Machine Learning. Machine Learning /15-781

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Machie Learig -75 75-78, 78, Sprig 8 Itroductio ad Desity Estimatio Eric Xig Lecture,

1 Machie Learig , 78, Sprig 8 Itroductio ad Desity Estimatio Eric Xig Lecture, Jauary 4, 8 Readig: Chap.,, CB Machie Learig Class webpage:

2 Logistics Tet book Chris Bishop, Patter Recogitio ad Machie Learig required Tom Mitchell, Machie Learig David Mackay, Iformatio Theory, Iferece, ad Learig Algorithms Mailig Lists: To cotact the istructors: Class aoucemets list: TA: Leo Gu, Wea Hall 84, , Office hours: Kyug-Ah Soh, Doherty Hall 43e, 8-39, Office hours: Thursday 3:-4: Class Assistat: Diae Stidle, Wea Hall 46, 8-99 Logistics 5 homework assigmets: 3% of grade Theory eercises Implemetatio eercises Fial project: % of grade Applyig PGM to your research area LP, IR, Computatioal biology, visio, robotics Theoretical ador algorithmic work a more efficiet approimate iferece algorithm a ew samplig scheme for a o-trivial model Two eams: 5% of grade each Theory eercises ador aalysis Policies

3 What is Learig Apoptosis Medicie Machie Learig 3

Machie Learig Machie Learig seeks to develop theories ad computer systems for

ad reactig to comple, real world data, based o the system's ow eperiece with data,

characterized ad aalyzed ca take ito accout huma prior kowledge ca geeralize ad

iterpreted ad perceived by huma. Where Machie Learig is beig used or ca be useful?

4 Machie Learig Machie Learig seeks to develop theories ad computer systems for represetig; classifyig, clusterig ad recogizig; reasoig uder ucertaity; predictig; ad reactig to comple, real world data, based o the system's ow eperiece with data, ad hopefully uder a uified model or mathematical framework, that ca be formally characterized ad aalyzed ca take ito accout huma prior kowledge ca geeralize ad adapt across data ad domais ca operate automatically ad autoomously ad ca be iterpreted ad perceived by huma. Where Machie Learig is beig used or ca be useful? Iformatio retrieval Speech recogitio Computer visio Games Robotic cotrol Pedigree Evolutio Plaig 4

5 atural laguage processig ad speech recogitio ow most pocket Speech Recogizers or Traslators are ruig o some sort of learig device --- the more you playuse them, the smarter they become! Object Recogitio Behid a security camera, most likely there is a computer that is learig ador checkig! 5

6 Robotic Cotrol I The best helicopter pilot is ow a computer! it rus a program that lears how to fly ad make acrobatic maeuvers by itself! o taped istructios, joysticks, or thigs like A. g 5 Robotic Cotrol II ow cars ca fid their ow ways! 6

7 Tet Miig We wat: Readig, digestig, ad categorizig a vast tet database is too much for huma! Uderstadig Brai Activities 7

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T years Q h Q m A C 8

9 Paradigms of Machie Learig Supervised Learig D { X, Y } f : Y f Give, lear, s.t. i i i X i Usupervised Learig { } f : Y f Give D X i, lear i X i, s.t. ew D { X } { Y } ew D { X } { Y } Reiforcemet Learig Give D { ev, actios, rewards, simulatortracereal game} j j j j lear policy : e, r a utility : a, e r, s.t. { ev,ew real game} a, a, K a 3 Active Learig ew Give ~ G, lear D ~ G' ad f, s.t. all D G', policy, { } D Y j Fuctio Approimatio ad Desity Estimatio 9

10 Fuctio Approimatio Settig: Set of possible istaces X Ukow target fuctio f: X Y Set of fuctio hypotheses H{ h h: X Y } Apoptosis Medicie Give: Traiig eamples {< i,y i >} of ukow target fuctio f Determie: Hypothesis h H that best approimates f Desity Estimatio A Desity Estimator lears a mappig from a set of attributes to a Probability Ofte kow as parameter estimatio if the distributio form is specified Biomial, Gaussia Three importat issues: ature of the data iid, correlated, Objective fuctio E, MAP, Algorithm simple algebra, gradiet methods, EM, Evaluatio scheme likelihood o test data, predictability, cosistecy,

11 Desity Estimatio Schemes Data Lear parameters Algorithm Score param Maimum likelihood Aalytical 5, K,,, K K M, K, M Bayesia Coditioal likelihood Margi Gradiet EM Samplig 3 5 K Parameter Learig from iid Data Goal: estimate distributio parameters from a dataset of idepedet, idetically distributed iid, fully observed, traiig cases D {,..., } Maimum likelihood estimatio E. Oe of the most commo estimators. With iid ad full-observability assumptio, write L as the likelihood of the data: L P,, K, ; P ; P ;, K, P i P i; 3. pick the settig of parameters most likely to have geerated the data we saw: ; * arg ma L arg ma log L

12 Eample: Beroulli model Data: We observed iid coi tossig: D{,,,, } Represetatio: Biary r.v: Model: {, } for P for How to write the likelihood of a sigle observatio i? P i P i i The likelihood of datasetd{,, }: i i i P,,..., P i i i i i i #head #tails Maimum Likelihood Estimatio Objective fuctio: h t l ; D log P D log log log We eed to maimize this w.r.t. Take derivatives wrt h h l h h h E or E i i Sufficiet statistics h, where k i, Frequecy as sample mea The couts, are sufficiet statistics of data D i

!! The rescue: "smoothig" head Where ' is kow as the pseudo- imagiary cout head But ca we make this more formal?

13 Overfittig Recall that for Beroulli Distributio, we have head What if we tossed too few times so that we saw zero head? We have head ad we will predict that the probability of, seeig a head et is zero!!! The rescue: "smoothig" head Where ' is kow as the pseudo- imagiary cout head But ca we make this more formal? head tail head ' head tail ' Bayesia Parameter Estimatio Treat the distributio parameters also as a radom variable The a posteriori distributio of after seem the data is: This is Bayes Rule p D p p D p D likelihood prior posterior margial likelihood p D p p D p d The prior p. ecodes our prior kowledge about the domai 3

Hece they have to come up with differet "objective" estimators ways of computig from data, istead of usig Bayes rule.

14 Frequetist Parameter Estimatio Two people with differet priors p will ed up with differet estimates p D. Frequetists dislike this subjectivity. Frequetists thik of the parameter as a fied, ukow costat, ot a radom variable. Hece they have to come up with differet "objective" estimators ways of computig from data, istead of usig Bayes rule. These estimators have differet properties, such as beig ubiased, miimum variace, etc. The maimum likelihood estimator, is oe such estimator. Discussio or p, this is the problem! Bayesias kow it 4

5 Bayesia estimatio for Beroulli Beta distributio: Whe is discrete Posterior distributio of : otice the isomorphism of the posterior to the prior, such a prior is called a cojugate prior ad are

15 5 Bayesia estimatio for Beroulli Beta distributio: Whe is discrete Posterior distributio of : otice the isomorphism of the posterior to the prior, such a prior is called a cojugate prior ad are hyperparameters parameters of the prior ad correspod to the umber of virtual headstails pseudo couts t h t h p p p P,...,,...,,..., Γ Γ Γ,, ; P B! Γ Γ Bayesia estimatio for Beroulli, co'd Posterior distributio of : Maimum a posteriori MAP estimatio: Posterior mea estimatio: Prior stregth: A A ca be iteroperated as the size of a imagiary data set from which we obtai the pseudo-couts t h t h p p p P,...,,...,,..., d C d D p h Bayes t h Bata parameters ca be uderstood as pseudo-couts,..., ma log arg MAP P

16 Effect of Prior Stregth Suppose we have a uiform prior, ad we observe v h, 8 t Weak prior A. Posterior predictio: v v p h h, t 8, '. 5 Strog prior A. Posterior predictio: v v p h h, t 8, '. 4 However, if we have eough data, it washes away the prior. e.g., v h, t 8. The the estimates uder weak ad strog prior are ad, respectively, both of which are close to. Eample : Gaussia desity Data: We observed iid real samples: Model: D{-.,,, -5.,, 3} P Log likelihood: π ep{ } E: take derivative ad set to zero: l ; D log P D log π l E l 4 E 6

17 7 E for a multivariate-gaussia It ca be show that the E for ad Σ is where the scatter matri is The sufficiet statistics are Σ ad Σ T. ote that X T XΣ T may ot be full rak eg. if <D, i which case Σ is ot ivertible S T E E Σ T T T S K M T T T X M Bayesia estimatio ormal Prior: Joit probability: Posterior: { } π ep P ~ ad, ~ where Sample mea { } π π ep ep, P { } π ~ ~ ep ~ P

18 8 Bayesia estimatio: ukow, kow The posterior mea is a cove combiatio of the prior ad the E, with weights proportioal to the relative oise levels. The precisio of the posterior is the precisio of the prior plus oe cotributio of data precisio for each observed data poit. Sequetially updatig the mea.8 ukow,. kow Effect of sigle data poit Uiformative vague flat prior, ~, Summary Machie Learig is Cool ad Useful!! Learig scearios: Data Objective fuctio Frequetist ad Bayesia Desity estimatio Typical discrete distributio Typical cotiuous distributio Cojugate priors

Elementary manipulations of probabilities

Elementary manipulations of probabilities Elemetary maipulatios of probabilities Set probability of multi-valued r.v. {=Odd} = +3+5 = /6+/6+/6 = ½ X X,, X i j X i j Multi-variat distributio: Joit probability: X true true X X,, X X i j i j X X