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Monte Carlo integration with Markov chain - Department of Statistics

1972 Z. Tan / Journal **of** Statistical Planning and Inference 138 (2008) 1967 – 1980 0.35 Scheme (i) 0.16 Scheme (ii) 0.3 0.14 0.25 0.12 0.2 0.1 log Z 0.15 log Z 0.08 0.06 0.1 0.04 0.05 0.02 0 0 0.05 0.02 Fig. 1. Boxplots **of** estimators **of** log Z. Left to right: basic estimator, subsampled estimator (b=10), and regulated estimator (δ=10 −4 ) and amplified estimator under subsampling (a = 1.5). 2.2. Regulation and amplification The performance **of** the estimator ˜Z or its subsampled version ˜Z b depends on the properties **of** the **Markov** **chain** such as how quickly the **Markov** **chain** mixes and how closely the stationary density p ∗ (x) matches the integrand q(x). For Gibbs sampling, the stationary density is made perfectly proportional to the integrand, and the problem **of** speeding up the Gibbs sampler has been studied in the MCMC literature (e.g. Gilks et al., 1996; Liu, 2001). We address one additional factor that affects the performance **of** the estimator ˜Z or its subsampled version ˜Z b . For example, let Ξ and X be the real line and q(x) be exp(−x 2 /2)/ √ 2π. Consider two sampling schemes, where the sequence (x 1 ,...,x n ) converges to the standard normal distribution N(0, 1): (i) ξ t |x t−1 ∼ N(ρx t−1 , 1 − ρ 2 ) and x t |ξ t ∼ N(ρξ t , 1 − ρ 2 ). (ii) ξ t ∼ N(0, 1) and x t |ξ t ∼ N(ρξ t , 1 − ρ 2 ). Scheme (ii) **with** ρ ≈ 1 is an extreme **of** those situations where the **Markov** **chain** [(ξ 1 ,x 1 ),...,(ξ n ,x n )] mixes well but the transition density p(·; ξ) is narrowly spread; see a generalized Gibbs sampling example in Section 3.2. Under scheme (i), as ρ increases to one, the **Markov** **chain** mixes more slowly. The estimator ˜Z has larger variance and more serious bias. Under scheme (ii), the **Markov** **chain** mixes perfectly and the estimator ˜Z is unbiased for any 0 < ρ < 1. But, the variance **of** ˜Z depends on the value **of** ρ. Ifρ is close to one, the estimator ˜Z has a skewed distribution **with** a heavy right-tail. Now consider the subsampled estimator ˜Z b . Under scheme (i), the subsampled sequence becomes approximately independent for a large subsampling interval b. The estimator ˜Z b has reduced bias but increased variance, compared **with** the estimator ˜Z.Ifρ is near one, the estimator ˜Z b tends to yield large overestimates. Under scheme (ii), the estimator ˜Z b has an even skewed distribution, compared **with** the estimator ˜Z. These different performances are illustrated in Fig. 1, which is based on 5000 simulations **of** size 1000 and ρ = .9. The preceding discussion makes it clear that a poor performance may be caused by the narrow spread **of** the transition density p(·; ξ). We propose two modifications **of** the basic ˜μ or the subsampled ˜μ b . Only those **of** ˜μ are presented and those **of** ˜μ b should be understood in a similar manner. As seen from Fig. 1, these techniques are helpful for variance reduction by removing those extreme estimates. First, the spread **of** the transition density p(·; ξ) is relevant because it affects how uniformly the average n −1∑ n j=1 p(x; ξ j ) converges to the stationary density p ∗ (x) on a multiplicative scale for x ∈ X. Nonuniformity

Z. Tan / Journal **of** Statistical Planning and Inference 138 (2008) 1967 – 1980 1973 is most likely to occur on X where p ∗ (x) is close to zero. We consider the regulated estimator ˜μ δ ({x}) = ˆ P({x}) δ ∨[n −1∑ n j=1 p(x; ξ j )] , by censoring n −1∑ n j=1 p(x; ξ j ) from below at δ0. For a real-valued function q(x), the estimator ∫ q(x)d ˜μ δ has asymptotic bias [ ] [ ( q(x) 1 E p∗ − Z = E p∗ q(x) δ ∨ p ∗ (x) δ − 1 ) ] 1 {p∗ (x)

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