliminate all hidden variables 20

Operation 1: Join Factors  First basic operation: joining factors  Combining factors: Just like a database join  Get all factors over the joining variable  Build a new factor over the union of the  variables involved  Example: Join on R Computation for each entry: pointwise  products R +r 0.1 +r +t 0.8 +r +t 0.08 R,T -r 0.9 +r -t 0.2 +r -t 0.02 -r +t 0.1 -r +t 0.09 T -r -t 0.9 -r -t 0.81 21

Example: Multiple Joins 22

Example: Multiple Joins +r 0.1 R -r 0.9 Join R Join T +r +t 0.08 R, T, L +r -t 0.02 -r +t 0.09 +r +t 0.8 T R, T -r -t 0.81 +r -t 0.2 -r +t 0.1 0.024 +r +t +l -r -t 0.9 0.056 +r +t -l L L 0.002 +r -t +l 0.018 +r -t -l +t +l 0.3 +t +l 0.3 0.027 -r +t +l +t -l 0.7 +t -l 0.7 0.063 -r +t -l -t +l 0.1 -t +l 0.1 0.081 -r -t +l -t -l 0.9 -t -l 0.9 0.729 -r -t -l 23

Operation 2: Eliminate  Second basic operation: marginalization  Take a factor and sum out a variable Shrinks a factor to a smaller one  A projection operation   Example: +r +t 0.08 +t 0.17 +r -t 0.02 -t 0.83 -r +t 0.09 -r -t 0.81 24

Multiple Elimination R, T, L T, L L 0.024 +r +t +l Sum Sum 0.056 +r +t -l out R out T 0.002 +r -t +l 0.018 +r -t -l +t +l 0.051 +l 0.134 0.027 -r +t +l +t -l 0.119 -l 0.886 0.063 -r +t -l -t +l 0.083 0.081 -r -t +l -t -l 0.747 0.729 -r -t -l 25

Thus Far: Multiple Join, Multiple Eliminate (= Inference by Enumeration) 26

Inference by Enumeration vs. Variable Elimination  Idea: interleave joining and marginalizing!  Why is inference by enumeration so slow?  Called “ Variable Elimination ” You join up the whole joint distribution before   you sum out the hidden variables Still NP-hard, but usually much faster than inference by enumeration  First we ’ ll need some new notation: factors 27

Traffic Domain R  Variable Elimination  Inference by Enumeration T L Join on r Join on r Join on t Eliminate r Eliminate r Join on t Eliminate t Eliminate t 28

Marginalizing Early (= Variable Elimination) 29

Marginalizing Early! (aka VE) Join R Sum out T Sum out R Join T +r +t 0.08 +r 0.1 +r -t 0.02 +t 0.17 -r 0.9 -r +t 0.09 -t 0.83 -r -t 0.81 R T T, L R, T L +r +t 0.8 +r -t 0.2 -r +t 0.1 T L -r -t 0.9 L +t +l 0.051 +l 0.134 +t -l 0.119 -l 0.866 -t +l 0.083 L +t +l 0.3 +t +l 0.3 -t -l 0.747 +t +l 0.3 +t -l 0.7 +t -l 0.7 +t -l 0.7 -t +l 0.1 -t +l 0.1 -t +l 0.1 -t -l 0.9 -t -l 0.9 -t -l 0.9 30

Evidence  If evidence, start with factors that select that evidence  No evidence uses these initial factors: +r 0.1 +r +t 0.8 +t +l 0.3 -r 0.9 +r -t 0.2 +t -l 0.7 -r +t 0.1 -t +l 0.1 -r -t 0.9 -t -l 0.9  Computing , the initial factors become: +r 0.1 +r +t 0.8 +t +l 0.3 +r -t 0.2 +t -l 0.7 -t +l 0.1 -t -l 0.9  We eliminate all vars other than query + evidence 31

Evidence II  Result will be a selected joint of query and evidence E.g. for P(L | +r), we would end up with:  Normalize +r +l 0.026 +l 0.26 +r -l 0.074 -l 0.74  To get our answer, just normalize this!  That ’ s it! 32

Distribution of products on sums  Exploiting the factorization properties to allow sums and products to be interchanged needs three operations while 𝑏 × (𝑐 + 𝑑)  𝑏 × 𝑐 + 𝑏 × 𝑑 requires two 33