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Formulate this problem as a Markov decision process by identifying the states and decisions and then finding the Cik.
Prove that, during a first-order phase transition-The entropy of the entire system is a linear function of the total volume.
Using Stirling's approximation and the method of Lagrangian multipliers, render In OBE a maximum,subject to the equations of constraint ?Ni = N = const.
In the region of moderate pressures, the equation of state of 1mol of gas may be written Pv = RT + B’P + C’P2 .
Assuming the density and isothermal compressibility to remain constant at values of 8.90 x 103 kg/m3 and 6.75 x 10-12 Pa-1, respectively.
However, when looking directly at the likelihood, using optimise eventually produces a diverging sequence since the likelihood gets too small around n = 300 .
Show that the random walk has no stationary distribution. Give the distribution of X(t) for t = 104 and t = 106 when X(0) = 0.
Compute the value of the marginal density of y for the swiss dataset.
Evaluate the impact of parameterization on gelman.diag for the model of Example 1 using the same MCMC sample .
Show that if ?(t) ~ pt and if the stationary distribution is the posterior density associated with f(x|?) and p(?).
The gamma distribution with parameters a and b, G(a, b), has density baxa-1e-bx/G(a).
For Z ~ N (0, 1) with truncation (i) -1 3, generate 1000 random variables and compare the histograms with the density functions.
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.