The Average Estimation Of Functions On Markov Chain

Suppose that Ω finit set,π is a probability measure on Ω and f is a real-valued function on Ω.In many practical problems,we often need to calculate Eπ(f)=∑x∈Ωf(x)π(x).But when th number of elements in Ω is too large,it is impossible to calculate Eπ(f).So we can use the Markov chain Monte Carlo method to estimate Eπ(f).Namely,we pick Random variablesξ1,ξ2,...,ξN which are i.i.d.And 1/NΣNi=1 f(ξi)is a approxi-mate estimation of Eπ(f).But in many situations,it is hard to get ξi(i = 1,2,...,N)by computer.Then we can get a ergodic Markov chain {η0,η1,η2,...}.And the stationary distribution of the chain is π.Then 1/N ΣNi=1 f(ξi)is a estimation of Eπ(f).Because of the stationary distributions of the first samples of the Markov chain is different fromπ,it is better to use 1/N-sΣN0=s+1f(ηi)to estimate Eπ(f).This article introduce how to determine the s and N when given the precision requirement(Proposition5.10)and related preliminary knowledge in detail.