Chapter 4 Dynamic Bayesian Networks Fall Jin Gu, Michael Zhang

Transcription

1 Chapter 4 Dynamic Bayesian Networks 2016 Fall Jin Gu, Michael Zhang

2 Reviews: BN Representation Basic steps for BN representations Define variables Define the preliminary relations between variables Check the dependences and independences Define local conditional probability distributions Mixture models Latent factor analysis

3 Mixture Model Cancer is a general disease category consisting of many different types of diseases. For example, breast cancer has four major subtypes: normal-like, basal, luminal A and luminal B, with distinct clinical outcomes. Each subtype has different gene expression patterns. We need to infer the subtypes based on the observed gene expressions. Variables: subtypes (s), gene expressions (e) Relation: subtypes define the patterns of expressions s e N 1) Circles represent variables. Shadow/white circles represent observed/unobserved variables. 2) Rectangle indicates the variables in the same logic layer. The number (N) at the right-bottom corner represents the number of samples. 3

4 Variables: subtypes (s), gene expressions (e) Relation: subtypes define the patterns of expressions Important: carefully check the independences in your model s e N π μ s e Σ N Consider the local CPDs Different subtypes have different occurrence probability (s) K P s k, 1 i k1 Each expression pattern (indicated by i) independently follows different Gaussian distributions given the cancer subtype (e s) e, p s k k i k 4

5 Latent Factor Model Cancer is usually caused by multiple driving processes (factors), such as sustainable proliferation, resistance to cell death, immune escape and promoted vascular growth, etc. These driving processes have combined effects on the gene expressions patterns. We want to infer the driving processes based on large-scale gene expression datasets. Variables: driving factors (z i ), gene expressions (e j ) Relations: z determines the distribution of e Local CPDs Driving process z i follows a standard Gaussian distribution p z 0,1, i 1,, K i Expression e j also follows a Gaussian distribution but its mean is determined by a linear model of z p e z z z i1 K 2 j 1,..., k 0 i i, j 5

6 Variables: independent driving processes (z i ), gene expressions (e j ) Relations: z determines the distribution of e Local CPDs Driving process z i follows a standard Gaussian distribution p z 0,1, i 1,, K e 2 i Expression e j also follows a Gaussian distribution but its mean is determined by a linear model of z e 1 α 1 τ 1 z 1 z K e L N α 2 τ 2 α L τ L p e z z z i1 K 2 j 1,..., k 0j ij i, j 6

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8 Reviews: BN Representation Basic steps for BN representations Define variables Define the preliminary relations between variables Check the dependences and independences Define local conditional probability distributions Questions A vector or scalar for a variable? how to deal with the variables with unknown independences? (Recall assignment #2)

9 Outlines Stochastic Process and Markov property Dynamic Bayesian Networks Hidden Markov Models (HMMs) State-Observation Models A Few Examples of HMM Representations Calculations for HMMs Generate Simulated Data Extended Models

10 Markov / Memoryless Property Add the time dimension to variables X X (t) Assumptions X 0 X 1 X 2 X 3 Time can be discretized into interesting points t 1,t 2,...t n ( ) n P X P( X ( t) X (1),... X ( t 1)) Markov assumption holds P( X ) t 1 n P( X ( t) X ( t t Distribution is stationary (time invariant or homogeneous) 1 1)) i, j : P( X ( i) X ( i 1)) P( X ( j) X ( j 1))

11 Example: Random Walk Define X (t) as the coordinate of particle at time point t. The particle can randomly move to neighbor variables (nodes) with fixed probability. Then, P(X (t+1) ) is only determined by X (t). lysing-in-vivo-cell-migration-data/

12 Dynamic Bayesian Networks for Time Invariant Process/System

13 Consider a smoke detection tracking application, where we have 3 rooms connected in a row. Each room has a true smoke level (X) and a measured smoke level (Y) by a smoke detector situated in the middle of the room. Which of the below are the proper DBN structure(s) for this problem? X X 1 1 X 1 X 1 Y 1 Y 1 X 2 X 2 X 2 X 2 Y 2 Y 2 X 3 X 3 X 3 X 3 Y 3 Y 3 X 1 X 1 X 1 X 1 Y 1 Y 1 X 2 X 2 X 2 X 2 Y 2 Y 2 X 3 X 3 X 3 X 3 Y 3 Y 3

14 The Parameterization of Dynamic BNs Usually assume time-invariant systems (parameters are constant over time) The parameters in each time slice The parameters between internal state variables The emission or observation parameters between state variables and observation variables The transition parameters between two consecutive time slices

15 Dice problem You stay in a casino for a long time. You find that the probability of dice is equal for each of its six faces (1-6). But you doubt that there must be some trick on the dice. So you record the series of the dice number of each play, and want to find the problem... Hidden Markov Models Probabilistic Graphic Models, Fall

16 Dice problem Hidden Markov Models Probabilistic Graphic Models, Fall

17 Hidden Markov Models S 0 S 1 S 2 S 3 State variable S (dice index 1~K) O 1 O 2 O 3 Observation variable O (dice number 1~6) Local structure #1 Hidden states (cannot be observed) are linked as Markov chain: P(S T S t<t ) = P(S T S T-1 ) Local structure #2 Observable variables are only determined by the hidden state: P(O T S t<=t ) = P(O T S T )

18 State-Observation Models If the hidden variables can be discretized into several states, HMM can be represented as State- Observation Models. S 0 S 1 S 2 O 1 O 2 S 3 O 3 Hidden states (cannot be observed) are linked as Markov chain: P(S T S t<t ) = P(S T S T-1 ) Observable variables are only determined by the hidden state: P(O T S t<=t ) = P(O T S T )

19 State-Observation Models State-State Transmission Matrix T (n,n) S t = TS t-1 State-Observation Emission Matrix E (m,n) O t = ES t

20 Example: Stock Market Index We can define three different states: UP/DOWN/UNCHANGE Hidden Markov Models Probabilistic Graphic Models, Fall

21 Calculations in HMMs Problem 1: P X θ, given the model and the observation sequence, infer the probability of getting that observation sequence from the model Problem 2: argmax Y P(X, Y θ), given the model and the observation sequence, infer the hidden labels of the sequence Problem 3: argmax θ P(X θ), if parameters are un known, learn them from the observation sequence

22 Question #1: P X θ Inference

23 Formal Model Representation HMMs (discretized) have the below parameters and the observation sequences X T ti, j, i, j 1... N, E ei, j, i 1... N, j 1... K,, 1... i i N X x, t 1,..., T t Hidden Markov Models Probabilistic Graphic Models, Fall

24 Compute P(X θ) i P x,...,, 1 x y i t t t Forward/Backward algorithm Initialization: Induction: i e i i x1 1, N t 1 i t j t j, i ei, x j1 Termination: t 1 P X N T i1 i Y 1 X 1 Y 2 X 2 Y 3 X 3

25 i P x,,, 1 x y i Compute P(X θ) t t T t Forward/Backward algorithm Initialization: Induction: Termination: T N i 1 i t e j t i, j j, xt 1 t1 j1 Y 1 Y 2 X 1 X 2 P X i e j N 0 j j, x1 1 j1 Y 3 X 3

26 Full Chain Local Transition (forward) Local Transition (Backward) Figures from Rabiner LR. Proceedings of the IEEE 1989, 77(2):

27 Question #2: argmax Y P(X, Y θ) Inference

28 Full Chain argmax Y P(X, Y θ) Figures from Rabiner LR. Proceedings of the IEEE 1989, 77(2): Infer the most possible states

29 Compute argmax Y P(X, Y θ) HMM inference (Viterbi algorithm) Known transition matrix and emission matrix Infer the maximum probability STATE series The probability at time 0 with observation x 0 For Y i e 1 : 1, i i i, x The probability at time 1 with observation x 1 e max e t e max t 2, i i, x y y, x y, i i, x 1, y y, i y 1,, N y 1,, N i arg max y ty i 2 1, 1 1, y 1,, N 1 Hidden Markov Models Probabilistic Graphic Models, Fall

30 Compute argmax Y P(X, Y θ) HMM inference (Viterbi algorithm) The probability at time t with observation x t t, i i, x t1, y y, i y 1,, N i arg max 1, t, For the last observation e t t1 t1 t1 t t yt1 yt1 i y 1,, N t1 max t e max t T, i i, x 1,, 1,, T y N y y i T T 1 T 1 T 1 i arg max 1, t, T T yt 1 yt 1 i y 1,, N T 1 y * T arg max y T 1,, N T, y T Hidden Markov Models Probabilistic Graphic Models, Fall

31 Compute argmax Y P(X, Y θ) HMM inference (Viterbi algorithm) After y t is inferred, trace back to get other y i y * T arg max y T 1,, N We can infer other y i * * y i y arg max t T T Hidden Markov Models Probabilistic Graphic Models, Fall T, y i arg max 1, t, t t yt 1 yt 1 i y 1,, N t1 T 1 T T T 1, y 1 y 1, i y 1,, N T 1 arg max y i y t * * t1 t t y 1,, N t 1, yt 1 yt 1, i t1 T

32 Question #3: argmax θ P(X θ) Learning

33 Compute argmax θ P(θ X) Baum Welch algorithm Problem 3: if all parameters are unknown, we need to learn the model from observations Unknown variables The probability of y=i for time point t and y=j for the following time point t+1 i, j P y i, y j X, t t t1 Hidden Markov Models Probabilistic Graphic Models, Fall

34 Baum Welch algorithm T Use forward and backward algorithm to calculate probability i, j P y i, y j X, t t t1 i j t1, t Y Y i1 j1 i t e t i, j j, x t1 i t, E, t i, j t1 j, x j j e t1 0 i P x 1,, x, y i t t t 0 i P x 1,, x y i, t t T t Hidden Markov Models Probabilistic Graphic Models, Fall

35 Compute argmax θ P(X θ) Baum Welch algorithm The probability of Y=i for time point t: t i Y t j1 i, j So expected times for state stayed in Y=i and the expected times for state transition i-j: T 1 t i, t i j t0 T 1 t0 Hidden Markov Models Probabilistic Graphic Models, Fall

36 Compute argmax θ P(X θ) Baum Welch algorithm Re-estimate all parameters (maximize) t T 1 t t0 i, j T 1 t0 t i, j i t0 Repeat above steps until convergence e ix, T x x i T t t0 0 P X P X log log t i t Hidden Markov Models Probabilistic Graphic Models, Fall

37 Gibbs Sampling for Learning You can read textbooks and supplementary materials. Understand the prior setting of unknown parameters. Tobias Ryden. EM verus Markov chain Monte Carlo for Estimation of Hidden Markov Models: A Computational Perspective. Bayesian Analysis 2008, 3(4): Zoubin Grahramani, Michael I Jordan. Factorial Hidden Markov Models. Machine Learning 1997, 29(2-3):

38 MCMC & Gibbs Sampling In statistics and in statistical physics, Gibbs sampling or a Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm for obtaining a sequence of random samples from multivariate probability distribution (i.e. from the joint probability distribution of two or more random variables), when direct sampling is difficult. Its theory and detailed methods will be introduced in the following sections.

39 Generate the hidden variables using current parameter Y t ~P Y X, θ t Step 1: Inference (Gibbs Sampling) Initialization: generate the labeling sequencing according to the prior probability Y y, 1 t T t Re-generate Y t according to the initial setting P y Y, X, P y y, y, x, t t t1 t1 t P y, x y, y, P y y, t1 t t t1 t t1 t t1 t t t1 t t1 t t1 t1 t t t t t1 y, y y, e y, y P y, x y, P y y, P x y, P y y, t e t i We get a score proportional to the probability. So we can randomly generate Y t based on these scores.

40 Step 2: Parameter Re-Estimation Simply update the parameter in transition and emission probability matrix t i,j = e k,j = T t=1 T t=1 I Y t =j,y t 1 =i T I Y t =j,x t =k T Note: I is an indicator function If the chain is not long enough, run multiple runs of step 1 and then do step 2

41 Generate simulated data using HMMs Find the initial probability for hidden states Markov chain Monte Carlo (MCMC) They are a class of algorithms for sampling from probability distributions based on constructing a Markov chain that has the desired distribution as its equilibrium distribution. The state of the chain after a large number of steps is then used as a sample of the desired distribution. Typical use of MCMC sampling can only approximate the target distribution, as there is always some residual effect of the starting position. More sophisticated MCMC-based algorithms such as coupling from the past can produce exact samples. A good chain will have rapid mixing (the stationary distribution is reached quickly starting from an arbitrary position) described further under Markov chain mixing time. Hidden Markov Models Probabilistic Graphic Models, Fall

42 Extended Models Factorial HMMs Hierarchical HMMs (GeneScan) Bayesian HMMs Two HMMs comparisons Max Entropy Markov Models Conditional Random Fields Hidden Markov Models Probabilistic Graphic Models, Fall

43 Factorial HMMs Hierarchical HMMs What ever complex structure if you can solve it. Tree Structured HMMs

44 Max Entropy Markov Models In machine learning, a maximum-entropy Markov model (MEMM), or conditional Markov model (CMM), is a graphical model for sequence labeling that combines features of hidden Markov models (HMMs) and maximum entropy (MaxEnt) models. An MEMM is a discriminative model that extends a standard maximum entropy classifier by assuming that the unknown values to be learned are connected in a Markov chain rather than being conditionally independent of each other. Y 0 Y 1 Y 2 Y 3 Y 0 Y 1 Y 2 Y 3 X 1 X 2 X 3 X 1 X 2 X 3 HMM MEMM Hidden Markov Models Probabilistic Graphic Models, Fall

45 Max Entropy Markov Models Hidden Markov Models Probabilistic Graphic Models, Fall

46 Supplementary Materials [1] Lawrence R Rabiner. A tutorial on Hidden Markov Models and selected applications in speech recognition. Proceedings of the IEEE 1989, 77(2): [2] Zoubin Ghahramani. An introduction to Hidden Markov Models and Bayesian Networks. International Journal of Pattern Recognition and Artificial Intelligence 2001, 15(1):9-42.

47 The End of Chapter 4 HMM is a generative model HMM is powerful for modeling hidden mechanisms Hidden Markov Models Probabilistic Graphic Models, Fall

48 Practice for HMMs Two different parts: Part I is recommended to be finished in three weeks); and, Part II can be finished later The full report should be submitted before December 5

49 Literature Survey Neural networks & deep learning Deep belief networks Convolutional neural networks Recurrent neural networks Residual neural networks Bayesian induction for concept learning Latent Dirichlet allocation & Dirichlet process Conditional random fields

50 Course Project 1 st choice: make your own proposal before Oct 17 (recommended!) Please send s to Dongfang Wang Requirements Clearly describe your problems & aims The availability of data The source of your project (by your supervisor or an international competition or your own interest?) 50

51 Course Project 2 nd choice: object recognition of sundial ( 日晷 ). Do image segmentation and target object recognition. (Major model: MRFs) 3 rd choice: molecular subtyping of liver cancer. Identify consensus gene expression patterns from multiple datasets. (Major model: hierarchical models / LDA) You will be organized into groups, but every member needs to write his own project report