Monte Carlo Standard Error Definition
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that we use for the average. A possible measure of the error standard error of monte carlo simulation is the ``variance'' defined by: (269) where and The
Monte Carlo Error Analysis
``standard deviation'' is . However, we should expect that the error decreases with monte carlo simulation error the number of points , and the quantity defines by (271) does not. Hence, this cannot be a good measure of the error. monte carlo error definition Imagine that we perform several measurements of the integral, each of them yielding a result . Thes values have been obtained with different sequences of random numbers. According to the central limit theorem, these values whould be normally dstributed around a mean . Suppouse that
How To Calculate Monte Carlo Standard Error
we have a set of of such measurements . A convenient measure of the differences of these measurements is the ``standard deviation of the means'' : (270) where and Although gives us an estimate of the actual error, making additional meaurements is not practical. instead, it can be proven that (271) This relation becomes exact in the limit of a very large number of measurements. Note that this expression implies that the error decreases withthe squere root of the number of trials, meaning that if we want to reduce the error by a factor 10, we need 100 times more points for the average. Subsections Exercise 10.1: One dimensional integration Exercise 10.2: Importance of randomness Next: Exercise 10.1: One dimensional Up: Monte Carlo integration Previous: Simple Monte Carlo integration Adrian E. Feiguin 2009-11-04
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Monte Carlo Error Propagation
ListHHS Author ManuscriptsPMC3337209 Am Stat. Author manuscript; available in PMC 2012 Apr 25.Published in final edited form as:Am http://www.northeastern.edu/afeiguin/phys5870/phys5870/node71.html Stat. 2009 May 1; 63(2): 155–162. doi: 10.1198/tast.2009.0030PMCID: PMC3337209NIHMSID: NIHMS272824On the Assessment of Monte Carlo Error in Simulation-Based Statistical AnalysesElizabeth Koehler, Biostatistician, Elizabeth Brown, Assistant Professor, and Sebastien J.-P. A. Haneuse, Associate Scientific InvestigatorElizabeth Koehler, Department https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3337209/ of Biostatistics, Vanderbilt University, Nashville, TN 37232;Contributor Information.Elizabeth Koehler: ude.tlibrednav@relheok.e; Elizabeth Brown: ude.notgnihsaw@bazile; Sebastien J.-P. A. Haneuse: gro.chg@s.esuenah Author information ► Copyright and License information ►Copyright notice and DisclaimerSee other articles in PMC that cite the published article.AbstractStatistical experiments, more commonly referred to as Monte Carlo or simulation studies, are used to study the behavior of statistical methods and measures under controlled situations. Whereas recent computing and methodological advances have permitted increased efficiency in the simulation process, known as variance reduction, such experiments remain limited by their finite nature and hence are subject to uncertainty; when a simulation is run more than once, different result
Pseudo-failures to replicate » Markov chain Monte Carlo standard errors Posted byAndrew on 2 April 2007, 12:36 am Galin Jones sent me this paper (by James Flegal, Murali Haran, and himself) which he said started with a suggestion I once http://andrewgelman.com/2007/04/02/markov_chain_mo/ made to him long ago. That's pretty cool! Here's the abstract: Current reporting of results based on Markov chain Monte Carlo computations could be improved. In particular, a measure of the accuracy of the resulting estimates is rarely reported in the literature. Thus the reader has little ability to objectively assess the quality of the reported estimates. This paper is an attempt to address this issue in that we discuss why monte carlo Monte Carlo standard errors are important, how they can be easily calculated in Markov chain Monte Carlo and how they can be used to decide when to stop the simulation. We compare their use to a popular alternative in the context of two examples. This is a clear paper with some interesting results. My main suggestion is to distinguish two goals: estimating a parameter in a model and estimating an expectation. monte carlo error To use Bayesian notation, if we have simulations theta_1,…,theta_L from a posterior distribution p(theta|y), the two goals are estimating theta or estimating E(theta|y). (Assume for simplicity here that theta is a scalar, or a scalar summary of a vector parameter.) Inference for theta or inference for E(theta) When the goal is to estimate theta, then all you really need is to estimate theta to more accuracy than its standard error (in Bayesian terms, its posterior standard deviation). For example, if a parameter is estimated at 3.5 +/- 1.2, that's fine. There's no point in knowing that the posterior mean is 3.538. To put it another way, as we draw more simulations, we can estimate that "3.538" more precisely-our standard error on E(theta|y) will approach zero-but that 1.2 ain't going down much. The standard error on theta (that is, sd(theta|y)) is what it is. This is a general issue in simulation (not just using Markov chains), and we discuss it on page 277 of Bayesian Data Analysis (second edition): if the goal is inference about theta, and you have 100 or more independent simulation draws, then the Monte Carlo error adds almost nothing to the uncertainty coming from the actual posterior variance. On the other hand, if your goal is to estimate
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