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CMU CS 10708 - Parameter Learning 2

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11Parameter Learning 2Graphical Models – 10708Carlos GuestrinCarnegie Mellon UniversitySeptember 27th, 2006Readings:K&F: 14.1, 14.2, 14.3, 14.4, 15.1, 15.2, 15.3.1, 15.4.110-708 –Carlos Guestrin 20062Your first learning algorithm Set derivative to zero:210-708 –Carlos Guestrin 20063Learning the CPTsx(1)…x(m)DataFor each discrete variable Xi10-708 –Carlos Guestrin 20064Maximum likelihood estimation (MLE) of BN parameters – General case Data: x(1),…,x(m) Restriction: x(j)[PaXi] → assignment to PaXiin x(j) Given structure, log likelihood of data:310-708 –Carlos Guestrin 20065Taking derivatives of MLE of BN parameters – General case10-708 –Carlos Guestrin 20066General MLE for a CPT Take a CPT: P(X|U) Log likelihood term for this CPT Parameter θX=x|U=u:410-708 –Carlos Guestrin 20067Announcements Late homeworks: 3 late days for the semester  one late day corresponds to 24 hours! (i.e., 3 late days due Saturday by noon) Give late homeworks to Monica Hopes, Wean Hall 4619  If she is not in her office, time stamp (date and time) your homework, sign it, and put it under her door After late days are used up: Half credit within 48 hours Zero credit after 48 hours All homeworks must be handed in, even for zero credit Homework 2 out later today Recitation tomorrow: review perfect maps, parameter learning10-708 –Carlos Guestrin 20068mCan we really trust MLE? What is better? 3 heads, 2 tails 30 heads, 20 tails 3x1023heads, 2x1023tails Many possible answers, we need distributions over possible parameters510-708 –Carlos Guestrin 20069Bayesian Learning Use Bayes rule: Or equivalently:10-708 –Carlos Guestrin 200610Bayesian Learning for Thumbtack Likelihood function is simply Binomial: What about prior? Represent expert knowledge Simple posterior form Conjugate priors: Closed-form representation of posterior (more details soon) For Binomial, conjugate prior is Beta distribution610-708 –Carlos Guestrin 200611Beta prior distribution – P(θ) Likelihood function: Posterior:10-708 –Carlos Guestrin 200612Posterior distribution Prior: Data: mHheads and mTtails Posterior distribution:710-708 –Carlos Guestrin 200613Conjugate prior Given likelihood function P(D|θ) (Parametric) prior of the form P(θ|α) is conjugate to likelihood function if posterior is of the same parametric family, and can be written as:  P(θ|α’), for some new set of parameters α’ Prior: Data: mHheads and mTtails (binomial likelihood) Posterior distribution: 10-708 –Carlos Guestrin 200614Using Bayesian posterior Posterior distribution:  Bayesian inference: No longer single parameter: Integral is often hard to compute810-708 –Carlos Guestrin 200615Bayesian prediction of a new coin flip Prior:  Observed mHheads, mTtails, what is probability of m+1 flip is heads?10-708 –Carlos Guestrin 200616Asymptotic behavior and equivalent sample size Beta prior equivalent to extra thumbtack flips: As m → ∞, prior is “forgotten” But, for small sample size, prior is important! Equivalent sample size: Prior parameterized by αH,αT, or m’ (equivalent sample size) and αFix m’, change αFix α, change m’910-708 –Carlos Guestrin 200617Bayesian learning corresponds to smoothing m=0 ⇒ prior parameter m→∞ ⇒ MLE m10-708 –Carlos Guestrin 200618Bayesian learning for multinomial What if you have a k sided coin??? Likelihood function if multinomial: Conjugate prior for multinomial is Dirichlet: Observe m data points, mifrom assignment i, posterior: Prediction:1010-708 –Carlos Guestrin 200619Bayesian learning for two-node BN Parameters θX, θY|X Priors: P(θX): P(θY|X):10-708 –Carlos Guestrin 200620Very important assumption on prior:Global parameter independence Global parameter independence: Prior over parameters is product of prior over CPTs1110-708 –Carlos Guestrin 200621Global parameter independence, d-separation and local predictionFluAllergySinusHeadacheNose Independencies in meta BN: Proposition: For fully observable data D, if prior satisfies global parameter independence, then 10-708 –Carlos Guestrin 200622Within a CPT Meta BN including CPT parameters: Are θY|X=tand θY|X=fd-separated given D? Are θY|X=tand θY|X=findependent given D? Context-specific independence!!! Posterior decomposes:1210-708 –Carlos Guestrin 200623Priors for BN CPTs(more when we talk about structure learning) Consider each CPT: P(X|U=u) Conjugate prior: Dirichlet(αX=1|U=u,…, αX=k|U=u) More intuitive: “prior data set” D’ with m’ equivalent sample size “prior counts”: prediction:10-708 –Carlos Guestrin 200624An example1310-708 –Carlos Guestrin 200625What you need to know about parameter learning MLE: score decomposes according to CPTs optimize each CPT separately Bayesian parameter learning: motivation for Bayesian approach Bayesian prediction conjugate priors, equivalent sample size Bayesian learning ⇒ smoothing  Bayesian learning for BN parameters Global parameter independence Decomposition of prediction according to CPTs Decomposition within a CPT10-708 –Carlos Guestrin 200626Where are we with learning BNs? Given structure, estimate parameters Maximum likelihood estimation Bayesian learning What about learning structure?1410-708 –Carlos Guestrin 200627Learning the structure of a BN Constraint-based approach BN encodes conditional independencies Test conditional independencies in data Find an I-map Score-based approach Finding a structure and parameters is a density estimation task Evaluate model as we evaluated parameters Maximum likelihood Bayesian  etc. Data<x1(1),…,xn(1)>…<x1(m),…,xn(m)>FluAllergySinusHeadacheNoseLearn structure andparameters10-708 –Carlos Guestrin 200628Remember: Obtaining a P-map? Given the independence assertions that are true for P Obtain skeleton Obtain immoralities From skeleton and immoralities, obtain every (and any) BN structure from the equivalence class Constraint-based approach: Use Learn PDAG algorithm Key question: Independence test1510-708 –Carlos Guestrin 200629Independence tests


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CMU CS 10708 - Parameter Learning 2

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