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EM algorithm LING 572 Fei Xia 03/02/06
Outline • The EM algorithm • EM for PM models • Three special cases • Inside-outside algorithm • Forward-backward algorithm • IBM models for MT
Basic setting in EM • X is a set of data points: observed data • Θ is a parameter vector. • EM is a method to find θML where • Calculating P(X | θ) directly is hard. • Calculating P(X,Y|θ) is much simpler, where Y is “hidden” data (or “missing” data).
The basic EM strategy • Z = (X, Y) • Z: complete data (“augmented data”) • X: observed data (“incomplete” data) • Y: hidden data (“missing” data) • Given a fixed x, there could be many possible y’s. • Ex: given a sentence x, there could be many state sequences in an HMM that generates x.
The log-likelihood function • L is a function of θ, while holding X constant:
The iterative approach for MLE In many cases, we cannot find the solution directly. An alternative is to find a sequence: s.t.
Jensen’s inequality log is a concave function
Maximizing the lower bound The Q function
The Q-function • Define the Q-function (a function of θ): • Y is a random vector. • X=(x1, x2, …, xn) is a constant (vector). • Θt is the current parameter estimate and is a constant (vector). • Θ is the normal variable (vector) that we wish to adjust. • The Q-function is the expected value of the complete data log-likelihood P(X,Y|θ) with respect to Y given X and θt.
The inner loop of the EM algorithm • E-step: calculate • M-step: find
L(θ) is non-decreasing at each iteration • The EM algorithm will produce a sequence • It can be proved that
The inner loop of the Generalized EM algorithm (GEM) • E-step: calculate • M-step: find
Idea #2: find the θt sequence No analytical solution iterative approach, find s.t.
Idea #3: find θt+1 that maximizes a tight lower bound of a tight lower bound
Idea #4: find θt+1 that maximizes the Q function Lower bound of The Q function
The EM algorithm • Start with initial estimate, θ0 • Repeat until convergence • E-step: calculate • M-step: find
Important classes of EM problem • Products of multinomial (PM) models • Exponential families • Gaussian mixture • …
PM models Where is a partition of all the parameters, and for any j
PCFG • PCFG: each sample point (x,y): • x is a sentence • y is a possible parse tree for that sentence.
Maximizing the Q function Maximize Subject to the constraint Use Lagrange multipliers
Optimal solution Expected count Normalization factor
PM Models is rth parameter in the model. Each parameter is the member of some multinomial distribution. Count(x,y, r) is the number of times that is seen in the expression for P(x, y | θ)
The EM algorithm for PM Models • Calculate expected counts • Update parameters
PCFG example • Calculate expected counts • Update parameters
The EM algorithm for PM models // for each iteration // for each training example xi // for each possible y // for each parameter // for each parameter
Inner loop of the Inside-outside algorithm Given an input sequence and • Calculate inside probability: • Base case • Recursive case: • Calculate outside probability: • Base case: • Recursive case:
Inside-outside algorithm (cont) 3. Collect the counts 4. Normalize and update the parameters
Expected counts for PCFG rules This is the formula if we have only one sentence. Add an outside sum if X contains multiple sentences.
Relation to EM • PCFG is a PM Model • Inside-outside algorithm is a special case of the EM algorithm for PM Models. • X (observed data): each data point is a sentence w1m. • Y (hidden data): parse tree Tr. • Θ (parameters):
The inner loop for forward-backward algorithm Given an input sequence and • Calculate forward probability: • Base case • Recursive case: • Calculate backward probability: • Base case: • Recursive case: • Calculate expected counts: • Update the parameters:
Relation to EM • HMM is a PM Model • Forward-backward algorithm is a special case of the EM algorithm for PM Models. • X (observed data): each data point is an O1T. • Y (hidden data): state sequence X1T. • Θ (parameters): aij, bijk, πi.
Expected counts for (f, e) pairs • Let Ct(f, e) be the fractional count of (f, e) pair in the training data. Alignment prob Actual count of times e and f are linked in (E,F) by alignment a
Relation to EM • IBM models are PM Models. • The EM algorithm used in IBM models is a special case of the EM algorithm for PM Models. • X (observed data): each data point is a sentence pair (F, E). • Y (hidden data): word alignment a. • Θ (parameters): t(f|e), d(i | j, m, n), etc..
Summary • The EM algorithm • An iterative approach • L(θ) is non-decreasing at each iteration • Optimal solution in M-step exists for many classes of problems. • The EM algorithm for PM models • Simpler formulae • Three special cases • Inside-outside algorithm • Forward-backward algorithm • IBM Models for MT
Relations among the algorithms The generalized EM The EM algorithm PM Inside-Outside Forward-backward IBM models Gaussian Mix