Today. CS 188: Artificial Intelligence Spring Probabilities. Uncertainty. Probabilistic Models. What Are Probabilities?

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1 C 188: Atificial Intelligence ping 2006 Lectue 8: oaility 2/9/2006 an Klein UC Bekeley Many slides fom eithe tuat Russell o Andew Mooe oday Uncetainty oaility Basics Joint and Condition istiutions Models and Independence Bayes Rule Estimation Utility Basics Value unctions Expectations Uncetainty oailities Let action A t = leave fo aipot t minutes efoe flight Will A t get me thee on time? olems: patial osevaility (oad state, othe dives' plans, etc.) noisy sensos (KCB taffic epots) uncetainty in action outcomes (flat tie, etc.) immense complexity of modeling and pedicting taffic A puely logical appoach eithe Risks falsehood: A 25 will get me thee on time o Leads to conclusions that ae too weak fo decision making: A 25 will get me thee on time if thee's no accident on the idge, and it doesn't, and my ties emain intact, etc., etc.'' A 1440 might easonaly e said to get me thee on time ut I'd have to stay ovenight in the aipot oailistic appoach Given the availale evidence, A 25 will get me thee on time with poaility 0.04 (A 25 no epoted accidents) = 0.04 oailities change with new evidence: (A 25 no epoted accidents, 5 a.m.) = 5 (A 25 no epoted accidents, 5 a.m., ing) = 0.08 i.e., oseving evidence causes eliefs to e updated oailistic Models What Ae oailities? Cs: Vaiales with domains Constts: map fom assignments to tue/false Ideally: only cetain vaiales diectly inteact oailistic models: (Random) vaiales with domains Joint distiutions: map fom assignments (o outcomes) to positive numes Nomalized: sum to 1.0 Ideally: only cetain vaiales ae diectly coelated A A B B Ojectivist / fequentist answe: Aveages ove epeated expeiments E.g. empiically estimating () fom histoical osevation Assetion aout how futue expeiments will go (in the limit) New evidence changes the efeence class Makes one think of inheently andom events, like olling dice ujectivist / Bayesian answe: egees of elief aout unoseved vaiales E.g. an agent s elief that it s ing, given the tempeatue Often estimate poailities fom past expeience New evidence updates eliefs Unoseved vaiales still have fixed assignments (we just don t know what they ae) 1

2 oailities Eveywhee? istiutions on Random Vas Not just fo games of chance! I m snuffling: am I sick? contains REE! : is it spam? ooth huts: have cavity? afe to coss steet? 60 min enough to get to the aipot? Root otated wheel thee times, how fa did it advance? Why can a andom vaiale have uncetainty? Inheently andom pocess (dice, etc) Insufficient o weak evidence Unmodeled vaiales Ignoance of undelying pocesses he wold s just noisy! Compae to fuzzy logic, which has degees of tuth, o soft assignments A joint distiution ove a set of andom vaiales: is a map fom assignments (o outcome, o atomic event) to eals: ize of distiution if n vaiales with domain sizes d? Must oey: o all ut the smallest distiutions, impactical to wite out Examples Maginalization An event is a set E of assignments (o outcomes) om a joint distiution, we can calculate the poaility of any event oaility that it s AN ny? oaility that it s? oaility that it s OR ny? Maginalization (o summing out) is pojecting a joint distiution to a su-distiution ove suset of vaiales 0.6 Conditional oailities Conditioning Conditional o posteio poailities: E.g., (cavity toothache) = 0.8 Given that toothache is all I know Conditioning is fixing some vaiales and enomalizing ove the est: Notation fo conditional distiutions: (cavity toothache) = a single nume (Cavity, oothache) = 4-element vecto summing to 1 (Cavity oothache) = wo 2-element vectos, each summing to 1 If we know moe: (cavity toothache, catch) = 0.9 (cavity toothache, cavity) = 1 Note: the less specific elief emains valid afte moe evidence aives, ut is not always useful New evidence may e ielevant, allowing simplification: (cavity toothache, taffic) = (cavity toothache) = 0.8 his kind of infeence, sanctioned y domain knowledge, is cucial elect Nomalize

3 Infeence y Enumeation Infeence y Enumeation (R)? (R winte)? (R winte,)? summe summe summe summe winte winte winte winte R Geneal case: Evidence vaiales: Quey vaiales: Hidden vaiales: We want: he equied summation of joint enties is done y summing out H: hen enomalizing All vaiales Ovious polems: Wost-case time complexity O(d n ) pace complexity O(d n ) to stoe the joint distiution he Chain Rule I ometimes joint (X,Y) is easy to get ometimes easie to get conditional (X Y) Example: (un,y)? Lewis Caoll's ack olem ack contains a ed o lue all, 50/50 We add a ed all If we daw a ed all, what s the chance of dawing a second ed all? Vaiales: ={,} is the oiginal all ={,} is the all we daw Quey: (= =) R 0.8 dy dy dy dy Lewis Caoll's ack olem Independence Now we have (,) Want ( =) wo vaiales ae independent if: his says that thei joint distiution factos into a poduct two simple distiutions Independence is a modeling assumption Empiical joint distiutions: at est close to independent What could we assume fo {un, y, oothache, Cavity}? How many paametes in the full joint model? How many paametes in the independent model? Independence is like something fom Cs: what? 3

4 Example: Independence Example: Independence? N fai, independent coins: H H H Aitay joint distiutions can e (pooly) modeled y independent factos 0.6 Conditional Independence (oothache,cavity,catch) has 2 3 = 8 enties (7 independent enties) If I have a cavity, the poaility that the poe catches in it doesn't depend on whethe I have a toothache: (catch toothache, cavity) = (catch cavity) Conditional Independence Unconditional independence is vey ae (two easons: why?) Conditional independence is ou most asic and oust fom of knowledge aout uncetain envionments: he same independence holds if I haven't got a cavity: (catch toothache, cavity) = (catch cavity) Catch is conditionally independent of oothache given Cavity: (Catch oothache, Cavity) = (Catch Cavity) Equivalent statements: (oothache Catch, Cavity) = (oothache Cavity) (oothache, Catch Cavity) = (oothache Cavity) (Catch Cavity) What aout this domain: affic Umella Raining What aout fie, smoke, alam? he Chain Rule II Can always facto any joint distiution as a poduct of incemental conditional distiutions he Chain Rule III Wite out full joint distiution using chain ule: (oothache, Catch, Cavity) = (oothache Catch, Cavity) (Catch, Cavity) = (oothache Catch, Cavity) (Catch Cavity) (Cavity) = (oothache Cavity) (Catch Cavity) (Cavity) Why? his actually claims nothing What ae the sizes of the tales we supply? Cav (Cavity) Cat (oothache Cavity) (Catch Cavity) Gaphical model notation: Each vaiale is a node he paents of a node ae the othe vaiales which the decomposed joint conditions on MUCH moe on this to come! 4

5 Bayes Rule wo ways to facto a joint distiution ove two vaiales: Moe Bayes Rule iagnostic poaility fom causal poaility: hat s my ule! ividing, we get: Example: m is meningitis, s is stiff neck Why is this at all helpful? Lets us invet a conditional distiution Often the one conditional is ticky ut the othe simple oundation of many systems we ll see late (e.g. AR, M) Note: posteio poaility of meningitis still vey small Note: you should still get stiff necks checked out! Why? In the unning fo most impotant AI equation! Comining Evidence (Cavity toothache, catch) = α (toothache, catch Cavity) (Cavity) = α (toothache Cavity) (catch Cavity) (Cavity) his is an example of a naive Bayes model: C Expectations Real valued functions of andom vaiales: Expectation of a function a andom vaiale Example: Expected value of a fai die oll otal nume of paametes is linea in n! We ll see much moe of naïve Bayes next week E 1 E 2 E n X f Expectations Utilities Expected seconds wasted ecause of spam filte spam spam ham ham tict ilte B f Lax ilte B lock 5 0 spam lock 5 allow 0 10 spam allow 0 lock ham lock 0.02 allow 0 0 ham allow 3 f eview of utility theoy (late) Utilities: unction fom events to eal numes (payoffs) E.g. spam E.g. aipot We ll use the expected cost of actions to dive classification, decision netwoks, and einfocement leaning 5

6 Estimation How to estimate the a distiution of a andom vaiale X? Maximum likelihood: Collect osevations fom the wold o each value x, look at the empiical ate of that value: his estimate is the one which maximizes the likelihood of the data Elicitation: ask a human! Hade than it sounds E.g. what s (ing )? Usually need domain expets, and sophisticated ways of eliciting poailities (e.g. etting games) Estimation olems with maximum likelihood estimates: If I flip a coin once, and it s heads, what s the estimate fo (heads)? What if I flip it 50 times with 27 heads? What if I flip 10M times with 8M heads? Basic idea: We have some pio expectation aout paametes (hee, the poaility of heads) Given little evidence, we should skew towads ou pio Given a lot of evidence, we should listen to the data How can we accomplish this? tay tuned! 6