Graphical Event Models and Causal Event Models. Chris Meek Microsoft Research

Transcription

1 Graphical Even Models and Causal Even Models Chris Meek Microsof Research

2 Graphical Models Defines a join disribuion P X over a se of variables X = X 1,, X n A graphical model M =< G, Θ > G =< X, E > is a direced acyclic graph. Θ = {Θ 1,, Θ n } where Θ i defines he condiional disribuion P(X i π i ) where π i are he parens of X i in G. Learning: Assume we see many draws from P X. X 1 X 2 X 3 P X 1 = f 1 X 1, Θ 1 P X 2 = f 2 X 2, Θ 2 P X 3 X 1, X 2 = f 3 (X 3, X 1, X 2, Θ 3 )

3 Graphical Models Explaining away ype reasoning Wha is probabiliy of Burglary given AlarmSound? Wha is probabiliy of Burglary given AlarmSound and a NewsRepor of an earhquake? Earhquake? Burglary? NewsRepor? AlarmSound?

4 Graphical Models Explaining away ype reasoning Wha is probabiliy of Burglary given AlarmSound? Wha is probabiliy of Burglary given AlarmSound and a NewsRepor of an earhquake? Wha if he NewsRepor said he earhquake was afer he Alarm wen off?

5 Ouline Temporal Even Sequences Graphical Even Models Learning Graphical Even Models Learning Causal Dependencies Causal Even Model Graphical Even Models

6 Modeling emporal even sreams A emporal even sream is a ime-samped sream of labeled evens. This ype of daa is pervasive: daacener even logs, search queries, Wan o model: Wha evens will happen when, based on wha evens have happened when.

7 Temporal Even Sequences: Even Logs from a Daacener LinkFail SQLTimeou CDriveFail LinkFail SQLTimeou LinkFail HomailAuhFail LinkFail QueueOverflow SQLTimeou QueueOverflow QueueOverflow QueueOverflow CDriveFail QueueOverflow ReimageFail SendTech 2:15 pm 2:30 pm 2:45 pm 3:00 pm L is he se of possible evens (i.e., hings ha can happen) D = 1, l 1, 2, l 2, 3, l 3, n, l n where l L and i < i+1

8 Marked poin processes Trea daa as a realizaion of a marked poin process: x = 1, l 1,, n, l n Forward in ime likelihood: p x = n i=1 p( i, l i h i ) where he hisory h i = h i (x) = 1, l 1,, i 1, l i 1 Any p( i, l i h i ) can be represened via condiional inensiies λ l i h i : p i, l i h i = l l l=l λ l i h i i e i λ 0 l τ h i dτ

9 Proof skech Given pdf p() define: λ p() 1 0 p τ dτ P() Then, Calculus: P = λ 1 P P = 1 e 0 λ τ dτ λ τ dτ p = λ e 0 Given p, l = p p(l ) define λ l λ p l Then p, l = p p l = λ l e l 0 λl τ dτ

10 Condiional inensiies p i, l i h i = l λ l i h l l=l i i e i 0 λ l τ h i dτ

11 Condiional inensiies λ h λ h p i, l i h i = l λ l i h l l=l i i e i 0 λ l τ h i dτ

12 Condiional inensiies λ h λ h p i, l i h i = l λ l i h l l=l i i e i 0 λ l τ h i dτ

13 Condiional inensiies λ h λ h p i, l i h i = l λ l i h l l=l i i e i 0 λ l τ h i dτ

14 Condiional inensiies λ h λ h p i, l i h i = l λ l i h l l=l i i e i 0 λ l τ h i dτ

15 Condiional inensiies λ h λ h p i, l i h i = l λ l i h l l=l i i e i 0 λ l τ h i dτ

16 Condiional inensiies λ h λ h p i, l i h i = l λ l i h l l=l i i e i 0 λ l τ h i dτ

17 Condiional inensiies λ h λ h p i, l i h i = l λ l i h l l=l i i e i 0 λ l τ h i dτ

18 Condiional inensiies λ h λ h p i, l i h i = l λ l i h l l=l i i e i 0 λ l τ h i dτ

19 Condiional inensiies λ h λ h p i, l i h i = l λ l i h l l=l i i e i 0 λ l τ h i dτ

20 Condiional inensiies λ h λ h p i, l i h i = l λ l i h l l=l i i e i 0 λ l τ h i dτ

21 Condiional inensiies λ h λ h p i, l i h i = l λ l i h l l=l i i e i 0 λ l τ h i dτ

22 Condiional inensiies λ h λ h 1) Behavior of differen even ypes specified separaely 2) Highlighs dependency of each even ypes on hisory

23 Even Sequence Noaion Example: L = {a, b, c} D = 1, l 1, n, l n e.g., 1, a, 3, b, 3 = 5, a, 8, c h i = h(, D) is he hisory up o i h 3 = 1, a, 3, b, (5, a) h A is he filered hisory for A L D a = { 1, a, 5, a }

24 Graphical Even Models A Graphical Even Model (GEM) is a pair G, Θ Verices for each even ype L = a, b, c Edges represen poenial dependencies Θ l Θ parameerizes inensiy funcion for l λ a h, Θ a λ b h, Θ b λ c h, Θ c = λ a h πa, Θ a = λ b h πb, Θ b = λ c h πc, Θ c

25 Learning Graphical Even Models Specify funcional form(s) for inensiies λ l wih separae parameers for each even Likelihood facors according o L so we can learn each inensiy funcion separaely. Bayesians also require facorizaion of prior Search over space of direced graphs Add/remove parens ha improve he score

26 Piecewise-Consan CIMs (PCIM) Idea: resric λ l i h i o be piecewise consan in for all even sequences A sae funcion σ(, h) maps hisories o a discree se of saes Σ A PCIM is a pair M = S, Θ where Srucure S= σ l, h, Σ l l L Parameers Θ = Θ l l L and Θ l = λ ls s Σl

27 Piecewise-Consan CIMs σ a (, h) σ b (, h) σ c (, h) a: b: c: Sae Funcions (coloring) λ a ( ) = 0 λ a ( ) = 0.1 λ a ( ) = 10. λ c ( ) = 10

28 Piecewise-Consan CIM A PCIM (CIM) M = S, Θ (where Θ = {Θ 1,, Θ L } and Θ l = λ ls s Σl ) has likelihood c(l,s) p D M = λ ls e λ ls d(l,s) l L s Σ l c(l, s) is he coun of even l when σ l, h = s in D d(l, s) is he oal duraion of σ l, h = s in D

29 Piecewise-Consan CIM Produc of Gammas is conjugae prior, even hough he likelihood isn a produc of exponenials! p λ ls α ls, β ls = β ls α λ ls 1 Γ α ls e β ls λ ls ls Closed-form poserior: p λ ls α ls, β ls, D, S = p λ ls α ls + c(l, s), β ls + d(l, s) Closed-form marginal likelihood: p D S = γ ls D ls α ls γ ls D = β ls Γ α ls Γ α ls + c(l, s) β ls + d(l, s) α ls+c(l,s)

30 Piecewise-Consan CIM Defining PCIM Srucures Example Le B = f 1, h,, f n (, ) where f i, h is a basis sae funcion (BSF) A family of srucures S(B) is obained by combining BSFs. We use decision rees bu one could use decision graphs. no σ l, h = r λ lr f i, h = s i? no σ l, h = s λ ls yes f j, h = s j? yes σ l, h = λ l

31 Piecewise-Consan CIM Example Types of basis sae funcions f, h Even-ype specific sae funcions f, h = f(, h l ) depends only on he hisory of a specific even ype Windowed sae funcions f, h = f, h s, e depends only on he hisory during a window relaive o ime. Hisorical sae funcions f, h depends on he las evens ha have happened bu no heir imes.

32 Piecewise-Consan CIM σ a (, h) σ b (, h) σ c (, h)

33 Piecewise-Consan CIM:Learning We use a Bayesian Model selecion approach o choose S S B For each l L Sar wih empy decision ree (i.e.,, h σ l, h = k) For each leaf in decision ree Evaluae each possible spli (f B, s) Choose spli ha mos improves he marginal likelihood Alernaively one could use MCMC o average over S B.

34 Example: λ RebooFail decision ree false IniReboo [30 min] rue 216 Even ypes λ = 0 false SuccReboo [30min] rue false Unhealh VersionCheck [30 min] rue λ = 0 λ = 3.36 false Healhy VersionCheck [30 min] rue λ = 41.4 λ = 0

35 Learning Causal Graphical Models Direced Acyclic Assumpion (DAG) causal model can be represened by a direced acyclic graph G = X, E Daa Assumpion: world is described by some P X and he observed world is some P O O X Reliable Informaion Assumpion (Reliable) he world provides reliable informaion abou independencies among observed variables O X. I(A, C, B) means A is independen of B given C

36 Learning Causal Graphical Models Assumpions ha connec observed world P and causal model G Causal Markov Assumpion (CMA): If d G A, B, C I P A, B, C Noe 1: d G (A, B, C) is d-separaion: a verex separaion crierion Noe 2: A graphical model non-causal Markov w.r.. P X i defines Causal Faihfulness Assumpion (CFA): If I P A, B, C d G (A, B, C)

37 Learning Causal Graphical Models Learning Scenarios Complee daa: All variables observed (X = O) Causal sufficiency: No pair of observed variables have unobserved common ancesors. General case: O X Goal: In each learning scenario, use independence facs o idenify causal informaion common o every graph wih hose independencies/separaion facs.

38 Learning Causal Graphical Models Manra: Correlaion does no imply causaion Complee daa: Indisinguishable A B A B General case: Also indisinguishable H H H A B A B A B

39 Learning Causal Graphical Models Ineresing general case resul Under he assumpions DAG, Reliable, CMA and CFA If he only independence facs we observe o hold are I A,, B, I A, C, D, I(B, C, D) hen C is a cause of D. A C D B

40 Learning Causal GEMs Sep 1: Change he separaion crierion from d-separaion o δ-separaion δ(a, C, B) in G = L, E if and only if d(a, C, B) in G B where G B = L, E B and E B = l 1, l 2 E l 1 B Sep 2: Change from independence ess o facorizaion (process independence) ess Sep 3: Assume analog of CMA, CFA, Reliable Sep 4: Prove hings

41 Learning Causal GEMs Learning Scenarios Complee daa: All variables observed (X = O) Resul: Can recover he srucure. Causal sufficiency: No pair of observed variables have unobserved common ancesors. Resul: Can recover he srucure over O. (all causes) General case: O X In Progress: Some sufficien condiions for cause Some sufficien condiions for non-cause Some sufficien condiions for exisence of unmeasured common cause

42 Open Issues Characerize wha one can learn in he general case Jusificaion of using Process Independence o learn cause (e.g., compleeness of δ-separaion) Principled approaches o esing process independence saemens. Consisency of score learning for process independence. Relaxing assumpions such as he reliabiliy assumpion