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Evaluate the impact of the lag G on the shape of those graphs. (Try G = 2, 5, 10, 100, 1000.)
This is a sequential graph in that each dot is based on the sample of ß(t)'s produced at time T and then thinned by a lag of G = 10 and separated.
Find sets of parameters (d, s, y) for which b these two phenomena occur.
Compare those assessments with an on-line evaluation of the integral R f(x) d x based on the MCMC sample thus produced.
Find a thinning lag G large enough that the distribution of the Kolmogorov-Smirnov p-values has no visible pattern.
The historical example of Hastings (1970) considers the formal problem of generating the normal distribution N (0, 1) based on a random walk proposal equal .
What is the acceptance rate for the Metropolis-Hastings algorithm with candidate L(3)?
Run the Gibbs sampler for the pump failure data and get 95% posterior credible intervals for the parameters ?i.
Deduce a converging estimator of m(x) based on the Rao-Blackwellized estimate of the posterior density p(?|x) .
Construct an EM algorithm for this model, and derive the maximum likelihood estimators of the parameters for the sample .
Calculate the mean of a t distribution with ? = 4 degrees of freedom using a Metropolis- Hastings algorithm with candidate density.
For each of the following cases, generate random variables Xi and Yi and compare the empirical average and Rao-Blackwellized estimator.
Write an R code that truly produces a sample of 400 observations from (equ.) instead of setting the normal subsample sizes to 100 and 300.
Design a Monte Carlo experiment that studies the variation of the EM solutions around this numerical optimum.
When using nlm, the modes are obtained within a few iterations, depending on the starting points, and the intermediate values of the Newton-Raphson sequence.
Examine the performance of this algorithm in terms of acceptance probability when using a simulated sample with the same parameters as in Figure .
The sequences represented in Figure all end up in one of the two modes, but with highly nonlinear patterns.
Deduce from the acceptance rate an estimator of the normalizing constant of f for each of the instrumental densities.
Derive an estimate of E[X|X>K] based on a sample from F.
Show that the sum of the weights ?i = f(Xi)/g(Xi) is only equal to n in expectation and deduce that the weights need to be renormalized .
Monte Carlo marginalization is a technique for calculating a marginal density when simulating from a joint density.
Deduce the regular and the self-normalized sequences of estimators of E[exp(X)].
Show that the bias due to the replacement of µ by x¯ n is of the order of a X2n term, which can thus be corrected directly in dn.
Show that Sigma=cov(matrix(rnorm(30),nrow=10)) defines a proper covariance matrix.
Show that the probability of acceptance in an Accept-Reject algorithm with upper bound M on the density ratio f /g is 1/M.