Monte carlo integration, importance sampling, basic idea of markov chain monte carlo
- 1. UNIVERSITY OF RWANDA DEPARTMENT OF APPLIED STATISTICS Monte Carlo integration, importance sampling, basic idea of Markov chain Monte Carlo. BIZIMANA Appolinaire Student in UR-College of Business and Economics
- 2. This presentation is made up of 7 key points I. Definition II. History of Monte Carlo integration III. The use of Monte Carlo integration IV. How to apply of Monte Carlo integration V. Importance sampling VI. Basic idea of Markov chain Monte Carlo (MCMC). VII.Exercises of application to take a look at them on word document.
- 3. I. Definition It is somehow cumbersome to define Monte carlo without taking into consideration on its methods and domain of application, So There is no consensus on how Monte Carlo should be defined. By definition In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral.
- 4. II. History of Monte Carlo integration Monte Carlo methods were originally practiced under more generic names such as ”statistical sampling”. The name ”Monte Carlo” was popularized by physics researchers S. Ulam, E. Fermi,J. von Neumann, and N. Metropolis; The name is a reference to a famous casino in Monaco where Ulam’s uncle would borrow money to gamble. This name and the systematic development of Monte Carlo methods dates from about 1944.
- 5. III.The use of Monte Carlo integration The real use of Monte Carlo methods as a research tool stems from work on the atomic bomb during the second world war. The most common use for Monte Carlo methods is the evaluation of integrals. IV. How to apply of Monte Carlo integration There are a number of practical methods that use this principle to attempt to achieve better estimates of the mean with fewer random samples
- 6. Cont’d We call these methods variance reduction techniques. Those are practical techniques including adaptive sampling, stratification, importance sampling, and combined sampling. Monte-Carlo methods generally follow the following steps Determine the statistical properties of possible inputs Generate many sets of possible inputs which follows the above properties Perform a deterministic calculation with these sets Analyze statistically the results.
- 7. V. Importance sampling By definition is a general technique for estimating properties of a particular distribution, while only having samples generated from a different distribution from the distribution of interest. Importance sampling reduces variance by observing that we have the freedom to choose the PDF used during integration. By choosing samples from a distribution pd f (x), which has a similar shape as the function f (x) being integrated, variance is reduced.
- 8. VII. Basic idea of Markov chain Monte Carlo (MCMC). In terms of meaning Markov chain Monte is a mathematical model for stochastic system that generates random variable X1, X2, X3,..........., Xt . P(Xt/ X1, X2, X3,.............,Xt-1). The distribution of the next random variable depends on only the current variable which the reason why of saying chain. So the Basic idea of Markov chain Monte Carlo is the following, To construct a Markov Chain Such that :
- 9. Cont’d Have parameters as the state space, and the stationary distribution is posterior probability distribution f the parameters. Simulate the chain Treat the realisation as a sample from the posterior probability distribution MCMC is a general purpose technique for generating fair samples from a probability in high dimension space, using random numbers draw from uniform probability in certain range.
- 10. Conclusion Monte Carlo simulations are an important tool in modern-day studies of many physical systems. Where unlikely events are to be simulated, the importance sampling technique can considerably ease the processing burden, without compromising statistical significance. Here a comparison of importance sampling and standard Monte Carlo simulations is given. Emphasis is on variance reduction, and on the simulation gain of importance sampling.
Editor's Notes
- #11: Reference Introduction to Monte Carlo, Astro 542 Princeton University ,Shirley Ho Markov chain monte carlo (mcmc) methods (Mark Denny 2001 Eur. J. Phys. 22 403. doi:10.1088/0143-0807/22/4/315Received 23 March 2001. ) MONTE%20CARLO%20TP%20(1)%20(2).pdf https://www.academia.edu/1703285/MCMC_-_Markov_Chain_Monte_Carlo_One_of_the_top_ten_algorithms_of_the_20th_century http:// en.wikipedia.org/wiki/Monte_Carlo_integration