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

Stabler - Lx 185/209 2003
8.1.5 Random variables
(44) A random (or stochastic) variable on probability space (Ω, 2Ω ,P) is a function X : Ω → R.
(45) Any set of numbers A ∈ 2R determines (or “generates”) an event, a set of outcomes, namely X−1 (A) =
{e| X(e) ∈ A}.
(46) So then, for example, P(X−1 (A)) = P({e| X(e) ∈ A}) is the probability of an event, as usual.
(47) Many texts use the notation X ∈ A for an event, namely, {e| X(e) ∈ A}).
So P(X ∈ A) is just P(X−1 (A)), whichisjustP({e| X(e) ∈ A}).
Sometimes you also see P{X ∈ A}, with the same meaning.
(48) Similarly, for some a ∈ R, itiscommontoseeP(X = s),whereX = s is the event {e| X(e) = s}).
(49) The range of X is sometimes called the sample space of the stochastic variable X, ΩX.
X is discrete if ΩX is finite or countably infinite. Otherwise it is continuous.
• Why do things this way? What is the purpose of these functions X?
The answer is: the functions X just formalize the classification of events, the sets of outcomes that we are
interested in, as explained in (45) and (48).
This is a standard way to name events, and once you are practiced with the notation, it is convenient.
The events are classified numerically here, that is, they are named by real numbers, but when the set of events
ΩX is finite or countable, obviously we could name these events with any finite or countable set of names.
8.1.6 Stochastic processes and n-gram models of language users
(50) A stochastic process is a function X from times (or “indices”) to random variables.
If the time is continuous, then X : R → [Ω → R], where[Ω→R] is the set of random variables.
If the time is discrete, then X : N → [Ω → R]
(51) For stochastic processes X, instead of the usual argument notation X(t), we use subscripts Xt, toavoid
confusion with the arguments of the random variables.
So Xt is the value of the stochastic process X at time t, a random variable.
When time is discrete, for t = 0, 1, 2,... we have the sequence of random variables X0,X1,X2,...
(52) We will consider primarily discrete time stochastic processes, that is, sequences X0,X1,X2,...of random
variables.
So Xi is a random variable, namely the one that is the value of the stochastic process X at time i.
(53) Xi = q is interpreted as before as the event (now understood as occurring at time i) whichisthesetof
outcomes {e| Xi(e) = q}.
So, for example, P(Xi = q) is just a notation for the probability, at time i, of an outcome that is named
q by Xi, thatis,P(Xi = q) is short for P({e| Xi(e) = q}).
(54) Notice that it would make perfect sense for all the variables in the sequence to be identical, X0 = X1 =
X2 = .... In that case, we still think of the process as one that occurs in time, with the same classification
of outcomes available at each time.
Let’s call a stochastic process time-invariant (or stationary) iff all of its random variables are the same
function. That is, for all q, q ′ ∈ N,Xq = Xq ′.
(55) A finite stochastic process X is one where sample space of all the stochastic variables,
is finite.
ΩX = ∞ i=0 ΩXi
The elements of ΩX name events, as explained in (45) and (48), but in this context the elements of ΩX
are often called states.
Markov chains
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