# Monte Carlo Methods, Second Edition

A Primer for the Monte Carlo Method

In principle, Monte Carlo methods can be used to solve any problem having a probabilistic interpretation. By the law of large numbers , integrals described by the expected value of some random variable can be approximated by taking the empirical mean a. That is, in the limit, the samples being generated by the MCMC method will be samples from the desired target distribution.

In other problems, the objective is generating draws from a sequence of probability distributions satisfying a nonlinear evolution equation. These flows of probability distributions can always be interpreted as the distributions of the random states of a Markov process whose transition probabilities depend on the distributions of the current random states see McKean—Vlasov processes , nonlinear filtering equation.

These models can also be seen as the evolution of the law of the random states of a nonlinear Markov chain. In contrast with traditional Monte Carlo and MCMC methodologies these mean field particle techniques rely on sequential interacting samples. The terminology mean field reflects the fact that each of the samples a.

When the size of the system tends to infinity, these random empirical measures converge to the deterministic distribution of the random states of the nonlinear Markov chain, so that the statistical interaction between particles vanishes. For example, consider a quadrant circular sector inscribed in a unit square. In this procedure the domain of inputs is the square that circumscribes the quadrant. We generate random inputs by scattering grains over the square then perform a computation on each input test whether it falls within the quadrant. Uses of Monte Carlo methods require large amounts of random numbers, and it was their use that spurred the development of pseudorandom number generators , which were far quicker to use than the tables of random numbers that had been previously used for statistical sampling.

Before the Monte Carlo method was developed, simulations tested a previously understood deterministic problem, and statistical sampling was used to estimate uncertainties in the simulations. Monte Carlo simulations invert this approach, solving deterministic problems using a probabilistic analog see Simulated annealing. In the s, Enrico Fermi first experimented with the Monte Carlo method while studying neutron diffusion, but did not publish anything on it.

The modern version of the Markov Chain Monte Carlo method was invented in the late s by Stanislaw Ulam , while he was working on nuclear weapons projects at the Los Alamos National Laboratory. In , physicists at Los Alamos Scientific Laboratory were investigating radiation shielding and the distance that neutrons would likely travel through various materials. Despite having most of the necessary data, such as the average distance a neutron would travel in a substance before it collided with an atomic nucleus, and how much energy the neutron was likely to give off following a collision, the Los Alamos physicists were unable to solve the problem using conventional, deterministic mathematical methods.

Ulam had the idea of using random experiments. He recounts his inspiration as follows:. The first thoughts and attempts I made to practice [the Monte Carlo Method] were suggested by a question which occurred to me in as I was convalescing from an illness and playing solitaires.

The question was what are the chances that a Canfield solitaire laid out with 52 cards will come out successfully?

## Introducing Monte Carlo Methods with R

After spending a lot of time trying to estimate them by pure combinatorial calculations, I wondered whether a more practical method than "abstract thinking" might not be to lay it out say one hundred times and simply observe and count the number of successful plays. This was already possible to envisage with the beginning of the new era of fast computers, and I immediately thought of problems of neutron diffusion and other questions of mathematical physics, and more generally how to change processes described by certain differential equations into an equivalent form interpretable as a succession of random operations.

Later [in ], I described the idea to John von Neumann , and we began to plan actual calculations. Being secret, the work of von Neumann and Ulam required a code name. Though this method has been criticized as crude, von Neumann was aware of this: he justified it as being faster than any other method at his disposal, and also noted that when it went awry it did so obviously, unlike methods that could be subtly incorrect.

Monte Carlo methods were central to the simulations required for the Manhattan Project , though severely limited by the computational tools at the time.

In the s they were used at Los Alamos for early work relating to the development of the hydrogen bomb , and became popularized in the fields of physics , physical chemistry , and operations research. The Rand Corporation and the U. Air Force were two of the major organizations responsible for funding and disseminating information on Monte Carlo methods during this time, and they began to find a wide application in many different fields. The theory of more sophisticated mean field type particle Monte Carlo methods had certainly started by the mids, with the work of Henry P.

McKean Jr. Harris and Herman Kahn, published in , using mean field genetic -type Monte Carlo methods for estimating particle transmission energies. The origins of these mean field computational techniques can be traced to and with the work of Alan Turing on genetic type mutation-selection learning machines  and the articles by Nils Aall Barricelli at the Institute for Advanced Study in Princeton, New Jersey.

Quantum Monte Carlo , and more specifically diffusion Monte Carlo methods can also be interpreted as a mean field particle Monte Carlo approximation of Feynman — Kac path integrals. Resampled or Reconfiguration Monte Carlo methods for estimating ground state energies of quantum systems in reduced matrix models is due to Jack H. Hetherington in  In molecular chemistry, the use of genetic heuristic-like particle methodologies a.

Rosenbluth and Arianna. The use of Sequential Monte Carlo in advanced signal processing and Bayesian inference is more recent. It was in , that Gordon et al. The authors named their algorithm 'the bootstrap filter', and demonstrated that compared to other filtering methods, their bootstrap algorithm does not require any assumption about that state-space or the noise of the system. Particle filters were also developed in signal processing in — by P. Del Moral, J. Noyer, G. Rigal, and G. From to , all the publications on Sequential Monte Carlo methodologies, including the pruning and resample Monte Carlo methods introduced in computational physics and molecular chemistry, present natural and heuristic-like algorithms applied to different situations without a single proof of their consistency, nor a discussion on the bias of the estimates and on genealogical and ancestral tree based algorithms.

The mathematical foundations and the first rigorous analysis of these particle algorithms are due to Pierre Del Moral   in Del Moral, A. Guionnet and L. There is no consensus on how Monte Carlo should be defined. For example, Ripley  defines most probabilistic modeling as stochastic simulation , with Monte Carlo being reserved for Monte Carlo integration and Monte Carlo statistical tests.

Sawilowsky  distinguishes between a simulation , a Monte Carlo method, and a Monte Carlo simulation: a simulation is a fictitious representation of reality, a Monte Carlo method is a technique that can be used to solve a mathematical or statistical problem, and a Monte Carlo simulation uses repeated sampling to obtain the statistical properties of some phenomenon or behavior. Kalos and Whitlock  point out that such distinctions are not always easy to maintain. For example, the emission of radiation from atoms is a natural stochastic process.

It can be simulated directly, or its average behavior can be described by stochastic equations that can themselves be solved using Monte Carlo methods. The main idea behind this method is that the results are computed based on repeated random sampling and statistical analysis. The Monte Carlo simulation is, in fact, random experimentations, in the case that, the results of these experiments are not well known.

Monte Carlo simulations are typically characterized by many unknown parameters, many of which are difficult to obtain experimentally. The only quality usually necessary to make good simulations is for the pseudo-random sequence to appear "random enough" in a certain sense. What this means depends on the application, but typically they should pass a series of statistical tests. Testing that the numbers are uniformly distributed or follow another desired distribution when a large enough number of elements of the sequence are considered is one of the simplest and most common ones.

Sawilowsky lists the characteristics of a high-quality Monte Carlo simulation: . Pseudo-random number sampling algorithms are used to transform uniformly distributed pseudo-random numbers into numbers that are distributed according to a given probability distribution. Low-discrepancy sequences are often used instead of random sampling from a space as they ensure even coverage and normally have a faster order of convergence than Monte Carlo simulations using random or pseudorandom sequences.

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Methods based on their use are called quasi-Monte Carlo methods. In an effort to assess the impact of random number quality on Monte Carlo simulation outcomes, astrophysical researchers tested cryptographically-secure pseudorandom numbers generated via Intel's RdRand instruction set, as compared to those derived from algorithms, like the Mersenne Twister , in Monte Carlo simulations of radio flares from brown dwarfs.

RdRand is the closest pseudorandom number generator to a true random number generator.

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No statistically significant difference was found between models generated with typical pseudorandom number generators and RdRand for trials consisting of the generation of 10 7 random numbers. There are ways of using probabilities that are definitely not Monte Carlo simulations — for example, deterministic modeling using single-point estimates. Each uncertain variable within a model is assigned a "best guess" estimate. Scenarios such as best, worst, or most likely case for each input variable are chosen and the results recorded. By contrast, Monte Carlo simulations sample from a probability distribution for each variable to produce hundreds or thousands of possible outcomes.

The results are analyzed to get probabilities of different outcomes occurring. The samples in such regions are called "rare events". Monte Carlo methods are especially useful for simulating phenomena with significant uncertainty in inputs and systems with many coupled degrees of freedom. Areas of application include:. Monte Carlo methods are very important in computational physics , physical chemistry , and related applied fields, and have diverse applications from complicated quantum chromodynamics calculations to designing heat shields and aerodynamic forms as well as in modeling radiation transport for radiation dosimetry calculations.

In astrophysics , they are used in such diverse manners as to model both galaxy evolution  and microwave radiation transmission through a rough planetary surface. Monte Carlo methods are widely used in engineering for sensitivity analysis and quantitative probabilistic analysis in process design. The need arises from the interactive, co-linear and non-linear behavior of typical process simulations. For example,. The Intergovernmental Panel on Climate Change relies on Monte Carlo methods in probability density function analysis of radiative forcing. The PDFs are generated based on uncertainties provided in Table 8.

The combination of the individual RF agents to derive total forcing over the Industrial Era are done by Monte Carlo simulations and based on the method in Boucher and Haywood PDF of the ERF from surface albedo changes and combined contrails and contrail-induced cirrus are included in the total anthropogenic forcing, but not shown as a separate PDF. We currently do not have ERF estimates for some forcing mechanisms: ozone, land use, solar, etc.

Monte Carlo methods are used in various fields of computational biology , for example for Bayesian inference in phylogeny , or for studying biological systems such as genomes, proteins,  or membranes. Computer simulations allow us to monitor the local environment of a particular molecule to see if some chemical reaction is happening for instance.

Path tracing , occasionally referred to as Monte Carlo ray tracing, renders a 3D scene by randomly tracing samples of possible light paths. Repeated sampling of any given pixel will eventually cause the average of the samples to converge on the correct solution of the rendering equation , making it one of the most physically accurate 3D graphics rendering methods in existence.

The standards for Monte Carlo experiments in statistics were set by Sawilowsky. Monte Carlo methods are also a compromise between approximate randomization and permutation tests. An approximate randomization test is based on a specified subset of all permutations which entails potentially enormous housekeeping of which permutations have been considered. The Monte Carlo approach is based on a specified number of randomly drawn permutations exchanging a minor loss in precision if a permutation is drawn twice—or more frequently—for the efficiency of not having to track which permutations have already been selected.

Monte Carlo methods have been developed into a technique called Monte-Carlo tree search that is useful for searching for the best move in a game. Possible moves are organized in a search tree and many random simulations are used to estimate the long-term potential of each move.

A black box simulator represents the opponent's moves. The net effect, over the course of many simulated games, is that the value of a node representing a move will go up or down, hopefully corresponding to whether or not that node represents a good move. Monte Carlo methods are also efficient in solving coupled integral differential equations of radiation fields and energy transport, and thus these methods have been used in global illumination computations that produce photo-realistic images of virtual 3D models, with applications in video games , architecture , design , computer generated films , and cinematic special effects.

Each simulation can generate as many as ten thousand data points that are randomly distributed based upon provided variables. Ultimately this serves as a practical application of probability distribution in order to provide the swiftest and most expedient method of rescue, saving both lives and resources. Monte Carlo simulation is commonly used to evaluate the risk and uncertainty that would affect the outcome of different decision options. Monte Carlo simulation allows the business risk analyst to incorporate the total effects of uncertainty in variables like sales volume, commodity and labour prices, interest and exchange rates, as well as the effect of distinct risk events like the cancellation of a contract or the change of a tax law.

Monte Carlo methods in finance are often used to evaluate investments in projects at a business unit or corporate level, or to evaluate financial derivatives. They can be used to model project schedules , where simulations aggregate estimates for worst-case, best-case, and most likely durations for each task to determine outcomes for the overall project. A Monte Carlo approach was used for evaluating the potential value of a proposed program to help female petitioners in Wisconsin be successful in their applications for harassment and domestic abuse restraining orders.

It was proposed to help women succeed in their petitions by providing them with greater advocacy thereby potentially reducing the risk of rape and physical assault. However, there were many variables in play that could not be estimated perfectly, including the effectiveness of restraining orders, the success rate of petitioners both with and without advocacy, and many others. The study ran trials that varied these variables to come up with an overall estimate of the success level of the proposed program as a whole.

In general, the Monte Carlo methods are used in mathematics to solve various problems by generating suitable random numbers see also Random number generation and observing that fraction of the numbers that obeys some property or properties. The method is useful for obtaining numerical solutions to problems too complicated to solve analytically. The most common application of the Monte Carlo method is Monte Carlo integration.

Deterministic numerical integration algorithms work well in a small number of dimensions, but encounter two problems when the functions have many variables.

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First, the number of function evaluations needed increases rapidly with the number of dimensions. For example, if 10 evaluations provide adequate accuracy in one dimension, then 10 points are needed for dimensions—far too many to be computed. This is called the curse of dimensionality. Second, the boundary of a multidimensional region may be very complicated, so it may not be feasible to reduce the problem to an iterated integral. Monte Carlo methods provide a way out of this exponential increase in computation time. As long as the function in question is reasonably well-behaved , it can be estimated by randomly selecting points in dimensional space, and taking some kind of average of the function values at these points.

A refinement of this method, known as importance sampling in statistics, involves sampling the points randomly, but more frequently where the integrand is large. To do this precisely one would have to already know the integral, but one can approximate the integral by an integral of a similar function or use adaptive routines such as stratified sampling , recursive stratified sampling , adaptive umbrella sampling   or the VEGAS algorithm.

A similar approach, the quasi-Monte Carlo method , uses low-discrepancy sequences. These sequences "fill" the area better and sample the most important points more frequently, so quasi-Monte Carlo methods can often converge on the integral more quickly. Another class of methods for sampling points in a volume is to simulate random walks over it Markov chain Monte Carlo. Another powerful and very popular application for random numbers in numerical simulation is in numerical optimization. The problem is to minimize or maximize functions of some vector that often has many dimensions.

Many problems can be phrased in this way: for example, a computer chess program could be seen as trying to find the set of, say, 10 moves that produces the best evaluation function at the end. In the traveling salesman problem the goal is to minimize distance traveled. There are also applications to engineering design, such as multidisciplinary design optimization. It has been applied with quasi-one-dimensional models to solve particle dynamics problems by efficiently exploring large configuration space.

Reference  is a comprehensive review of many issues related to simulation and optimization. The traveling salesman problem is what is called a conventional optimization problem. That is, all the facts distances between each destination point needed to determine the optimal path to follow are known with certainty and the goal is to run through the possible travel choices to come up with the one with the lowest total distance. However, let's assume that instead of wanting to minimize the total distance traveled to visit each desired destination, we wanted to minimize the total time needed to reach each destination.

This goes beyond conventional optimization since travel time is inherently uncertain traffic jams, time of day, etc. As a result, to determine our optimal path we would want to use simulation - optimization to first understand the range of potential times it could take to go from one point to another represented by a probability distribution in this case rather than a specific distance and then optimize our travel decisions to identify the best path to follow taking that uncertainty into account.

Probabilistic formulation of inverse problems leads to the definition of a probability distribution in the model space. This probability distribution combines prior information with new information obtained by measuring some observable parameters data. As, in the general case, the theory linking data with model parameters is nonlinear, the posterior probability in the model space may not be easy to describe it may be multimodal, some moments may not be defined, etc.

When analyzing an inverse problem, obtaining a maximum likelihood model is usually not sufficient, as we normally also wish to have information on the resolution power of the data. In the general case we may have many model parameters, and an inspection of the marginal probability densities of interest may be impractical, or even useless. But it is possible to pseudorandomly generate a large collection of models according to the posterior probability distribution and to analyze and display the models in such a way that information on the relative likelihoods of model properties is conveyed to the spectator.

This can be accomplished by means of an efficient Monte Carlo method, even in cases where no explicit formula for the a priori distribution is available. The best-known importance sampling method, the Metropolis algorithm, can be generalized, and this gives a method that allows analysis of possibly highly nonlinear inverse problems with complex a priori information and data with an arbitrary noise distribution.

### Recensie(s)

This introduction to Monte Carlo methods seeks to identify and study the unifying elements This is the second, completely revised and extended edition of the. This introduction to Monte Carlo methods seeks to identify and study the unifying elements that underlie their effective application.

Main article: Stochastic optimization. Statistics portal. October The Journal of Chemical Physics. Bibcode : JChPh.. Bibcode : Bimka.. Journal of the American Statistical Association. Computational Statistics.

## Monte Carlo theory, methods and examples

Nonlinear Markov processes. Cambridge Univ. Mean field simulation for Monte Carlo integration. Xiphias Press.

Lecture 37- Introduction to Monte Carlo Simulation

Retrieved Bibcode : PNAS LIX : — Methodos : 45— Methodos : — Feynman—Kac formulae. Genealogical and interacting particle approximations. Probability and Its Applications. Lecture Notes in Mathematics. Atlantic 3. This accessible new edition explores the major topics in Monte Carlo simulation Simulation and the Monte Carlo Method , Second Edition reflects the latest developments in the field and presents a fully updated and comprehensive account of the major topics that have emerged in Monte Carlo simulation since the publication of the classic First Edition over twenty-five years ago.

While maintaining its accessible and intuitive approach, this revised edition features a wealth of up-to-date information that facilitates a deeper understanding of problem solving across a wide array of subject areas, such as engineering, statistics, computer science, mathematics, and the physical and life sciences. The book begins with a modernized introduction that addresses the basic concepts of probability, Markov processes, and convex optimization.

Subsequent chapters discuss the dramatic changes that have occurred in the field of the Monte Carlo method, with coverage of many modern topics including: Markov Chain Monte Carlo Variance reduction techniques such as the transform likelihood ratio method and the screening method The score function method for sensitivity analysis The stochastic approximation method and the stochastic counter-part method for Monte Carlo optimization The cross-entropy method to rare events estimation and combinatorial optimization Application of Monte Carlo techniques for counting problems, with an emphasis on the parametric minimum cross-entropy method An extensive range of exercises is provided at the end of each chapter, with more difficult sections and exercises marked accordingly for advanced readers.

A generous sampling of applied examples is positioned throughout the book, emphasizing various areas of application, and a detailed appendix presents an introduction to exponential families, a discussion of the computational complexity of stochastic programming problems, and sample MATLAB programs. Requiring only a basic, introductory knowledge of probability and statistics, Simulation and the Monte Carlo Method , Second Edition is an excellent text for upper-undergraduate and beginning graduate courses in simulation and Monte Carlo techniques.

The book also serves as a valuable reference for professionals who would like to achieve a more formal understanding of the Monte Carlo method. Have doubts regarding this product? Post your question.