Notes on computational linguistics.pdf - UCLA Department of ...

Stabler - Lx 185/209 2003
added probabilities, is such that the probability of generating (or accepting) an a next, after generating
acccc is P(a|acccc) = 0.8, and the probability of generating (or accepting) a b next, after generating
bcccc is P(b|bcccc) = 0.8. That is, the strings show a kind of history dependence.
0.2
b 0.4
stop
s 1 s2
s 3
c 0.2
a 0.4
a 0.8
b 0.8
However the corresponding state sequences of this same machine are Markovian in some sense: they are
not history dependent in the way the strings seem to be. That is, we can have both P(q1|q1q2q2q2q2) =
P(q1|q2) = 0.8 andP(q1|q1q3q3q3q3) = P(q1|q3) = 0.8 since these involve transitions from different
states.
We will make this idea clearer in the next section.
8.1.7 Markov models
(70) A Markov chain can be specified by an initial distribution and state-state transition probabilities can
be augmented with stochastic outputs, so that we have in addition an initial output distribution and
state-output emission probabilities.
One way to do this is to is to define a Markov model as a pair X,O where X is a Markov chain X : N →
[Ω → R] and O : N → [Ω → Σ] where the latter function provides a way to classify outcomes by the
symbols a ∈ Σ that they are associated with.
In a Markov chain, each number n ∈ R names an event under each Xi, namely{e| Xi(e) = n}.
In a Markov model, each output symbol a ∈ Σ names an event in each Oi, namely{e| 0i(e) = a}.
(71) In problems concerning Markov models where the state sequence is not given, the model is often said
to be “hidden,” a hidden Markov model (HMM).
See, e.g., Rabiner (1989) for an introductory survey on HMMs. Some interesting recent ideas and applications appear
in, e.g., Jelinek and Mercer (1980), Jelinek (1985), De Mori, Galler, and Brugnara (1995), Deng and Rathinavelu
(1995), Ristad and Thomas (1997b), Ristad and Thomas (1997a).
(72) Let’s say that a Markov model (X, O) is a Markov source iff the functions X and O are “aligned” in the
following sense: 35
c 0.2
∀e ∈ Ω, ∀i ∈ N, ∀n ∈ R, ∃a ∈ Σ, Xi(e) = n implies Oi(e) = a
Then for every Xi, foralln ∈ ΩXi , there is a particular output a ∈ Σ such that P(Oi = a|Xi = n) = 1.
35 This is nonstandard. I think “Markov source” is usually just another name for a Markov model.
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