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1 Markov Models Lecture : Markov and Hidden Markov Models PSfrag Use past replacements as state. Next output depends on previous output(s): y t = f[y t, y t,...] order is number of previous outputs y t y t y t y t y t y t k Sam Roweis March 8, 00 Add noise to make the system probabilistic: p(y t y t, y t,..., y t k ) Markov models have two problems: need big order to remember past events output noise is confounded with state noise y t Probabilistic Models for Time Series Generative models for -series: To get interesting variability need noise. To get correlations across, need some system state. internal state Time: discrete States: discrete or continuous Outputs: discrete or continuous noise sources outputs Today: discrete state similar to finite state automata; Moore/Mealy machines Learning Markov Models The ML parameter estimates for a simple Markov model are easy: p(y, y,..., y T ) = p(y... y k ) p(y t y t, y t,..., y t k ) log p({y}) = log p(y... y k ) + t=k+ T log p(y t y t, y t,..., y t k ) t=k+ Each window of k + outputs is a training case for the model p(y t y t, y t,..., y t k ). Example: for discrete outputs (symbols) and a nd-order markov model we can use the multinomial model: p(y t = m y t = a, y t = b) = α mab The maximum likelihood values for α are: α mab = num[t s.t. y t = m, y t = a, y t = b] num[t s.t. y t = a, y t = b]
2 PSfrag replacements Hidden Markov Models (HMMs) Add a latent (hidden) variable x t to improve the model. HMM probabilistic function of a Markov chain :. st-order Markov chain generates hidden state sequence (path): P(x t+ = j x t = i) = S ij P(x = j) = π j. A set of output probability distributions A j ( ) (one per state) converts state path into sequence of observable symbols/vectors P(y t = y x t = j) = A j (y) (state transition diagram) Even though hidden state seq. is st-order Markov, the output process is not Markov of any order [ex.... ] x y HMM Graphical Model Hidden {x t }, outputs {y t } Joint probability factorizes: x y x y P({x}, {y}) = P(x t x t )P(y t x t ) t= T = π x S xt,x t+ A xt (y t ) t= t= NB: Data are not i.i.d. There is no easy way to use plates to show this model. (Why?) y T PSfrag replacements Applications of HMMs Speech recognition. Language modeling. Information retrieval. Motion video analysis/tracking. Protein sequence and genetic sequence alignment and analysis. Financial series prediction.... Links to Other Models You can think of an HMM as: A Markov chain with stochastic measurements. PSfrag replacements x y or A mixture model with coupled across. x y x y y T x x x y y y y T The future is independent of the past given the present. However, conditioning on all the observations couples hidden.
3 Probability of an Observed Sequence To evaluate the probability P({y}), we want: P({y}) = {x} P({x}, {y}) P(observed sequence) = P( observed outputs, state path ) all paths Looks hard! ( #paths = # T ). But joint probability factorizes: P({y}) = P(x t x t )P(y t x t ) x x t= = P(x )P(y x ) P(x x )P(y x ) x x P( )P(y T ) By moving the summations inside, we can save a lot of work. Naive algorithm: Bugs on a Grid. start bug in each state at t= holding value 0. move each bug forward in : make copies & increment the value of each copy by transition prob. + output emission prob.. go to until all bugs have reached T. sum up values on all bugs We want to compute: The forward (α) recursion L = P({y}) = {x} P({x}, {y}) There is a clever forward recursion to compute the sum efficiently. α j (t) = P( y t, x t = j ) α j () = π j A j (y ) α k (t + ) = { α j (t)s jk }A k (y t+ ) j Bugs on a Grid - Trick Clever recursion: adds a step between and above which says: at each node, replace all the bugs with a single bug carrying the sum of their values Notation: x b a {x a,..., x b }; y b a {y a,..., y b } This enables us to easily (cheaply) compute the desired likelihood L since we know we must end in some possible state: L = k α k (T ) α This trick is called dynamic programming, and can be used whenever we have a summation, search, or maximization problem that can be set up as a grid in this way. The axes of the grid don t have to be and.
4 Inference of Hidden States What if we we want to estimate the hidden given observations? To start with, let us estimate a single hidden state: p(x t {y}) = γ(x t ) = p({y} x t)p(x t ) p({y}) = p(yt x t)p(y T t+ x t)p(x t ) p(y T ) = p(yt, x t)p(y T t+ x t) p(y T ) p(x t {y}) = γ(x t ) = α(x t)β(x t ) p(y T ) where α j (t) = p( y t, x t = j ) β j (t) = p(y T t+ x t = j ) γ i (t) = p(x t = i y T ) Forward-Backward Algorithm α i (t) gives total inflow of prob. to node (t, i) β i (t) gives total outflow of prob. α Bugs again: we just let the bugs run forward from 0 to t and backward from T to t. In fact, we can just do one forward pass to compute all the α i (t) and one backward pass to compute all the β i (t) and then compute any γ i (t) we want. Total cost is O(K T ). β Forward-Backward Algorithm We compute these quantites efficiently using another recursion. Use total prob. of all paths going through state i at t to compute the conditional prob. of being in state i at t: where we defined: γ i (t) = p(x t = i y T ) = α i (t)β i (t)/l β j (t) = p(y T t+ x t = j ) There is also a simple recursion for β j (t): β j (t) = i S ji β i (t + )A i (y t+ ) Likelihood from Forward-Backward Algorithm Since x t γ(x t ) =, we can compute the likelihood at any using the results of the α β recursions: L = p({y}) = x t α(x t )β(x t ) In the forward calculation we proposed originally, we did this at the final step t = T : L = α( ) because β T =. β j (T ) = α i (t) gives total inflow of prob. to node (t, i) β i (t) gives total outflow of prob.
5 Using HMMs for Recognition Profile (String-Edit) HMMs Use many HMMs for recognition by:. training one HMM for each class (requires labelled training data). evaluating probability of an unknown sequence under each HMM. classifying unknown sequence: HMM with highest likelihood d i d i d i dt it m m m mt L L This requires the solution of two problems:. Given model, evaluate prob. of a sequence. (We can do this exactly & efficiently.). Give some training sequences, estimate model parameters. (We can find the local maximum of parameter space nearest our starting point using Baum-Welch (EM).) Lk i = insert d = delete m = match (state transition diagram) A profile HMM or string-edit HMM is used for probabilistically matching an observed input string to a stored template pattern with possible insertions and deletions. Three kinds of : match, insert, delete. m n use position n in the template to match an observed symbol i n insert extra symbol(s) observations after template position n d n delete (skip) template position n Viterbi Decoding The numbers γ j (t) above gave the probability distribution over all at any. By choosing the state γ (t) with the largest probability at each, we can make a best state path. This is the path with the maximum expected number of correct. But it is not the single path with the highest likelihood of generating the data. In fact it may be a path of prob. zero! To find the single best path, we do Viterbi decoding which is just Bellman s dynamic programming algorithm applied to this problem. The recursions look the same, except with max instead of. Bugs once more: same trick except at each step kill all bugs but the one with the highest value at the node. How do we fill in the costs for a DP grid using a string-edit HMM? Almost the same as normal except: Now the grid is s its normal height. It is possible to move down without moving right if you move into a deletion state. DP for Profile HMMs eg: template length=, test sequence length= deletions insertions matches y y y y Test Sequence Position y (emit on arrival)
6 String-Edit HMM Grid Costs C x x = log T x,x log A x (y t ) if x is match or insert C x x = log T x,x if x is a delete state d i Profile HMMs have Linear Costs d d dt i i it State x {m n, i n, d n } has nonzero transition probabilities only to x {m n+, i n, d n+ }. deletions insertions matches y y y y Test Sequence Position y (emit on arrival) m m m mt i = insert d = delete m = match number of = (length template) (state transition diagram) Only insert and match can generate output symbols. Once you visit or skip a match state you can never return to it. At most destination from any state, so S ij very sparse. Storage/Time cost linear in #, not quadratic. State variables and observations no longer in sync. (e.g. y:m ; d ; y:i ; y:i ; y:m ;...) Forward-Backward for Profile HMMs The equations for the delete in profile HMMs need to be modified slightly, since they don t emit any symbols. For delete k, the forward equations become: α k (t) = α j (t)s jk j which should be evaluated after the insert and match state updates. For all, the backward equations become: β k (t) = S ki β i (t + )A i (y t+ ) + S kj β j (t) i match,ins j del which should be evaluated first for delete k; then for the rest. The gamma equations remain the same: γ i (t) = p(x t = i y T ) = α i(t)β i (t)/l Notice that each summation above contains only three terms, regardless of the number of! Initializing Forward-Backward for Profile HMMs The initialization equations for Profile HMMs also need to be fixed up, to reflect the fact that the model can only begin in m, i, d and can only finish in m N, i N, d N. In particular, π j = 0 if j is not one of m, i, d. When initializing α k (), delete k have zeros, and all other have the product of the transition probabilities through only delete up to them, plus the final emission probability. When initializing β k (T ), the same kind of adjustment must be made.
7 Some HMM History Markov ( ) and later Shannon ( 8, ) studied Markov chains. Baum et. al (BP, BE, BS 8, BPSW 0, B ) developed much of the theory of probabilistic functions of Markov chains. Viterbi ( ) (now Qualcomm) came up with an efficient optimal decoder for state inference. Applications to speech were pioneered independently by: Baker ( ) at CMU (now Dragon) Jelinek s group ( ) at IBM (now Hopkins) communications research division of IDA (Ferguson unpublished) Dempster, Laird & Rubin ( ) recognized a general form of the Baum-Welch algorithm and called it the EM algorithm. A landmark open symposium in Princeton ( 80) hosted by IDA reviewed work till then. Example Problem on HMM Inference Consider an HMM with and a sequence of length. As you know, HMM inference can be converted into path problems on discrete grids and thus made isomorphic to dynamic programming. The rows of the grid correspond to and the columns to steps. Any state sequence is a path through the grid that starts at the leftmost column and ends in the rightmost column, going through exactly one node per column. The cost of moving from one node to another corresponds to negative log probability. Such a grid is shown below, with all the costs filled in. (a) Write an expression for the cost of moving from node (i, t ) to node (j, t) in terms of the probabilities S ij of transitioning from state i to state j and the probabilities A j(y t) of emitting observation y t from state j (t > ). (b) Write an expression for the cost of starting at any node (k, ) in the leftmost column in terms of the probability of the initial distribution π k amd the probability of emitting the first symbol y from state k. (c) Find the log probability (negative of the total cost) of the sequence in the grid below. (d) Find the most likely hidden state sequence (smallest cost) and its cost given the grid below. Use {A,B,C} to denote the A B C
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