CS 3710 Advanced Topics in AI Lecture 17. Density estimation. CS 3710 Probabilistic graphical models. Administration

Transcription

1 CS 37 Avace Topcs AI Lecture 7 esty estmato Mlos Hauskrecht mlos@cs.ptt.eu 539 Seott Square CS 37 robablstc graphcal moels Amstrato Mterm: A take-home exam week ue o Weesay ovember 5 before the class epes o the materal covere so far: Exact fereces Mote-Carlo samplg Varatoal approxmato You wll be evaluate o the correctess a clarty of your aswers Be eat a expla clearly your otatos a solutos CS 37 robablstc graphcal moels

2 esty estmato ata: {.. } x a vector of attrbute values Attrbutes: moele by raom varables X { X X K X } wth: Cotuous values screte values E.g. bloo pressure wth umercal values or chest pa wth screte values [o-pa ml moerate strog] Uerlyg true probablty strbuto: px CS 37 robablstc graphcal moels ata: esty estmato {.. } x a vector of attrbute values Objectve: try to estmate the uerlyg true probablty strbuto over varables X px usg examples true strbuto samples p X.. } { estmate pˆ X Staar assumptos: Samples are epeet of each other come from the same etcal strbuto fxe px CS 37 robablstc graphcal moels

3 esty estmato Types of esty estmato: arametrc the strbuto s moele usg a set of parameters Θ p X Θ Example: mea a covaraces of multvarate ormal Estmato: f parameters Θˆ that ft the ata the best o-parametrc The moel of the strbuto utlzes all examples As f all examples were parameters of the strbuto Examples: earest-eghbor Sem-parametrc CS 37 robablstc graphcal moels arametrc esty estmato arametrc esty estmato Basc settgs: A set of raom varables X { X X K X } A moel of the strbuto over varables X wth parameters Θ ata.. } { Objectve: f parameters Θˆ that escrbe p X Θ the best CS 37 robablstc graphcal moels

4 arameter learg What s the best set of parameters? Maxmum lkelhoo ML estmates maxmze p Θ - represets pror backgrou kowlege Maxmum a posteror probablty MA estmate maxmze p Θ Selects the moe of the posteror p Θ p Θ p Θ p CS 37 robablstc graphcal moels arameter learg Both ML or MA pck oe parameter value Is t always the best soluto? Bayesa approach Remees the lmtato of oe choce Keeps a uses complete posteror strbuto Optmzato s replace wth tegrato p Θ How s t use? Assume we wat: x Coser all parameter settgs a averages the result x x p Example: prect the result of the outcome x x CS 37 robablstc graphcal moels

5 Beroull strbuto. x Outcomes: wth values or hea or tal ata: a sequece of outcomes x Moel: probablty of a outcome probablty of x x x Beroull strbuto CS 37 robablstc graphcal moels Maxmum lkelhoo ML estmate. Lkelhoo of ata: x Maxmum lkelhoo estmate ML Optmze log-lkelhoo arg max x x l log log x log x log log x - umber of s see - umber of s see x log x CS 37 robablstc graphcal moels

6 Maxmum lkelhoo ML estmate. Optmze log-lkelhoo l log log Set ervatve to zero Solvg l ML Soluto: ML CS 37 robablstc graphcal moels Maxmum a posteror estmate Maxmum a posteror estmate Selects the moe of the posteror strbuto MA arg max p How to choose the pror probablty? p p va Bayes rule p - s the lkelhoo of ata x x - s the pror probablty o CS 37 robablstc graphcal moels

7 CS 37 robablstc graphcal moels ror strbuto p Choce of pror: strbuto strbuto fts bomal samplg - cojugate choces MA p MA Soluto: Why? CS 37 robablstc graphcal moels strbuto β.5.5 β.5.5 β5

8 CS 37 robablstc graphcal moels Bayesa approach osteror probablty: robablty of a outcome the ext tral Equvalet to the expecte value of the parameter expectato s take wth regar to the posteror strbuto p x p x x E p p CS 37 robablstc graphcal moels Bayesa learg Expecte value of the parameter E ote:

9 Expecte value rectve probablty of a outcome x the ext tral x E Substtutg the results for p We get x E Istea of MA a ML choce of the parameter we ca use the expecte value of the parameter ˆ E CS 37 robablstc graphcal moels