Ch 7 Probability Theory and Stochastic Simulation Frequentist statistics Probability of observing E Joint Probability Expectation NE N p E1 E2 p E1 p E2 E1 1 N exp E W W 1 W N exp p E Bayes Theorem p E1 p E2 E1 p E2 p E1 E2 Bayes Theorem is general Definitions variance var W E W E W E W 2 E W X Y independent var X Y var X var Y standard deviation var W covariance cov X Y E X E X Y E Y for two random variables X and Y covariance matrix 2 2 Important Probability Distributions Definitions Discrete random variable o For X i X 1 X 2 X m o N X i number of observations of Xi M N X j T o T is the total number of observations i 1 o Probability is definied by P X i o Normalization is defined by P M i 1 N X i T X 1 j 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 o Expectation Cumulative Probability distribution o Basis of RAND in matlab xhi xlo p x dx 1 xhi E x x xp x dx xlo o F X M p x dx u x xlo o u is defined as uniformly distributed 0 u 1 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 x 0 x 2 nl 2 Binomial Distribution n n n n probability distribution P n nH p H H 1 p H H nH n n binomial coefficient nH nH n nH 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 n n 1 o p x n l n x l n x l 2 2 2 Evaluate in limit that n take natural log and use Stirling s approximation Algebra and taylor expand around the ln terms Taking the exponential and normalizing such that P x n l dx 1 x2 1 exp 2 P x 2 nl 2 2 2 Binomial Distribution of random walk reduces to Gaussian Distribution as n Central Limit Theorem sequence of random variables which are not distributed normally the statistic 1 N j j o Sn n j 1 j o random variable j with mean j and variance j is normally distributed in the limit that n with variance 1 S 2 1 o P S n exp n 2 2 Non zero Mean basis of randn x 2 1 2 o N exp 2 2 2 2 Multivariate Gaussian Distribution use of covariance matrix o Covariance Matrix cov ij E i E i j E j o Covariance Matrix is always symmetric and positive definite o For independent components cov 2 I o cov cov A x A cov x AT o if is a random vector and c is a constant vector o var c var c c cov c c cov c o P T 1 2 N 2 T 1 T exp 1 2 Poisson Distribution Poisson distribution can be used to determine probability of success if there are n trials derived in the limit as n Total number of successes in trial is a random variable which d Another limiting case of binomial distribution pn pn P n p e p probability of individual success n number of trials result if success or failure typically 1 0 with different probabilities Boltzmann Maxwell Distributions 1 E q o P q exp Q kT o Q is the normalization constant o Replacing E q for kinetic energy we arrive at Maxwell Distribution m 2 o P exp 2kT Brownian Dynamics and Stochastic Differential Equations velocity autocorrelation function o CV t 0 CV 0 e o Vx t Vx 0 2 D t x 2 R 2 v 9 x Dirac Delta Function t2 1 o t lim exp 2 0 2 2 o x t Vx f t t dt f 0 Langevin equation Wiener process Stochastic Differential equations o Explicit Euler SDE method 1 dU t 2 D 1 2 Wt o x t t x t dx x t 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 model parameters Main goal match model prediction to that of the observed response by selecting 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 k k k assume a linear model y k 0 1 x1 2 x2 M xM k the error in k is responsible for the difference between model and observed T define 0 1 2 M response is k y k x true k k true model prediction is y k x define design matrix X which contains all information about every experiment with different predictor variables x 1 2 x X N x vector of predicted responses y 1 2 y y X N y Linear Least Squares Regression 2 N S y k y k minimize sum of squared errors First derivative 0 2nd derivative is 0 using these conditions with above equation you can derive a linear system X X T LS X T y LS X T X 1 k 1 X T y 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 single response regression k true Begin with y k x k When we repeat this experiment multiple times we get a vector With Gauss Markov Conditions E k 0 cov k j kj 2 We also assume that our error is normally distributed Probability of observing some response y 1 N 1 p y exp 2 S 2 2 We use Bayes Theorem to get probability of and p y p Posterior density p y py o N p y is a normalizing factor we redefine p y to l y in the Bayesian framework we want to maximize posterior density p c Non informative priors p p p p 1 Nonlinear least squares the treatment via least squares still works we just use numerical optimization utilizing a cost function …