Unformatted text preview: v property :
p(θt |θt−1 , . . . , θ1 ) = p(θt |θt−1 ).
We will be interested in the (unconditioned) probability distribution over
states at time t, which we denote as π t (θ) := Pr(θt = θ). Under some
24 conditions that will be satisﬁed by all of the chains we are interested in,
the sequence of state distributions π 0 (θ), π 1 (θ), . . . will converge to a unique
distribution π (θ) which we call the stationary distribution, or equilibrium dis
tribution or steady-state distribution.
We deﬁne the transition kernel to be the probability of transitioning from
state θ to state θ' : K (θ, θ' ) = p(θ' |θ). We then have the following important
Fact: if the distribution π (·) satisﬁes the detailed balance equation :
K (θ, θ' )π (θ) = K (θ' , θ)π (θ' ), for all θ, θ' , (23) then π (·) is the stationary distribution. The interpretation of the detailed
balance equation is that the amount of mass transitioning from θ' to θ is the
same as the amount of mass transition back from θ to θ' . For a stationary
distribution, we cannot have mass going from state to...
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This note was uploaded on 03/24/2014 for the course MIT 15.097 taught by Professor Cynthiarudin during the Spring '12 term at MIT.
- Spring '12