MetropolisHastings: The modified Metropolis-Hastings algorithm
In clemlaflemme/test: Methods in Structural Reliability

Description Usage Arguments Details Value

The function implements the specific modified Metropolis-Hastings algorithm as described first by Au \& Beck and including another scaling parameter for an extended search in initial steps of the SMART algorithm.

MetropolisHastings(
  x0,
  eval_x0 = -1,
  chain_length,
  modified = TRUE,
  sigma = 0.3,
  proposal = "Uniform",
  lambda = 1,
  limit_fun = function(x) {     -1 },
  burnin = 20,
  thinning = 4
)

`x0`	the starting point of the Markov chain
`eval_x0`	the value of the limit-state function on `x0`
`chain_length`	the length of the Markov chain. At the end the chain will be chain_length + 1 long
`modified`	a boolean to use either the original Metropolis-Hastings transition kernel or the coordinate-wise one
`sigma`	a radius parameter for the Gaussian or Uniform proposal
`proposal`	either "Uniform" for a Uniform random variable in an interval [-sigma, sigma] or "Gaussian" for a centred Gaussian random variable with standard deviation sigma
`lambda`	the coefficient to increase the likelihood ratio
`limit_fun`	the limite-state function delimiting the domain to sample in
`burnin`	a burnin parameter, ie a number of initial discards samples
`thinning`	a thinning parameter, ie that one sample over `thinning` samples is kept along the chain

The modified Metropolis-Hastings algorithm is supposed to be used in the Gaussian standard space. Instead of using a proposed point for the multidimensional Gaussian random variable, it applies a Metropolis step to each coordinate. Then it generates the multivariate candidate by checking if it lies in the right domain.

This version proposed by Bourinet et al. includes an scaling parameter lambda. This parameter is multiplied with the likelihood ratio in order to increase the chance of accepting the candidate. While it biases the output distribution of the Markov chain, the authors of SMART suggest its use (lambda > 1) for the exploration phase. Note such a value disable to possiblity to use the output population for Monte Carlo estimation.

A list containing the following entries:

`points`	the generated Markov chain
`eval`	the value of the limit-state function on the generated samples
`acceptation`	the acceptation rate
`Ncall`	the total number of call to the limit-state function
`samples`	all the generated samples
`eval_samples`	the evaluation of the limit-state function on the `samples` samples