# COL333/671: Introduction To AI

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COL333/671: Introduction to AISemester I, 2021Probabilistic ReasoningRohan Paul1

Outline Last Class Reinforcement Learning This Class Probabilistic Reasoning Reference Material AIMA Ch. 13 and 142

AcknowledgementThese slides are intended for teaching purposes only. Some materialhas been used/adapted from web sources and from slides by DoinaPrecup, Dorsa Sadigh, Percy Liang, Mausam, Dan Klein, AncaDragan, Nicholas Roy and others.3

Uncertainty in AI Uncertainty: Observed variables (evidence): Agent knows certain things aboutthe state of the world (e.g., sensor measurements or symptoms) Unobserved variables: Agent needs to reason about other aspects(e.g. what disease is present, is the car operational, location of theburglar) Model: Agent knows something about how the known variablesrelate to the unknown variables Probabilistic reasoning gives us a framework for managingour beliefs and knowledge.I hear an unusual sound and aburning smell in my car, whatfault is there in my engine?I have fever, loss of smell, lossof taste, do I have Covid?I hear some footsteps in myhouse, where is the burglar?4

Random Variables A random variable is some aspect of the world about whichwe (may) have uncertainty R Do I have Covid?T Engine is faulty or working?D How long will it take to drive to IIT?L Where is the person? Domains R in {true, false} (often write as { r, -r})T in {faulty, working}D in [0, )L in possible locations in a grid {(0,0), (0,1), }I hear an unusual sound and aburning smell in my car, whatfault is there in my engine?I have fever, loss of smell, lossof taste, do I have Covid?I hear some footsteps in myhouse, where is the burglar?5

Joint Distributions A joint distribution over a set of random variables:specifies a real number for each assignment (or outcome): Must te: Joint distribution can answer all probabilistic queries.Problem: Table size is dn.6

Events An event is a set E of outcomes From a joint distribution, we can calculate theprobability of any event Probability that it’s hot AND sunny? .4 Probability that it’s hot? Probability that it’s hot OR sunny?.4 .1.4 .1 .2TWPhotsun0.4hotrain0.1coldsun0.2coldrain0.37

Marginalization§ From a joint distribution ( 1 variable) reduce it to a distribution over a smaller set of variables§ Called marginal distributions are sub-tables which eliminate variables§ Marginalization (summing out): Combine collapsed rows by adding n0.3TPhot0.5cold0.5WPsun0.6rain0.48

Conditional ProbabilitiesJoint Distribution§ Conditional distributions are probability distributionsover some variables given fixed values of .2coldrain0.3Conditional DistributionsWPWPsun0.4sun0.8rain0.6rain90.2

Inference by Enumeration otrain0.05wintercoldsun0.15wintercoldrain0.20

Inference by Enumeration P(W)?P(sun) .3 .1 .1 .15 rain0.05wintercoldsun0.15wintercoldrain0.20

Inference by Enumeration P(W)?P(sun) .3 .1 .1 .15 .65P(rain) 1-.65 rain0.05wintercoldsun0.15wintercoldrain0.20

Inference by Enumeration P(W winter, otrain0.05wintercoldsun0.15wintercoldrain0.20

Inference by Enumeration P(W winter, hot)?P(sun winter,hot) .1P(rain winter,hot) .05P(sun winter,hot) 2/3P(rain winter,hot) rain0.05wintercoldsun0.15wintercoldrain0.20

Product Rule Marginal and a conditional provides the joint distribution. yrain0.0615

Chain RuleConstructing a larger distribution by simpler distribution.16

Bayes Rule Two ways to factor a joint distribution over two variables: Dividing, we get: Example: Diagnostic probability from causal probability:P (cause e ect) UsefulnessP (e ect cause)P (cause)P (e ect) Lets us build one conditional from its reverse. Often one conditional is difficult to obtain but the other one issimple.17

Independence Two variables are independent if: This says that their joint distribution factors into aproduct two simpler distributions Another form: We write:n smallerdistributionsH0.5T0.5H0.5T0.5H0.5T0.5 Example N-independent flips of a fair coin.18

Bayesian Networks Problem with using full joint distribution tables as our probabilistic models: Unless there are only a few variables, the joint is hard to represent explicitly. Bayesian Networks: A technique for describing complex joint distributions (models) using simple, localdistributions (conditional probabilities) Also known as probabilistic graphical models Encode how variables locally influence each other. Local interactions chain togetherto give global, indirect interactions19

Examples20

Bayesian Networks: Semantics 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 Bayesian Networks 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:

Example: The Alarm NetworkEP(E)0.001 e0.0020.999-e0.998BP(B) b-bBurglaryEarthquakeAlarmMaryCallsJohnCallsBEAP(A B,E) b e a0.95 b e-a0.05 b-e a0.94AJP(J A)AMP(M A) b-e-a0.06 a j0.9 a m0.7-b e a0.29 a-j0.1 a-m0.3-b e-a0.71-a j0.05-a m0.01-b-e a0.001-a-j0.95-a-m0.99-b-e-a0.999

Example: The Alarm NetworkBP(B) b0.001-b0.999AJP(J A) a j aBEAEP(E) e0.002-e0.998AMP(M A)0.9 a m0.7-j0.1 a-m0.3-a j0.05-a m0.01-a-j0.95-a-m0.99JMBEAP(A B,E) b e a0.95 b e-a0.05 b-e a0.94 b-e-a0.06-b e a0.29-b e-a0.71-b-e a0.001-b-e-a0.999

Example: The Alarm NetworkBP(B) b0.001-b0.999AJP(J A) a j aBEAEP(E) e0.002-e0.998AMP(M A)0.9 a m0.7-j0.1 a-m0.3-a j0.05-a m0.01-a-j0.95-a-m0.99JMBEAP(A B,E) b e a0.95 b e-a0.05 b-e a0.94 b-e-a0.06-b e a0.29-b e-a0.71-b-e a0.001-b-e-a0.999

Inference by Enumeration in a Bayes Net Inference by enumeration is one way to performinference in a Bayesian Network (Bayes Net).BEP (B j, m) /B P (B, j, m) Xe,a XAP (B, e, a, j, m)JMP (B)P (e)P (a B, e)P ( j a)P ( m a)e,a P (B)P ( e)P ( a B, e)P ( j a)P ( m a) P (B)P ( e)P ( a B, e)P ( j a)P ( m a)P (B)P ( e)P ( a B, e)P ( j a)P ( m a) P (B)P ( e)P ( a B, e)P ( j a)P ( m a)

Bayesian Networks: Inference Bayesian Networks Implicitly encode a probability distribution As a product of local conditional distributions Variables Query variables Evidence variables Hidden variables Inference: What we want to estimate? Estimating some useful quantity from the jointdistribution. Posterior probability Most likely explanationAll variables

Inference by Enumeration Setup: A distribution over query variables (Q)given evidence variables (E) Select entries consistent with the evidence. E.g., Alarm rang, it is rainy, disease present Compute the joint distributionSum out (eliminate) the hidden variables (H)Normalize the distributionNext Introduce a notion called factors Understand this computation using joining andmarginalization of factors.

Example: Traffic Domain Random Variables R: Raining T: Traffic L: Late for classRTP (L) ? XLP (r, t, L)r,t Xr,tP (r)P (t r)P (L t) r-r0.10.9 r r-r-r t-t t-t0.80.20.10.9 t t-t-t l-l l-l0.30.70.10.9

Example: Traffic Domain What are factors? A factor is a function from some setof variables into a specific value.Traffic domainR r-r Initial factors Conditional probability tables (oneper node) Select the values consistent withthe evidence Inference by Enumeration Procedure that joins all the factorsand then sums out all the hiddenvariables. Define “joining” and “summing”next.0.10.9TL r r-r-r t-t t-t0.80.20.10.9 t t-t-t l-l l-l0.30.70.10.9 l l0.30.1applied to the initial factors r-r0.10.9 r r-r-r t-t t-t0.80.20.10.9 t-t

Operation I: Joining Factors Joining Get all the factors over the joining variables. Build a new factor over the union of variables involved. Computation for each entry: pointwise productsR r-rT0.10.9 r r-r-r t-t t-t0.80.20.10.9 r r-r-r t-t t-t0.080.020.090.81R,T

Joining FactorsSource: AIMA Ch 14.

Joining Multiple FactorsRTL r-r r r-r-r0.10.9 t-t t-t0.80.20.10.9 t l 0.3 t -l 0.7-t l 0.1-t -l 0.9Join R r t 0.08 r -t 0.02-r t 0.09-r -t 0.81Join TR, TL t l 0.3 t -l 0.7-t l 0.1-t -l 0.9R, T, L r r r r-r-r-r-r t t-t-t t t-t-t l-l l-l l-l l-l0.0240.0560.0020.0180.0270.0630.0810.729

Operation II: Eliminating Factors Marginalization Take a factor and sum out a variable Shrinks the factor to a smaller oneR, T, L r r r r-r-r-r-r t t-t-t t t-t-t l-l l-l l-l l-l r t 0.08 r -t 0.02-r t 0.09-r -t 0.81Sum out R t-tT, L0.0240.0560.0020.0180.0270.0630.0810.729Sum out R t l 0.051 t -l 0.119-t l 0.083-t -l 0.747LSum out T l-l0.1340.8660.170.83

Inference by EnumerationRMultiple join operations and multiple eliminate operationsTLP (L) ?

Variable Elimination Inference by Enumeration Problem: the whole distribution is “joined up“ before “sum out” the hidden variables Variable Elimination Interleaves joining and eliminating variables Does not create the full joint distribution in one go Key Idea: Picks a variable ordering. Picks a variable. Joins all factors containing that variable. Sums out the influence of the variable on new factor. Leverage the structure (topology) of the Bayesian Network Marginalize early (avoid growing the full joint distribution)

Inference by Enumeration vs. VariableEliminationRP (L) ?Inference by EnumerationT LXXtP (L t)P (r)P (t r)Variable Elimination XtrJoin on rJoin on tEliminate rEliminate tP (L t)XP (r)P (t r)rJoin on rEliminate rJoin on tEliminate t

Variable EliminationJoin R r-rRTL r r-r-r0.10.9 t-t t-t0.80.20.10.9 t l 0.3 t -l 0.7-t l 0.1-t -l 0.9Sum out R r t 0.08 r -t 0.02-r t 0.09-r -t 0.81 t-t0.170.83R, TTLL t l 0.3 t -l 0.7-t l 0.1-t -l 0.9Sum out TJoin T t l 0.3 t -l 0.7-t l 0.1-t -l 0.9T, L t l 0.051 t -l 0.119-t l 0.083-t -l 0.747L l-l0.1340.866

Incorporating Evidence Till Now, we computed P(Late)? What happens when P(Late Rain)? How to incorporate evidence in VariableElimination. Solution If evidence, then start with factors and selectthe evidence. After selecting evidence, eliminate all variablesother than query and evidence. r-r0.10.9 r r-r-r t-t t-t0.80.20.10.9 t t-t-t l-l l-l0.30.70.10.9 r0.1 r r t-t0.80.2 t t-t-t l-l l-l0.30.70.10.9Evidence incorporated in the initial factors

General Variable Elimination Query: Start with initial factors: Local conditional probability tables. Evidence (known) variables are instantiated. While there are still hidden variables (not Q or evidence): Pick a hidden variable H (from some ordering) Join all factors mentioning H Eliminate (sum out) H Join all the remaining factors and normalize

Example: Alarm Domainmarginal can be obtained from joint by summing outP (B j, m) / P (B, j, m) XP (B, j, m, e, a)e,aXP (B)P (e)P (a B, e)P (j a)P (m a)e,aXX P (B)P (e)P (a B, e)P (j a)P (m a) eXaP (B)P (e)f1 (j, m B, e)e P (B)XP (e)f1 (j, m B, e) P (B)fe2 (j, m B)use Bayes’ net joint distribution expressionuse x*(y z) xy xzjoining on a, and then summing out gives f1use x*(y z) xy xzjoining on e, and then summing out gives f2

Example: Alarm DomainChoose A

Example: Alarm DomainChoose EFinish with BNormalize

Variable Elimination: Structuring ComputationSource: AIMA Ch 14.

ExampleStart by inserting evidence, which gives the following initial factors:P (Z), P (X1 Z), P (X2 Z), P (X3 Z), P (y1 X1 ), P (y2 X2 ), P (y3 X3 )There are three variables to eliminate { X1, X2 and Z }. The Yvariables are observed (instantiated).

ExampleStart by inserting evidence, which gives the following initial factors:P (Z), P (X1 Z), P (X2 Z), P (X3 Z), P (y1 X1 ), P (y2 X2 ), P (y3 X3 )PEliminate X1 , this introduces the factor f1 (y1 Z) x1 P (x1 Z)P (y1 x1 ),and we are left with:P (Z), P (X2 Z), P (X3 Z), P (y2 X2 ), P (y3 X3 ), f1 (y1 Z)PEliminate X2 , this introduces the factor f2 (y2 Z) x2 P (x2 Z)P (y2 x2 ),and we are left with:P (Z), P (X3 Z), P (y3 X3 ), f1 (y1 Z), f2 (y2 Z)Eliminate Z, this introduces the factor f3 (y1 , y2 , X3 ) and we are left with:P (y3 X3 ), f3 (y1 , y2 , X3 )PzP (z)P (X3 z)f1 (y1 Z)f2 (y2 Z),No hidden variables left. Join the remaining factors to get:f4 (y1 , y2 , y3 , X3 ) P (y3 X3 ), f3 (y1 , y2 , X3 )Normalizing over X3 gives P (X3 y1 , y2 , y3 ) f4 (y1 , y2 , y3 , X3 )/Px3f4 (y1 , y2 , y3 , x3 )

ExampleStart by inserting evidence, which gives the following initial factors:P (Z), P (X1 Z), P (X2 Z), P (X3 Z), P (y1 X1 ), P (y2 X2 ), P (y3 X3 )PEliminate X1 , this introduces the factor f1 (y1 Z) x1 P (x1 Z)P (y1 x1 ),and we are left with:P (Z), P (X2 Z), P (X3 Z), P (y2 X2 ), P (y3 X3 ), f1 (y1 Z)PEliminate X2 , this introduces the factor f2 (y2 Z) x2 P (x2 Z)P (y2 x2 ),and we are left with:P (Z), P (X3 Z), P (y3 X3 ), f1 (y1 Z), f2 (y2 Z)Eliminate Z, this introduces the factor f3 (y1 , y2 , X3 ) and we are left with:P (y3 X3 ), f3 (y1 , y2 , X3 )PzP (z)P (X3 z)f1 (y1 Z)f2 (y2 Z),No hidden variables left. Join the remaining factors to get:f4 (y1 , y2 , y3 , X3 ) P (y3 X3 ), f3 (y1 , y2 , X3 )Normalizing over X3 gives P (X3 y1 , y2 , y3 ) f4 (y1 , y2 , y3 , X3 )/Px3f4 (y1 , y2 , y3 , x3 )Computational complexity Depends on the largest factorgenerated in VE. Factor size number of entriesin the table. In this example: each factor is ofsize 2 (only one variable). Notethat y is observed. X1, X2, Z, X3

Effect of Different Orderings For the query P(Xn y1, ,yn) Two different orderings as Eliminate Z first. Z, X1, , Xn-1 Eliminate Z last. X1, , Xn-1, Z. What is the size of the maximum factorgenerated for each of the orderings?

ExampleEliminate Z FirstP (Xn , y1 , y2 , . . . , yn ) P (Z)P (X1 Z)P (X2 Z), . . . , P (Xn Z)P (y1 X1 )P (y2 X2 ), . . . , P (yn Xn ) latexit sha1 base64 "Ly4zfsq4gYOMHOWlcBaWIWtuH5M " Oe7a sbmVmW7urO7t39QOzx6MnGqGe YmJYBQsFdQ uo1 2Ni7arqqw0WfuS8EvCXzVmpdG2aAa1utf2FoH/AlKAOiqiG9Q khS4YsuDJqnEEOP8W/BYaM5AZhZQpoW9K2ZTqikD 3lVOwTy p7rjEvXZvllLXKTzH6Ee4d9 od7qi /latexit sha1 base64 "xI2NPK8ymgrhn38KOHzUpeOU4dQ " AAACiXicbVFbS8MwFE7rfd7m5c2X4BA2GKMp3hCEgS8 TnA63EZNs9QF07Qkp0IZ/hd/k2/ G9NtlHk5kPCd73zfSXISplIY8Lwvx11aXlldW9 obG5t7 wa GoQTPL3yiAzPKXslb7wvoWKxtwMJ9NJvuMTy4xwlGi7FOApu wr35Bq8DvCI /latexit sha1 base64 "9vc4pWh5OHMfSSVEV5G wwwzCPo " hC04QmwciKnIjIspGuByb0X/pNe vfVHaCSdNdkDj33HOupKswlcKA57047oejj8efaif1xucvX0 bZ98eTJJpxkcskYmehNRwKRQfgQDJJ6nmNA4lH4frQVEf/ wKqDb/GMynRF8bD92MGFgez0u9Sv0spYtt1X5ZXJ2reEXxH vjGvjLZFJ2i2vJ5XBn4PyA60 t 27mm3KST/jSMgscJdouBbhk9x0bGhuTx6FVxhRW5rBWkP f28uh0COXzye/Dg94jXI3ek1b9C26ihH gCtRFBv1Af/UFDNELMOXa6zrXz0224xL1xf2 lrrPznKM34Q5eAU/9vVo /latexit This factor is2nf1 (X1 , X2 , . . . , Xn ) latexit sha1 base64 "pCpQ7Y/y6ncXDFBIdTDi7O3lAZc " Ylm2bb0Gx2Sd4K7bI/0IsHb/4KLx4UEUzrHtT6IGEyb YlGT AUCJO/GmtPQl7zjT67n/c4910ZE6g6mMR EdKREIBgFS3llFngpyapdj9Rx1zut4/4wAjPHqcpq BL3TRJ66SzDzeqsZjerxH07EfCsVl865 pWGVIYm7 ammDGz JRsC fvlZdA PhfH5LC07uOUS/qlD8AjlCrhQ /latexit sha1 base64 "zcRfo cv5LAfEBVaUNK9M1d97HM " AAACTHicbVDPSyMxGP2m7rrd7q Zt/hZc9rIhg uOwbX2Q8PK Z YLjVwxawfUT3GUM4OSK1FUhpkVKePX7FIMHNUsFnaUz8MoyC NeFteQte/ZgBaXtN SDr1A /latexit sha1 base64 "h169Sik4Mueb9yglVTZVFaJ0yPY " 1DJs1oMJMZkjtCO8y/ tv76xubVd2Nm9NlGiGW bpWHZblaJu3Yi4GG5Mneeumdpr1B0q 648DwgU1Cs7wfZd 0haHiFZzuHJSFXwCQ1pkPcGHop1SCY5Fm 08K FbNbqikDm3/ehkD fz5Lw5n87XRJpzpp49NFO55R/ZQ7GF /latexit PzP (z)P (X1 z), P (X1 z), . . . , P (X1 z)P (Xn , y1 , y2 , . . . , yn ) f1 (X1 , X2 , . . . , Xn )P (y1 X1 )P (y2 X2 ), . . . , P (yn Xn ) latexit sha1 base64 "CxJDQjCO iCmzd3sQ/Of5R/0GOY " AAACb3icbVFPS 7xZXPra rd/cS tUbq5gVshxxm NSpXggFTc/k/7x1FsTugIXUBnMTvBTw/3kxxcjU1AL iI62LKaRpXbI5 hmfSjavFvOb0CJkJTXOLywBdsK8TFc D1DwIVVeFcqptxyAfiL6iGw90/ CK57XRZ22T/WuTxtxrFF9slvckwYOSOX5C/pkwER5Mnb8/a9A /Z/ Uf nRp9b0m85O8KT94AUaktAY /latexit sha1 base64 "5oY2ocNIeECgCBDyAzqgqCGc/RM " 6TkptHIQho e/2Hj4 anrc N7S87u3vNr/s3Li cbvjEqV4IBU3LynvbMoNud0iC6gs5id46eD 1tlDsh/5QfPTTC1hg /latexit sha1 base64 "VuLMv10Zt18ws9Ruiq87gL1qCQ0 " maTQykEYPnv l7Wv6xubW43tbzu7e83v zcuL62QfZHr3EYJd1IrI/ugQMuosJJniZa3yf1Vrd8 SOtUbv7CtJCjjN8ZlSrBAam4 Uh7Z1FszukQXUCnMTvHTwf34xxcjU1AL mQ62LCaRpXbIZ NEXvTFFcmVmAnTC bISeBdFZEZ1gFagFsxJM0IibrbAdzot BmwJWt3DdF69uPmErUSZSQNCc BuicfZ KwUoKcIuLAK70rFhFsuAH9RPQT28cmfwU2nzcI2 8Na3QuyqE1yRH6SM8LIL9Il16RH kSQF2/fO/KOvVf/h3/i04XV95aZA/JP cEb7hu2vg /latexit f2 (X1 , X2 , . . . , Xn latexit sha1 base64 "t/hsUSnyDqQ7N4l6OyAcK1crcBw " AAACVXicdZHBSyMxFMYzs T/ay7H yF8FMOwe1 iDw43tf8pIvUaakxSD45/mfdnb3Pjf2m1 C31Oyy78ApjZPAmLP04pS3B FuQJMoYoYFtIIjmrpgHEj3V2B3zPDOLqPqEKgb5 70nf8ff21h9r95zTF6Vf/QMRymvEA /latexit sha1 base64 "ZBlxgg4Fsy1juw0bXI8rDxw5nPM " AAACVXicdZFdSyMxFIYz4 noSxMtW/cZaJScLutYwlZ issKHisOiWrWFI2zAMu20YT1O0FRf6hJbHcAEwtnkSFn cU5bg8vT7fBvmsWPot2YVwNg1x6VZCxvNoBPMBZ 7sXI4eaJcJOivlUSjhyzhTi1LilEebu x0FS6ydJZFLJgwf7MdaZX5VG Q6Afn/wZbrsdGnToDW32zshCm 9Tdjba/i /latexit sha1 base64 "Toix75VNBj0AM6aIYoscatq7 i4 " AAACVXicdZFNa lo00Lg3Gf7KXsv kl7KVkxz6OSB4eOedGWkUZUpaDIJ/nr yura sblV pkDxo10dwV wwzj6D6iWgJ9 T3cNlp06BNz2mje0wWsUn2yU/SJJSckC75S3qkTzi5J4 e5/neg/fkr/rrC6vvLWu k1fh7z4DiSixfA /latexit P (Xn , y1 , y2 , . . . , yn ) f2 (X1 , X2 , . . . , Xn latexit sha1 base64 "WDQU/wdiygvwsDvmGe lN34GANc " 701LrRyE4Z3nv3j5auf17pvg7bv3ex86 x8vXFFZIcei0IWNU 6kVkaOQYGWcWklz1MtL9PrH41 f/WjmOXfCZfSI8wckrOyU8yImMiyF/vwDvyet693/W/ oON1ffank/kUfkn/wDANLed /latexit sha1 base64 "Bl8sFGcK2hqbB8PDPv6a0xQB2Bo " lZZaOQjDG89/tPf4ydP9g DZ8xcvX3UOX5 Z1 o KUCvEHBhFZ6ViiW3XAB V/MI7O6V74OzaMjCITtl3dEn0tY eUvekR5h5AsZkR9kTCZEkD/esffB63l//a7/0R 0Vt/b9rwh/5X/ R/GwLkd /latexit sha1 base64 "dxL9fV9 G1Bp8OEAIMF/XSjRReo " AAACe3icbVFda9swFJW9revctcs 3gZDNIwlJQuW2eheBoG97DGDpTUkwciK3IjKspGuB8HkT yn9W3/ZC DXdshdO0uSBydc 7RV1pq5SAMf3n oAvoJmEjnCJcrwpwDTZD pkuuC7XnGZJHW3Rj6b4lilOavOebYeYhQG7KHR1RLQnouG tGVvd9Q8d26Tp jMOazdXa0h/6fNK8g t75fvgIhqzcMy sf7kA nqkLwmp2RAGDknE/KVTMmMCPLbe O98wbeH7/vn/mjzup7u56X5J/yP/4FZ7q6VQ /latexit 1) Pxnf1 (X1 , X2 , . . . , Xn1 , xn )P (yn xn )1 )P (y1 X1 )P (y2 X2 ), . . . , P (yn 1 Xn 1 )

ExampleEliminate Z LastP (Xn , y1 , y2 , . . . , yn ) P (Z)P (X1 Z)P (X2 Z), . . . , P (Xn Z)P (y1 X1 )P (y2 X2 ), . . . , P (yn Xn ) latexit sha1 base64 "Ly4zfsq4gYOMHOWlcBaWIWtuH5M " Oe7a sbmVmW7urO7t39QOzx6MnGqGe YmJYBQsFdQ uo1 2Ni7arqqw0WfuS8EvCXzVmpdG2aAa1utf2FoH/AlKAOiqiG9Q khS4YsuDJqnEEOP8W/BYaM5AZhZQpoW9K2ZTqikD 3lVOwTy p7rjEvXZvllLXKTzH6Ee4d9 od7qi /latexit sha1 base64 "xI2NPK8ymgrhn38KOHzUpeOU4dQ " AAACiXicbVFbS8MwFE7rfd7m5c2X4BA2GKMp3hCEgS8 TnA63EZNs9QF07Qkp0IZ/hd/k2/ G9NtlHk5kPCd73zfSXISplIY8Lwvx11aXlldW9 obG5t7 wa GoQTPL3yiAzPKXslb7wvoWKxtwMJ9NJvuMTy4xwlGi7FOApu wr35Bq8DvCI /latexit sha1 base64 "9vc4pWh5OHMfSSVEV5G wwwzCPo " hC04QmwciKnIjIspGuByb0X/pNe vfVHaCSdNdkDj33HOupKswlcKA57047oejj8efaif1xucvX0 bZ98eTJJpxkcskYmehNRwKRQfgQDJJ6nmNA4lH4frQVEf/ wKqDb/GMynRF8bD92MGFgez0u9Sv0spYtt1X5ZXJ2reEXxH vjGvjLZFJ2i2vJ5XBn4PyA60 t 27mm3KST/jSMgscJdouBbhk9x0bGhuTx6FVxhRW5rBWkP f28uh0COXzye/Dg94jXI3ek1b9C26ihH gCtRFBv1Af/UFDNELMOXa6zrXz0224xL1xf2 lrrPznKM34Q5eAU/9vVo /latexit This factor is size 2f1 (y1 Z) latexit sha1 base64 "XhoYpJqP0cCgLLeomjYCnWAUGKI " RVl14iJo65yCTzV4A22ZBG46dlEGmwueFmpaGC GyxquWbHq1gxwlaCCVECBpmu caJoSOaL/IRDFcG8OthnghLFJ5pgIpj egCVqAgBfwBqbg03g1Powv43seXTOKmRPwB8bPL2KRo8U /latexit sha1 base64 "shBTZfULs9qhEs o4r6VpgABk7I " Jk0kwbmnmQZKRl6A 58VdEcFERt/6GmbaL2nogcDjnXG7ucSPBlbbtsbWyura EJ 3q1Pqwv63saXbFmM0fwB9bPL37VpJc /latexit sha1 base64 "DcJ5lsBEHUVDQXsRz32dxSVP9oU " syZNJMG5p5kGSkZegPufFXRHBREbd oyjAWhVRLyUDRcLClnAa0qpjhtRIJi3 W07vZvUr/ RIVkYfCghhFt 7gbMI8RrLTkmLeek6BRYegg2NI5BR A MwYfxYrwbn8bXNJoxZjOH4A M71 kkKYx /latexit Px1P (y1 x1 )P (x1 Z)P (Xn , y1 , y2 , . . . , yn ) P (Z)f1 (y1 Z)P (X2 Z), . . . , P (Xn Z)P (y2 X2 ), . . . , P (yn Xn ) latexit sha1 base64 "Ppm2 Mn0OsRsn/H 84BgtqR/0zk " k0bWgSjKzIjagsG l6YEx R//X3vZjBr1OTcjaXpA499xzrqSrONfKQRj pygCWkash9sA83kGrsamS3/QKdf5gtNhcN lSVSxZYDKxoNU3WOwTtfmVetNVblWoX5TV6516Oi2MKJ2J WHDLBeCf1UNgr5/8FtwO izss1 sc3XRjGOPnJIzEhBGvpEr8pMMyYgI8s/reD3vi /7gf/VZy9S32s8n8l/4V8 A0nMuB8 /latexit sha1 base64 "72qIDP9/PCg9E6UaPXVunb0qGeU " 5msLRhSfBkRW5EZdlIxwVj8hx9r97tYQY9TkLI2h2w fWf7z ypTjXykEY/vH8nTdv3 189XA9y8DV2rTpdzrhOp9zOgh ph7l72avN/vXEBycW0UiYvQBqx2igpNIWM1rdBZ8pKAbpEwYVV K1UzLnlAvDO6kNgL3/5tbjpdVnYZT9Yq39GVrVHvpCvJCCMnJM uSYDMiSC/PVaXsf75vt 4J/6bIX63jrzmfxT/uUzUFi5nw /latexit sha1 base64 "F mpsp82caKr3Bs1IIN4kUZ7EAs " AAACgHicbVFda9swFJW9dW2zbk23vu1FLAwcyFIrFFoKg0Bf Kc60chOFvz3/1eu/N/sFh6 C/27vnLrw0S3hYjanbAfNqAvCduQzvA0aTCK2r wmyhSaUBo7tyUhTnMK25BCS1XrVnhZM7FA7 XU6SGp9LNq vh/4Zz5bW31vk/lI/oF/9RfxQ7rX /latexit Other steps are like the previous example. Each factor is of size 2 consisting of one variable.Variable ordering can have considerable impact.

Properties Variable elimination is dominated by the size of the largestfactor constructed during the operation of the algorithm. Depends on the structure of the network and order ofelimination of the variables. Finding the optimal ordering is intractable. Can pose the problem of finding good ordering as a search. Use heuristics. Min-fill heuristic Eliminate the variable that creates the smallest sized factor (greedyapproach). Min-neighbors Eliminate the variable that has the

summer cold rain 0.05 winter hot sun 0.10 winter hot rain 0.05 winter cold sun 0.15 winter cold rain 0.20 P(sun) .3 .1 .1 .15 .65. Inference by Enumeration P(W)? S T W P summer hot sun 0.30 summer hot rain 0.05 summer cold sun 0.10 summer cold rain 0.05 winter hot sun 0.10

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