Probabilistic inference

Transcription

1 robabilistic infrnc Suppos th agnt has to mak a dcision about th valu of an unobsrvd qury variabl X givn som obsrvd vidnc E = artially obsrvabl, stochastic, pisodic nvironmnt Eampls: X = {spam, not spam}, = mail mssag X = {zbra, giraff, hippo}, = imag faturs Bays dcision thory: Th agnt has a loss function, which is 0 if th valu of Xi is gussd corrctly and d1 othrwis Th stimat of X that minimizs pctd loss is th on that has th gratst postrior probability X = This is th Maimum a ostriori MA dcision

2 MA dcision Valu of that has th highst postrior probability givn th vidnc arg ma * = = arg ma

3 MA dcision Valu of that has th highst postrior probability givn th vidnc arg ma * = = arg ma liklihood prior postrior

4 MA dcision Valu of that has th highst postrior probability givn th vidnc arg ma * = = arg ma Maimum liklihood ML dcision: liklihood prior postrior arg ma * =

5 Eampl: Naïv Bays modl Suppos w hav many diffrnt typs of obsrvations symptoms, y faturs F 1,, F n that w want to us to obtain vidnc about an undrlying hypothsis H MA dcision involvs stimating H 1 F1, K, Fn F, K, Fn H H If ach fatur can tak on k valus, how many ntris ar in th joint probability tabl?

6 Eampl: Naïv Bays modl Suppos w hav many diffrnt typs of obsrvations symptoms, y faturs F 1,, F n that w want to us to obtain vidnc about an undrlying hypothsis H MA dcision involvs stimating H 1 F1, K, Fn F, K, Fn H H W can mak th simplifying assumption that th diffrnt faturs ar conditionally indpndnt givn th hypothsis: n F K 1,, Fn H = Fi H i= 1 If ach fatur can tak on k valus, what is th complity of storing th rsulting distributions?

7 Naïv Bays Spam Filtr MA dcision: to minimiz th probability of rror, w should classify a mssag as spam if spam mssag > spam mssag

8 Naïv Bays Spam Filtr MA dcision: to minimiz th probability of rror, w should classify a mssag as spam if spam mssag > spam mssag W hav spam mssag mssag spamspam W hav spam mssag mssag spamspam and spam mssag mssag spam spam

9 Naïv Bays Spam Filtr W nd to find mssag spam spam and mssag spam spam Th mssag is a squnc of words w 1,, w n Bag of words rprsntation Th ordr of th words in th mssag is not important Each word is conditionally indpndnt of th othrs givn mssag class spam or not spam

10 Naïv Bays Spam Filtr W nd to find mssag spam spam and mssag spam spam Th mssag is a squnc of words w 1,, w n Bag of words rprsntation Th ordr of th words in th mssag is not important Each word is conditionally indpndnt of th othrs givn mssag class spam or not spam mssag spam = w K n 1,, wn spam = wi spam i= 1 Our filtr will classify th mssag as spam if n spam wi spam > spam i= 1 n i= 1 w i spam

11 Bag of words illustration US rsidntial Spchs Tag Cloud

12 Bag of words illustration US rsidntial Spchs Tag Cloud

13 Bag of words illustration US rsidntial Spchs Tag Cloud

14 Naïv Bays Spam Filtr spam w w K spam 1,, n w i spam n i= 1 postrior prior liklihood

15 aramtr stimation In ordr to classify a mssag, w nd to know th prior spam and th liklihoods word spam and word spam Ths ar th paramtrs of th probabilistic modl How do w obtain th valus of ths paramtrs? prior spam: 0.33 spam: 0.67 word spam word spam

16 aramtr stimation How do w obtain th prior spam and th liklihoods word spam and word spam? Empirically: us training data word spam = # of word occurrncs in spam mssags total # of words in spam mssags This is th maimum liklihood lih ML stimat, t or stimat t that maimizs th liklihood of th training data: D n d d = 1 i= 1 w d, i classd, i d: ind of training documnt, i: ind of a word

17 aramtr stimation How do w obtain th prior spam and th liklihoods word spam and word spam? Empirically: us training data word spam = # of word occurrncs in spam mssags total # of words in spam mssags aramtr smoothing: daling with words that wr nvr sn or sn too fw tims Laplacian smoothing: prtnd you hav sn vry vocabulary word on mor tim than you actually did

18 Summary of modl and paramtrs Naïv Bays modl: n spam mssag spam spam mssag spam Modl paramtrs: i= 1 n w i= 1 i spam w i spam prior spam spam Liklihood of spam w 1 spam w 2 spam w n spam Liklihood of spam w 1 spam w 2 spam w n spam

19 Bag-of-word modls for imags Csurka t al. 2004, Willamowski t al. 2005, Grauman & Darrll 2005, Sivic t al. 2003, 2005

20 Bag-of-word modls for imags 1. Etract imag faturs

21 Bag-of-word modls for imags 1. Etract imag faturs

22 Bag-of-word modls for imags 1. Etract imag faturs 2. Larn visual vocabulary

23 Bag-of-word modls for imags 1. Etract imag faturs 2. Larn visual vocabulary 3. Map imag faturs to visual words

24 Baysian dcision making: Summary Suppos th agnt has to mak dcisions about th valu of an unobsrvd qury variabl X basd on th valus of an obsrvd vidnc variabl E Infrnc problm: givn som vidnc E =, what is X? Larning problm: stimat th paramtrs of th probabilistic bili i modl X E givn a training i sampl { 1, 1,, n, n }