Basics of Statistical Estimation
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1 Basics of Statistical Estimation Doug Downey, Nortwestern EECS 395/495, Spring 206 (several illustrations from P. Domingos, University of Wasington CSE
2 Bayes Rule P(A B = P(B A P(A / P(B Example: P(symptom disease = 0.95, P(symptom disease = 0.05 P(disease = P(disease symptom = P(symptom disease*p(disease P(symptom = 0.95*0.000 = * *0.9999
3 Bayes Rule P(A B = P(B A P(A / P(B Also: P(A B, C = P(B A, C P(A C / P(B C More generally: P(A B = P(B A P(A / P(B (Boldface indicates vectors of variables
4 Bayes Rule Wy is Bayes Rule so important? Often, we want to deduce P(Hidden state Data E.g., Hidden state = disease, Data = symptoms and te simplest way to express tat is in terms of causes of te model: P(Data Model E.g., ow common is a symptom, wit or witout a given disease times a prior belief about te model, P(Model E.g., probability of a disease
5 Terms for Bayes P(Model Data = P(Data Model P(Model / P(Data P(Model : Prior P(Data Model : Likeliood P(Model Data : Posterior
6 Probabilistic Models Joint Distribution can answer ueries P(symptoms, disease can be used to predict weter person as disease based on symptoms But: Were do te probabilities come from (learning? How do we represent a joint compactly using conditional independencies? (representation grapical models
7 Learning Probabilities:Classical Approac Simplest case: Flipping a tumbtack eads tails True probability is unknown Given: flips generated independently wit te same, (a.k.a. Independent and identically distributed data - iid, Estimate:
8 Estimating Probabilities Tree Metods: Maximum Likeliood Estimation (ML Bayesian Estimation Maximum A posteriori Estimation (MAP
9 Maximum Likeliood Principle Coose te parameters tat maximize te probability of te observed data
10 Tink/Pair/Sare If Data={ eads and t tails}, wat parameter θ maximizes te probability of Data? Tink Start End 0
11 Tink/Pair/Sare If Data={ eads and t tails}, wat parameter θ maximizes te probability of Data? Pair Start End
12 Tink/Pair/Sare If Data={ eads and t tails}, wat parameter θ maximizes te probability of Data? Sare 2
13 Maximum Likeliood Estimation p ( e a d s p ( tails ( p # ( t... t t t ( # t (Number of eads is binomial distribution
14 Computing te ML Estimate Use log-likeliood Differentiate wit respect to parameter(s Euate to zero and solve Solution: # # # t
15 Sufficient Statistics p ( t... ttt ( # # t (#,#t are sufficient statistics
16 Bayesian Estimation eads tails True probability is unknown Bayesian probability density for p( 0
17 Use of Bayes Teorem posterior prior likeliood p( eads p( p(eads p( p(eads d p ( p ( e a d s
18 Example: Observation of Heads" p( p(eads = p( eads prior likeliood posterior
19 Probability of Heads on Next Toss ( ( ( ( is t toss ( ( d d d d p N E d p d p X p n p
20 MAP Estimation Approximation: Instead of averaging over all parameter values Consider only te most probable value (i.e., value wit igest posterior probability Usually a very good approximation, and muc simpler MAP value Expected value MAP ML for infinite data (as long as prior 0 everywere
21 Prior Distributions for Direct assessment Parametric distributions Conjugate distributions (for convenience
22 Conjugate Family of Distributions (, Beta( ( t t p # ( # tails ead s, ( t t t p Beta distribution: Resulting posterior distribution: 0, t
23 Estimates Compared Prior prediction: Bayesian posterior prediction MAP estimate: ML estimate: t # + # # t E + ( # + # # t t t t + E # # # (
24 Intuition Te yperparameters and t can be tougt of as imaginary counts from our prior experience, starting from "pure ignorance" Euivalent sample size = + t ( euivalent in terms of effect on Bayesian estimate Te larger te euivalent sample size, te more confident we are about te true probability
25 Beta Distributions Beta(0.5, 0.5 Beta(, Beta(3, 2 Beta(9, 39
26 Assessment of a Beta Distribution Metod : Euivalent sample - assess and t - assess + t and /( + t p Metod 2: Imagined future samples ( e a d s 0.2 a nd p ( e a d s 3 e a d s 0.5, t 4 ceck : 0. 2 =,
27 Generalization to m Outcomes (Multinomial Distribution,, Diriclet( ( i i m i m,θ m, θ p m i N i m i i N, N p, ( Diriclet distribution: m i i i i E ( Properties: 0 i m i i
28 Oter Distributions Likelioods from te exponential family Binomial Multinomial Poisson Gamma Normal
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