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Matematik simulation and monte carlo with applications in finance and mcmc phần 6

Sử dụng 5000 lần nhắc lại, mỗi bản sao tạo ra một thời gian chờ đợi tiểu học và phản đối trong năm khách hàng, thời gian trung bình là 1,287 và sai số chuẩn là 0,00777. Đúng giảm tỷ lệ ước tính là khoảng 2,5. (b) | Markov chains and the MH algorithm 161 is to plot a several component s of the sequence x t . Another is to plot some function of X t for t 0 1 2 . For example it might be appropriate to plot h x t 1 2 . J. Whatever choice is made repeat for each of the K independent replications. Given that the initial state for each of these chains is different equilibrium is perhaps indicated when t is of a size that makes all K plots similar in the sense that they fluctuate about a common central value and explore the same region of the state space. A further issue is how many equilibrium observations n there should be in each realization. If the chain has strong positive dependence then the realization will move slowly through the states slow mixing and n will need to be large in order that the entire state space is explored within a realization. A final and positive observation relates to the calculation of a x y in Equation 8.6 . Since f appears in both the numerator and denominator of the right-hand side it need be known only up to an arbitrary multiplicative constant. Therefore it is unnecessary to calculate P D in Equation 8.1 . The original Metropolis Metropolis et al. 1953 algorithm took q y x q x y . Therefore min L fSỵlA a x y min 1 rr . f x J A suitable choice for q might be q y x A exp - y - x 2-1 y - x 8.7 that is given x Y N x 2 . How should 2 which controls the average step length be chosen Large step lengths potentially encourage good mixing and exploration of the state space but will frequently be rejected particularly if the current point x is near the mode of a unimodal density f. Small step lengths are usually accepted but give slow mixing long burn-in times and poor exploration of the state space. Clearly a compromise value for s is called for. Hastings 1970 suggested a random walk sampler that is given that the current point is x the candidate point is Y x W where W has density g. Therefore q ylx g y - x - This appears to be the most popular sampler at .