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made by Bouchra ZEITANE , Zaineb MOUNIR

under the supervision of Pr. Habib BENLAHMAR

This algorithm is an instance of a large class of sampling algorithms, known as Markov chain Monte Carlo (MCMC). These algorithms have played a significant role in statistics, econometrics, physics and computing science over the last two decades. There are several high-dimensional problems, such as computing the volume

of a convex body in d dimensions, for which MCMC simulation is the only known general approach for providing a solution within a reasonable time (polynomial in d) (Dyer,Frieze, & Kannan, 1991; Jerrum & Sinclair, 1996).

While convalescing from an illness in 1946, Stan Ulam was playing solitaire. It, then, occurred to him to try to compute the chances that a particular solitaire laid out with 52 cards would come out successfully (Eckhard, 1987). After attempting exhaustive combinatorial calculations, he decided to go for the more practical approach of laying out several solitaires at random and then observing and counting the number of successful plays. This idea of selecting a statistical sample to approximate a hard combinatorial problem by a much simpler problem is at the heart of modern Monte Carlo simulation.

### Stan Ulam soon realised that computers could be used in this fashion to answer questions of neutron diffusion and mathematical physics.

He contacted John Von Neumann, who understood the great potential of this idea. Over the next few years, Ulam and Von Neumann developed many Monte Carlo algorithms, including importance sampling and rejection sampling. Enrico Fermi in the 1930’s also used Monte Carlo in the calculation of neutron diffusion, and later designed the FERMIAC, a Monte Carlo mechanical device that performed calculations (Anderson, 1986). In the 1940’s Nick Metropolis, a young physicist, designed new controls for the state-of-the-art computer (ENIAC) with Klari Von Neumann, John’s wife. He was fascinated with Monte Carlo methods and this new computing device.

Soon he designed an improved computer, which he named the MANIAC in the hope that computer scientists would stop using acronyms.

During the time he spent working on the computing machines, many mathematicians and physicists (Fermi, Von Neumann, Ulam, Teller, Richtmyer, Bethe, Feynman, & Gamow) would go to him with their work problems.

Eventually in 1949, he published the first public document on Monte Carlo simulation with Stan Ulam (Metropolis & Ulam, 1949). This paper introduces, among other ideas, Monte Carlo particle methods, which form the basis of modern sequential Monte Carlo methods such as bootstrap filters, condensation, and survival of the fittest algorithms (Doucet, de Freitas, & Gordon, 2001). Soon after, he proposed the Metropolis algorithm with the Tellers and the Rosenbluths (Metropolis et al., 1953).

Many papers on Monte Carlo simulation appeared in the physics literature after 1953. From an inference perspective, the most significant contribution was the generalisation of the Metropolis algorithm by Hastings in 1970. Hastings and his student Peskun showed that Metropolis and the more general Metropolis-Hastings algorithms are particular instances of a large family of algorithms, which also includes the Boltzmann algorithm (Hastings, 1970; Peskun, 1973). They studied the optimality of these algorithms and introduced the formulation of the Metropolis-Hastings algorithm that we adopt in this paper. In the 1980’s, two important MCMC papers appeared in the fields of computer vision and artificial intelligence (Geman & Geman, 1984; Pearl, 1987). Despite the existence of a few MCMC publications in the statistics literature at this time, it is generally accepted that it was only in1990 that MCMC made the first significant impact in statistics (Gelfand & Smith, 1990).