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[MINIMUM,FVAL,EXITFLAG,OUTPUT] = fminmarkov(FUN,PARS,[OPTIONS],[CONSTRAINTS], ...) Markov Chain Monte Carlo optimizer


function [pars,fval,exitflag,output] = fminmarkov(varargin)


 [MINIMUM,FVAL,EXITFLAG,OUTPUT] = fminmarkov(FUN,PARS,[OPTIONS],[CONSTRAINTS], ...) Markov Chain Monte Carlo optimizer

 This minimization method uses a Markov Chain Monte Carlo, optionally 
 with constraints on function parameters.

 This is an implementation of the Goodman and Weare 2010 Affine
 invariant ensemble Markov Chain Monte Carlo (MCMC) sampler. MCMC sampling
 enables bayesian inference. The problem with many traditional MCMC samplers
 is that they can have slow convergence for badly scaled problems, and that
 it is difficult to optimize the random walk for high-dimensional problems.
 This is where the GW-algorithm really excels as it is affine invariant. It
 can achieve much better convergence on badly scaled problems. It is much
 simpler to get to work straight out of the box, and for that reason it
 truly deserves to be called the MCMC hammer.

 (This code uses a cascaded variant of the Goodman and Weare algorithm).
   fminmarkov(fun, pars) asks to minimize the 'fun' objective function with starting
     parameters 'pars' (vector)
   fminmarkov(fun, pars, options) same as above, with customized options (optimset)
   fminmarkov(fun, pars, options, fixed) 
     is used to fix some of the parameters. The 'fixed' vector is then 0 for
     free parameters, and 1 otherwise.
   fminmarkov(fun, pars, options, lb, ub) 
     is used to set the minimal and maximal parameter bounds, as vectors.
   fminmarkov(fun, pars, options, constraints) 
     where constraints is a structure (see below).
   fminmarkov(problem) where problem is a structure with fields
     problem.objective:   function to minimize
     problem.x0:          starting parameter values
     problem.options:     optimizer options (see below)
     problem.constraints: optimization constraints
   fminmarkov(..., args, ...)
     sends additional arguments to the objective function
       criteria = FUN(pars, args, ...)

   banana = @(x)100*(x(2)-x(1)^2)^2+(1-x(1))^2;
   [x,fval] = fminmarkov(banana,[-1.2, 1])

  FUN is the function to minimize (handle or string): criteria = FUN(PARS)
  It needs to return a single value or vector.

  PARS is a vector with initial guess parameters. You must input an
  initial guess. PARS can also be given as a single-level structure.

  OPTIONS is a structure with settings for the optimizer, 
  compliant with optimset. Default options may be obtained with
  options.MinFunEvals sets the minimum number of function evaluations to reach
  options.StepSize:  unit-less stepsize (default=2.5).
  options.ThinChain: Thin all the chains by only storing every N'th step (default=10)
  options.BurnIn:    fraction of the chain that should be removed. (default=0)
  An empty OPTIONS sets the default configuration.

  CONSTRAINTS may be specified as a structure
   constraints.min= vector of minimal values for parameters
   constraints.max= vector of maximal values for parameters
   constraints.fixed= vector having 0 where parameters are free, 1 otherwise
   constraints.step=  vector of maximal parameter changes per iteration
   constraints.eval=  expression making use of 'p', 'constraints', and 'options' 
                        and returning modified 'p'
                      or function handle p=@constraints.eval(p)
  An empty CONSTRAINTS sets no constraints.

  Additional arguments are sent to the objective function.

          MINIMUM is the solution which generated the smallest encountered
            value when input into FUN.
          FVAL is the value of the FUN function evaluated at MINIMUM.
          EXITFLAG return state of the optimizer
          OUTPUT additional information returned as a structure.

 Goodman & Weare (2010), Ensemble Samplers With Affine Invariance, Comm. App. Math. Comp. Sci., Vol. 5, No. 1, 65-80
 Foreman-Mackey, Hogg, Lang, Goodman (2013), emcee: The MCMC Hammer, arXiv:1202.3665

 By: Aslak Grinsted 2015 https://github.com/grinsted/gwmcmc

 Version: Nov. 26, 2018
 See also: fminsearch, optimset
 (c) E.Farhi, ILL. License: EUPL.


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