Stochastic simulation

Stochastic simulation involves generating a sample from the distribution of interest

The word ‘stochastic’ means random

The generated sample is then analysed to learn about the distribution from which the values were sampled

Stochastic simulation is often called Monte Carlo simulation The most commonly used technique for dealing with non-conjugate models is a stochastic simulation technique called Markov chain Monte Carlo, or MCMC


 * Monte Carlo standard error: sampling variability or Monte Carlo variability, The standard deviation of an estimator is known as the standard error of the estimator.

= non-conjugate models =

joint prior

= Metropolis–Hastings sampling =

The idea of Metropolis–Hastings sampling is to generate a Markov chain which has the target density f as its equilibrium density

To do this, the Metropolis–Hastings algorithm simulates a candidate value, x super *, from an arbitrary distribution for uppercase X subscript t +1 end | uppercase X sub t = x sub t called the proposal distribution, [The proposal distribution is also known as the candidate-generating distribution and the jumping distribution.] with associated proposal density

target density

Metropolis-Hastings algorithm
The Metropolis–Hastings algorithm can sample from a multivariate target distribution with density f.

"The algorithm was named after the leading American physicist and computer pioneer Nicholas Constantine Metropolis (1915–1999) and the Canadian statistician W. Keith Hastings (1930–)"

Symmetric random walk chains This special case of the Metropolis–Hastings algorithm is known as the Metropolis algorithm.

= Gibbs sampling =

= convergence diagnostics =

convergence diagnostics. They fall into three general categories: graphical methods, numerical methods, and a combination of both graphical and numerical methods.

dependence between sample values can be measured by the sample autocorrelation function.

Dependence
batch means method.