Monte-Carlo estimator

Valleau J P and Card D N 1972 Monte Carlo estimation of the free energy by multistage sampling J. Chem. Phys. 57 5457-62... 

The process is repeated vvitli another random number until tlie desired 15 simulated values of T have been obtained. The results are shown in Table 20.6.1. Tlie average value of the 15 simulated values of T is 1.02 years, a Monte Carlo estimate of the average life of the pmnp. The true average life of the pump is E(t), tlie expected value of T, obtained by application of Eq. (19.9.2) ... 

TABLE 20.6.1 Data for Monte Carlo Estimation of Average Time to... 

Monte Carlo sampling is discussed extensively in Hammersley and Morton (1956), Hammersley and Handscomb (1964), Kloek and Van Dijk (1978), and Wilson (1984). For Monte Carlo results to be believable, the convergence properties of the Monte Carlo estimators must be met. Several statistical and practical limitations exist in this regard. The most important practical limitations of Monte Carlo are the following ... 

Beckman RJ, McKay MD. 1987. Monte Carlo estimation under different distributions using the same simulation. Technometrics 29 153-160. 

Simulations [73] have recently provided some insights into the formal 5c —> 0 limit predicted by mean field lattice model theories of glass formation. While Monte Carlo estimates of x for a Flory-Huggins (FH) lattice model of a semifiexible polymer melt extrapolate to infinity near the ideal glass transition temperature Tq, where 5c extrapolates to zero, the values of 5c computed from GD theory are too low by roughly a constant compared to the simulation estimates, and this constant shift is suggested to be sufficient to prevent 5c from strictly vanishing [73, 74]. Hence, we can reasonably infer that 5 approaches a small, but nonzero asymptotic low temperature limit and that 5c similarly becomes critically small near Tq. The possibility of a constant... 

Comparison between asymptotic formulas based on direct enumeration and Monte Carlo estimates. Tetrahedral and square lattices. (Wall and Erpenbeck9). 

Our numerical implementation of the above scheme works as follows. We start off with an initial guesses kg for the many-body wavefunction and a corresponding guess for Vx. A fixed number Nc of statistically independent configurations Rt- are then sampled from Jq 2 and the Monte Carlo estimator of fi2 over these configurations is evaluated... 

Table 7.1 gives quasi Monte Carlo estimates of the Kn for the case of hard spheres. For the hard-sphere fluid, the predicted distributions for two densities are shown in Figs. 7.8 and 7.9. The predicted occupancy distributions are physically faithfiil to the data. 

Daub, C.D., Camp, P.J., and Patey, G.N. The constant-volume heat capacity of near-critical fluids with long-range interactions A discussion of different monte carlo estimates. J. Chem. Phys., 2003, 118, p. 4164—4168. 

Leonov, H., Mitchell, J. S. B., Arkin, 1. X. Monte Carlo estimation of the number of possible protein folds effects of sampling bias and folds distributions. Proteins 2003, 51, 352-359. 

The first three virial coefficients are easily computed, the fomrth with considerably more effort. - The fifth virial coefficients (except for the one-dimensional fluid) are Monte Carlo estimates. Nijboer and van Hove have also expanded g(r) in powers of p, up to and including the second power. 

The quantity in brackets, the variance, only depends on / Hence the standard deviation of the Monte Carlo estimate is 0(N 1/2). This is much worse than the bounds of Eq. (5.5) as a function of the number of points. It is this fact that has motivated the search for quasi-random points. 

The simplest method is the variational Monte Carlo method, discussed in the next section. Here an approximate expectation value is computed by employing an approximate eigenvector of G. Typically, this is an optimized trial state, say Mr>, in which case variational Monte Carlo yields X f° which is simply the expectation value of X in the trial state. Clearly, variational Monte Carlo estimates of X0 have both systematic and statistical errors. 

J. P. Valleau and D. N. Card, Monte Carlo Estimation of the Free Energy by Multistage Sampling, J. Chem. Phys. 57, 5457 (1972). 

Chen, M.-H. and Shao, Q.-M. (1999) Monte Carlo estimation of Bayesian credible and HPD intervals, Journal of Computational and Graphical Statistics 8,69-92. 

The statistical-mechanical applications of Monte Carlo nearly all involve special sampling methods known as importance sampling and ordinarily require a Markov chain of sample configurations rather than independent samples. In order to understand this it is helpful to begin by imagining a simpler Monte Carlo estimation of a quantity like U) of (2), and then to see why such an estimation would not be successful. 

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