Ch 7 Probability Theory and Stochastic Simulation: Frequentist statistics: - Probability of observing E: ()NNEpE≈ - Joint Probability: ()()()12121| EEpEpEEp=∩ - Expectation: ∑=≈=exp1exp1)(NWNWWEνν Bayes’ Theorem: ()()()()212121|| EEpEpEEpEp= - Bayes’ Theorem is general. Definitions: - variance: ()()()[]()()[]222var WEWEWEWEW −=−= - (X, Y independent, var(X+Y) = var(X) + var(Y)) - standard deviation: ()Wvar=σ - covariance ()()[]()[]{}YEYXEXEYX−−=,cov , for two random variables X and Y - covariance matrix Important Probability Distributions Definitions: - Discrete random variable o For {}miXXXX ,...,,21= o ()=iXN number of observations of Xi o T is the total number of observations ()TXNMij=∑=1 o Probability is definied by: ()()TXNXPii= o Normalization is defined by ()11=∑=MijXP - Continuous random variable o This is just the continuous version of the above, defined by integrals instead of limits, differentials instead of increments o Normalization condition: ()∫=hixloxdxxp 1 o Expectation () ()∫==hixloxdxxxpxxE - Cumulative Probability distribution o Basis of RAND in matlabo () ()udxxpXFxloxM==∫'' o u is defined as uniformly distributed 10≤≤u Bernoulli trials o Concept that observed error is the net sum of many small random errors Random Walk Problem - key point: independence of coin tosses - Main results: 0=x 22nlx = Binomial Distribution - probability distribution: () ( )()HnnHHnHHHppnnnnP−−= 1, - binomial coefficient: ()!!!HHHnnnnnn−= - BINORND Matlab to generate random number distributed using binomial distribution Gaussian (Normal) Distribution - Take binomial distribution, change into probability of observing net displacement after n steps of length l o ()()()nlxnlxnnlnxp−+=212/!2/!,; - Evaluate in limit that nÆ ∞, take natural log, and use Stirling’s approximation - Algebra, and taylor expand around the ln terms - Taking the exponential and normalizing such that: ()∫∞∞−=1,; dxlnxP - ()−=222exp21;σπσσxxP 22nl=σ - Binomial Distribution of random walk reduces to Gaussian Distribution as nÆ ∞ - Central Limit Theorem: sequence of random variables, which are not distributed normally, the statistic o ∑=−=NjjjjnnS1.1σµξo random variable: jξ with mean .jµ and variance 2jσ - is normally distributed in the limit that nÆ ∞, with variance = 1 o ()−=2exp212nnSSPπ - Non-zero Mean (basis of randn) o ()()−−=2222exp21,σµπσσµxN - Multivariate Gaussian Distribution (use of covariance matrix) o Covariance Matrix: ()[]()[]()[]{}jjiiijEEEννννν−−=cov o Covariance Matrix is always symmetric and positive definite o For independent components: ()I2covσν= o () ()()[]TAxAxA covcovcov ==ν o if ν is a random vector and c is a constant vector: o ()()()[]()[]ccccccTTννννcovcovvarvar ⋅===⋅ o ()()()()−Σ−−Σ=Σ−µνµνπµν12/21exp21,;TNP Poisson Distribution - Poisson distribution can be used to determine probability of success if there are n trials, derived in the limit as nÆ ∞ - Total number of successes in trial is a random variable, which d - Another limiting case of binomial distribution - ()()pnepnpnP−=!,;ξξξ - p = probability of individual success - n = number of trials - ξ = result if success or failure, typically {1,0} with different probabilities Boltzmann/Maxwell Distributions o ()()−=kTqEQqP exp1 o Q is the normalization constant o Replacing E(q) for kinetic energy we arrive at Maxwell Distribution o ()−∝kTmP2exp2νν Brownian Dynamics and Stochastic Differential Equations- velocity autocorrelation function o ()()xVxxtVVeCtCτ/00−≈≥ µρτ922Rxv= o () ( )()tDVtVxxδ20 = - Dirac Delta Function o ()−=→2202exp21limσπσδσtt o ()() ( )0fdtttf =∫∞∞−δ - Langevin equation - Wiener process - Stochastic Differential equations o Explicit Euler SDE method o ()()()()[]()ttxWDtdxdUtxttx ∆+∆−=−∆+2/121ζ - Ito’s Stochastic Calculus o Example: Black-Scholes o Fokker-Planck o Einstein Relation o Brownian motion in multiple dimensions - MCMC o Stat Mech example o Metropolis recipe (pg497) o Example: Ising Lattice o Field theory o Monte Carlo Integration o Simulated annealings o Genetic Programming Bayesian Statistics and Parameter Estimation Goal of this material is to draw conclusions from data (“statistical inference”) and estimate parameters. Basic definitions - Predictor variables: x = [x1 x2 x3… xM] - Response variable: y(R) = [y1 y2 y3 … yL] - θ: model parameters Main goal: match model prediction to that of the observed response by selecting θ. Single-Response Linear Regression- set of predictor variables, known a priori: x[k] = [x1[k] x2[k] x3[k]… xM[k]], for the kth experiment - measurement y[k] - assume a linear model: [][][][][]kkMMkkkxxxyεββββ+++++= ...22110 - the error in ε[k] is responsible for the difference between model and observed - define []TM210 ... ββββθ= - response is: [][]()[]ktruekkxyεθ+⋅= - model prediction is: [][]()truekkxyθ⋅=ˆ - define design matrix X, which contains all information about every experiment (with different predictor variables) −−−−−−−−−−−−−−−−−−=][]2[]1[:NxxxX - vector of predicted responses: ()()()()θθθθθXyyyyN==][]2[]1[:)))) Linear Least Squares Regression - minimize sum of squared errors: ()[] []()[]21∑=−=NkkkyySθθ) - First derivative = 0, 2nd derivative is > 0, using these conditions with above equation you can derive a linear system - ()yXXXTLST=θ Æ ()yXXXTTLS1−=θ (review point?) - XTX, contains information about experimental design to probe the parameter values - XTX is a real, symmetric matrix that is positive-semidefinite - Solving this is through standard linear solving, or QR decomposition or some other method - All this estimates parameters, but does not give us accuracy of our estimates Bayesian view of statistical inference- Statement of belief (especially in random number generators) Bayesian view of
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