pix ) pclx : Likelihood : arggrax Posterior piy.ES/pc5 ) - PowerPoint PPT Presentation

Inference Recap Bayesian : Prior pix ) pc§lx± : Likelihood : arggrax Posterior piy.ES/pc5 ) pcxly ) : = |dEpc5¥ ) Evidence pcj ) = : |dEp(y*lI)pcEl5 ( Marginal likelihood ) - MAP - - u plyix ) : pcxly ) anyynax = × Predictive pcy*iy ) ) : = = |dE ) 14 ] [ fk pcxoljlfk El Expectation ) :

Models Today : Graphical Reduce single high Exploit conditional independence dimensional : - Multiple integral lower integrals dimensional into -

Undirected Directed Graphs us . Directed Graph Graph Undirected j density Can define , b. ) pca d. same c. e ) later ( more

Cyclic Acyclic Graphs us Graph Directed Graph ( DAG ) Acyclic Directed ( cycle with ) No Definition node : be twice visited can following edges by

Connected Graphs Singly Multiply Connected Connected . a > > ) paths ) ( ( Multiple path One

Networks Graph ( DAG ) Acyclic Bayesian Directed Belief Netwonh@btso.B Example : Holmes B Sherlock apartment ' Bungled was Watson Alarm A The off went the heard alarm W f Gibbon Mrs the heard Alawn Conditional Dependencies p ( AIB ) A. B ) > ( B ) PCGIA ) ) ( PCWIA G. W p = , , B) / , .6 ) , 6) p( GW p ( Marginal w - PIBIW = , A. B over K Marginal A over

Networks Bayesian Structure relationships reflects causal

Networks Bayesian case : Sfompkfted Naive Bayes Example f classes : wo Bernoulli Yin n ) N ( i. i , ... , Plates ( Message ) is spam ( Ojc ) Bernoulli . it " Xij 14 i= ' : =c ~ . j=i ... ,D , ( j it Word in # Docs occurs message # Words

Networks Bayesian Naive Bayes Example , 4) : Independence Conditional phil El plxiillli p( X ) 1,1 = , , pl1il4i)pl4i_l Ml , p( Yi ,Xi ) ( does are = independent ) plllillli ) = PZXI ) ( not depend Does an ' =/ i ) other does i

Networks Bayesian Naive Bayes I params ) Example with unknown : / M Y ; 117in Bernoulli G) - ( Ojc ) Bernoulli Xij 4i=c ,Qi=Op 1 ~ " ) M Betak " , p ~ Qic Beta ( oo ,p° ) - On , pcxyliilpk µ , . ,u ;) Question : it

Networks Bayesian Naive Bayes I params ) Example with unknown : / M Dependencies Conditional pcy ) pl pcxiyl y ) x. = N |dn M ( y ) pcn ) plyiln ) p = it , Gul | 1¥ !%pCo ;) ;) pkijlbji do pcxiyi = , .

Colliders A Definition collider node is a c path there b when ← is a a → c It , . I I 4 - collider ) ) ( not Collider ( a

- Separation D D connection and - 2- be 4 X Definition Let disjoint and : , , 6 X 4 of nodes graph set th and a . by connected 7 d when are - U 's There exist path undirected 7 an . between 4th EX and some x 2 U For collider either c . every on 7- 7 of Th descendant is c or a C E No collider I is U A 3 in non - , d- separated 2- 4 X by all other and in cases are

- Separation D D connection and - Conditional Independence : XIY phiz 2- ) ply ylzl 177 I pix → = , Holds and whenever x are y Y " iden by separated d- 2- ✓ . ( a c- :& Ig amount Colliders a . et lb ? le ? No alle ate

- Separation D D connection and - AIB A # B 1C AIB AfDB . bi pca = , dcplcla.hn/pcalplb)pca.bt-/dcpla1c)plblclpc4=pCalplb play plb ) ) t K a ,b pl Ic ) b) P' a) P' b ) picta = , ,bk7= plaldplbk ) pea pic ) pcaicspcblc ) ⇐

✓ - Separation D D connection and - ID AIB ? No observed ( descendant ) of collide a ? IB A Yes . Ic ) collide observed ( non -

xIy ? For examples Exercise IZ which : Patni A " N ? iii. . ii " ' a . ygpathz ✓ ✓ . i ' I is . . . . is 2- connected d- descendant of W Yes No No No No

Exercise ? Far by which 17 examples x : No No Yes No No [ It links to/from to/from links Cut observed colliders unobserved collides non -

- membership Mixed Models Mixture Models Allocation Latent Dirichlet Hidden Models Markov

Conditional Models Independence Mixture in - { pink Generative ,n } , Lik Model O PIO ) - . ,k } define Will ) M pit ~ ) h , In PME later these Mh , ~ . , . . - Categorical ( Th Mk ) 7h17 Xu ~ . . , . , Normal ( fuk In ) 17h Xu n , Q 7- a Independence Conditional he # -117in I Xi 7in 7- O } # :N n n pttnlxn ,O ) I pttlx 7- Xun = # n , n , . I Xm¥n 10 Xu