Undirected Graphical Models II

Transcription

1 Readngs: K&F 4.4, 4.5, 4.6 Undrected Graphcal Models II Lecture 5 pr, 20 S 55, Statstcal Methods, Sprng 20 Instructor: Su-In Lee Unversty of Washngton, Seattle Last tme Markov networks representaton Local factor models (potentals) Independence propertes Global, parwse, local ndependences I-Map Factorzaton Today arameterzaton revsted ayesan nets and Markov nets artally drected graphs Inference 0 π [ D ],..., π n[ D n ( X,..., X n) π [ D ] ] S 55 Statstcal Methods Sprng 20 2

2 Factor Graphs From the Markov network structure, we are do not know how t s parameterzed. xample: fully connected graph may have parwse potentals or one large (exponental) potental over all nodes (,, ) π [ D ] mc(,, ) π [,, ] mc par (,, ) π [, ] π [, ] π [, ] par Markov network : Soluton: Factor Graphs Undrected graph Two types of nodes: Varable nodes, Factor nodes onnectvty? S 55 Statstcal Methods Sprng 20 3 Factor Graphs: example Two types of nodes: Varable nodes, Factor nodes onnectvty ach factor node s assocated wth exactly one factor π [D ] Scope of factor are all neghbor varables of the factor node mc(,, ) π mc [,, ] par (,, ) par π [, ] π [, ] π [, ] π π Markov network π Factor graph for jont parameterzaton π Factor graph for parwse parameterzaton 4 2

3 Local Structure: Feature Representaton Factor graphs stll encode complete tables X Y π XY [X,Y] x 0 y 0 00 x 0 y x y 0 x y 00 Y X feature φ[d] on varables D s an ndcator functon that for some d D: for example, when x y φ[ X, Y ] 0 otherwse Several features can be defned on one clque when x y when x > 50 φ[ D] φ2[ D] 0 otherwse 0 otherwse ny factor can be represented by features, where n general case, we defne a feature and weght for each entry n the factor pply log-transformaton: π [D] exp ( w φ [D] ) 5 Log-lnear model dstrbuton s a log-lnear model over H f t has Features φ [D ],...,φ k [D k ] where each D s a complete subgraph n H set of weghts w,...,w k such that k [ wφ[ D ] ( X,..., X n ) π [ D ] exp ] dvantages Log-lnear model s more compact for many dstrbutons especally wth large doman varables Representaton s ntutve and modular Features can be modularly added between any nteractng sets of varables S 55 Statstcal Methods Sprng

4 Markov Network arameterzatons hoce : Markov network roduct over potentals Rght representaton for dscussng ndependence queres hoce 2: Factor graph roduct over potentals Useful for nference (later) hoce 3: Log-lnear model roduct over feature weghts Useful for dscussng parameterzatons Useful for representng context specfc structures ll parameterzatons are nterchangeable 7 Outlne Markov networks representaton Local factor models π [ D ],..., π [ D n n] Independences global, parwse, local ndependences I-Map Factorzaton ( X,..., X n) π [ D ] Today arameterzaton revsted ayesan nets and Markov nets artally drected graphs Inference 0 S 55 Statstcal Methods Sprng

5 From ayesan nets to Markov nets Goal: buld a Markov network H capable of representng any dstrbuton that factorzes over G quvalent to requrng I(H) I(G) onstructon process ased on local Markov ndependences If X s connected wth Y n H, (X U-{X}-Y Y). onnect each X to every node n the smallest set Y s.t.: {(X U-{X}-Y Y) : X H} I(G) How can we fnd Y by queryng G? Y Markov blanket of X n G? S 55 Statstcal Methods Sprng 20 9 lockng aths ctve path: parents Y ctve path: descendants ctve path: v-structure X X X Y Y lock path: parents lock path: chldren lock path: chldren chldren s parents 5

6 From ayesan nets to Markov nets Goal: buld a Markov network H capable of representng any dstrbuton that factorzes over G quvalent to requrng I(H) I(G) onstructon process ased on local Markov ndependences If X s connected wth Y n H, (X U-{X}-Y Y). onnect each X to every node n the smallest set Y s.t.: {(X U-{X}-Y Y) : X H} I(G) How can we fnd Y by queryng G? Y Markov blanket of X n G (parents, chldren, chldren s parents) S 55 Statstcal Methods Sprng 20 Moralzed Graphs The Moral graph of a ayesan network structure G s the undrected graph that contans an undrected edge between X and Y f X and Y are drectly connected n G X and Y have a common chld n G ayesan network G Moralzed graph H S 55 Statstcal Methods Sprng

7 arameterzng Moralzed Graphs Moralzed graph contans a full clque for every X and ts parents a(x ) We can assocate Ds wth a clque Do we lose ndependence assumptons mpled by the graph structure? Yes, mmoral v-structures ( ) S 55 Statstcal Methods Sprng 20 3 From Markov nets to ayesan nets Transformaton s more dffcult and the resultng network can be much larger than the Markov network onstructon algorthm Use Markov network as template for ndependences I(H) Fx orderng of nodes dd each node along wth ts mnmal parent set accordng to the ndependences defned n the dstrbuton S 55 Statstcal Methods Sprng

8 From Markov nets to ayesan nets Markov network H ayesan network G Order:,,,D,,F D D F F S 55 Statstcal Methods Sprng 20 5 hordal Graphs Let X X 2... X k X be a loop n the graph chord n the loop s an edge connectng X and X j for two nonconsecutve nodes X and X j n undrected graph s chordal f any loop X X 2... X k X for k 4 has a chord That s, longest mnmal loop s a trangle hordal graphs are often called trangulated drected graph s chordal f ts underlyng undrected graph s chordal S 55 Statstcal Methods Sprng

9 From Markov Nets to ayesan Nets Theorem: Let H be a Markov network structure and G be any mnmal I-map for H. Then G s chordal. The process of turnng a Markov network nto a ayesan network s called trangulaton The process loses ndependences H Order:,,,D,,F G D D F (,F) F S 55 Statstcal Methods Sprng 20 7 Last tme Markov networks representaton Local factor models π [ D ],..., π [ D n n] Independences global, parwse, local ndependences I-Map Factorzaton ( X,..., X n) π [ D ] Today arameterzaton revsted ayesan nets and Markov nets artally drected graphs Inference 0 S 55 Statstcal Methods Sprng

10 ondtonal Random Felds (RFs) Specal case of partally drected models condtonal random feld s an undrected graph H whose nodes correspond to XUY; the network s annotated wth a set of factors φ (D ),, φ m (D m ) such that each D X. The network encodes a condtonal dstrbuton as follows: m ~ RF: example ( Y, X) φ ( D ) ~ ( Y X) ( Y, X) ( X) ~ ( X ) ( Y, X) Y Two varables n H are connected by an undrected edge whenever they appear together n the scope of some factor φ. X Y ~ ( Y, X) X 2 Y 2 k X 3 Y 3 φ ( Y, Y X 4 Y 4 X 5 Y 5 k + ) φ ( Y, X ) S 55 Statstcal Methods Sprng 20 9 Why ondtonal? Why (Y X), not (Y,X)? The network explctly does not encode any dstrbuton over the varables n X. One of the man strengths of the RF representaton. Ths flexblty allows us to do many thngs: Incorporatng nto the model a rch set of observed varables X whose dependences may be qute complex or even poorly understood. Includng contnuous varables X whose dstrbuton may not have a smple parametrc form Usng doman knowledge n order to defne a rch set of features characterzng our doman, wthout worryng about modelng ther jont dstrbuton. Many applcatons: omputer vson (detal later), text analyss, partof-speech labelng, many more S 55 Statstcal Methods Sprng

11 ondtonal Random Felds Drected and undrected dependences. RF defnes condtonal dstrbuton of Y on X, (Y X). It can be vewed as a partally drected graph, where we have an undrected component over Y, whch has the varables n X as parents. ny dfference wth ayesan networks? RF (partally drected) ondtonal ayesan network (fully drected) X X 2 X 3 X 4 X 5 X X 2 X 3 X 4 X 5 Y Y 2 Y 3 Y 4 Y 5 ~ ( Y, X) k φ ( Y, Y k + ) φ ( Y, X ) Y Y 2 Y 3 Y 4 Y 5 ( Y X) k ( Y k + Y ) ( Y X ) S 55 Statstcal Methods Sprng 20 2 han Networks ombnes Markov networks and ayesan networks artally drected graph (DG) s for undrected graphs, we have three dstnct nterpretatons for the ndependence assumptons mpled by a -DG xample: D S 55 Statstcal Methods Sprng 20 22

12 arwse Independences very node X s ndependent from any node whch s not ts descendant gven all non-descendants of X Formally: I (K) {(X Y ND(X)-{X,Y}) : X Y K, Y ND(X)} xample: (D,,) ( ) D S 55 Statstcal Methods Sprng Local Markov Independences Let oundary(x) be the unon of the parents of X and the neghbors of X Local Markov ndependences state that a node X s ndependent of ts non-descendants gven ts boundary Formally: I L (K) {(X ND(X)-oundary(X) oundary(x)) : X U} xample: (D,,) D S 55 Statstcal Methods Sprng

13 Global Independences I(K) {(X Y ) : X,Y,, X s c-separated from Y gven } X s c-separated from Y gven f X s separated from Y gven n the undrected moralzed graph M[K] The moralzed graph of a -DG K s an undrected graph M[K] by onnectng any par of parents of a gven node onvertng all drected edges to undrected edges For postve dstrbutons: I(K) I L (K) I (K) S 55 Statstcal Methods Sprng Doman pplcaton: Vson The mage segmentaton problem Task: artton an mage nto dstnct parts of the scene xample: separate water, sky, background S 55 Statstcal Methods Sprng

14 Markov Network for Segmentaton Grd structured Markov network (RF) Random varables (X, Y ) correspond to pxel X : Input mage for pxel (always gven) olor, texture, locaton Y : Doman s {,...K} (e.g. :road, 2:car, 3:bldg) (generally not gven) Value represents regon assgnment to pxel Neghborng pxels are connected n the network X X 2 X 3 X 4 Y Y 2 Y 3 Y 4 X 2 X 22 X 23 X 24 Y 2 Y 22 Y 23 Y 24 X 3 X 32 X 33 X 34 Y 3 Y 32 Y 33 Y 34 S 55 Statstcal Methods Sprng Markov Network for Segmentaton Node potentals (appearance dstrbuton) Introduce node potental exp(-w k {Y k}) w k extent to whch pxel fts regon k (e.g., based on X contanng varous nfo such as color, locaton, texture on pxel ) dge potentals (contguty preference) ncodes contguty preference by edge potental exp(λ{y Y j }) for λ>0 X X 2 X 3 X 4 Y Y 2 Y 3 Y 4 ppearance dstrbuton X 2 X 22 X 23 X 24 Y 2 Y 22 Y 23 Y 24 X 3 X 32 X 33 X 34 Y 3 Y 32 Y 33 Y 34 ontguty preference 28 4

15 Markov Network for Segmentaton ppearance dstrbuton π [Y, X ]exp(-w k {Y k}) Y Y 2 Y 3 2 Y 4 2 Y Y 2 Y 3 Y 4 Y 2 Y 22 2 Y 23 2 Y 24 3 Y 2 Y 22 Y 23 Y 24 Y 3 Y 32 2 Y 33 3 Y 34 3 Y 3 Y 32 Y 33 Y 34 Y: sky, Y2: car, Y3: buldng ontguty preference π,j [Y,Y j ] exp(λ{y Y j }) Soluton: nference on the parwse Markov network Fnd most lkely assgnment k (sky, buldng, etc) to Y varables ( Y X) π [, ] [, ( ) ] Y X π, j Y Yj X S 55 Statstcal Methods Sprng xample Results Input Input 2 aselne (a smple classfer): Result of segmentaton usng node potentals alone, so that each pxel s classfed ndependently Result of segmentaton usng a parwse Markov network encodng nteractons between adjacent pxels S 55 Statstcal Methods Sprng

16 Last tme Markov networks representaton Local factor models Independences π [ D ],..., π n[ D n global, parwse, local ndependences I-Map Factorzaton Today arameterzaton revsted ayesan nets and Markov nets artally drected graphs Inference 0 ] ( X,..., X n) π [ D ] S 55 Statstcal Methods Sprng 20 3 Inference Markov networks and ayesan networks represent a jont probablty dstrbuton Networks contan nformaton needed to answer any query about the dstrbuton Inference s the process of answerng such queres Drecton between varables does not restrct queres Inference combnes evdence from all network parts S 55 Statstcal Methods Sprng

17 7 Lkelhood Queres ompute probablty (lkelhood) of the evdence vdence: subset of varables and an assgnment e Task: compute (e) omputaton U e z e z ), ( ) ( S 55 Statstcal Methods Sprng ondtonal robablty Queres ondtonal probablty queres vdence: subset of varables and an assgnment e Query: a subset of varables Y Task: compute (Y e) pplcatons Medcal and fault dagnoss Genetc nhertance omputaton U Y U w e z e y w Y W e e y Y e y Y z ), ( ),, ( ) ( ), ( ) ( 34 S 55 Statstcal Methods Sprng 20

18 Maxmum osteror ssgnment Maxmum osteror ssgnment (M) vdence: subset of varables and an assgnment e Query: a subset of varables Y Task: compute M(Y e) argmax y (Yy e) Note : there may be more than one possble soluton Note 2: equvalent to computng argmax y (Yy, e) Why? (Yy e) (Yy, e) / (e) omputaton M( Y y e) argmax ( W w, Y y' e) y' w U Y S 55 Statstcal Methods Sprng Most robable ssgnment: M Most robable xplanaton (M) vdence: subset of varables and an assgnment e Query: all other varables Y (YU-) Task: compute M(Y e) argmax y (Yy e) Note: there may be more than one possble soluton pplcatons Decodng messages: fnd the most lkely transmtted bts Dagnoss: fnd a sngle most lkely consstent hypothess S 55 Statstcal Methods Sprng

19 Most robable ssgnment: M Note: We are searchng for the most lkely jont assgnment to all varables May be dfferent than most lkely assgnment (M) of each varable. ny example? Gven φ (a )>(a 0 ) M() a M(,) {a 0, b } (a 0, b 0 ) 0.04 (a 0, b ) 0.36 (a, b 0 ) 0.3 (a, b ) 0.3 () I a 0 a ( ) 0 a a S 55 Statstcal Methods Sprng xact Inference n Graphcal Models Graphcal models can be used to answer ondtonal probablty queres M queres M queres Naïve approach Generate jont dstrbuton Dependng on query, compute sum/max xponental blowup xplot ndependences for effcent nference S 55 Statstcal Methods Sprng

20 Summary: Markov network representaton Markov Networks undrected graphcal models Lke ayesan networks, defne ndependence assumptons Three defntons exst, all equvalent n postve dstrbutons Factorzaton s defned as product of factors over complete sub-graph lternatve parameterzatons Factor graphs Log-lnear models Relatonshp to ayesan networks Represent dfferent famles of ndependences Moralzaton transformng ayesan networks to Markov networks. Trangulaton transformng Markov networks to ayesan networks. artally drected graphs ondtonal random felds (RFs) pplcaton to mage segmentaton S 55 Statstcal Methods Sprng nnouncements Feedback on the course mal your comments anonymously. See the course webste. ddtonal OH Tuesday n the mornng 9-0am Slghtly modfed course outlne S 55 Statstcal Methods Sprng

21 Where are we? What next?. robablstc model representaton 2. xact nference n Ns -(Xx e)? 3. Learnng parameters/structure -Learnng Ds, structure from data 4. pproxmate nference -(Xx e)? 5. pplcatons - Decson makng, temporal processes cknowledgement These lecture notes were generated based on the sldes from rof ran Segal. S 55 Statstcal Methods Sprng