Introduction to Monte Carlo Methods

Compute tail probabilities for the standard normal with Monte Carlo by exploiting symmetry and splitting the unbounded region into negative and positive parts, estimating p-values from sample fractions above thresholds.

Explore importance sampling to improve tail probability estimates by weighting samples from a chosen proposal density g, such as a truncated exponential, to better estimate P(X>4) for a normal distribution.

Explore importance sampling to recycle samples with weights, using normalization and bootstrapping, and apply to truncated exponential and t distributions, plus a normal–cauchy posterior for convergent expectations.

Explore bootstrap methods, a non-parametric Monte Carlo technique that resamples observed data to estimate empirical and theoretical CDFs and derive confidence intervals.

Apply paired bootstrap to a linear regression of mpg on weight and am, resampling both dependent and independent variables to generate coefficient distributions and bias-corrected estimates.

Learn how to monitor convergence in Monte Carlo experiments by running many parallel trials, compare 95% confidence intervals, and use bootstrapping as a cheaper alternative.

Use antithetic variables to reduce variance in Monte Carlo estimates by pairing negatively correlated samples. This approach can outperform independent samples and save computational resources when experiments are costly.

Estimate E[H(x)] for x normal using simple Monte Carlo, compare with aesthetic variable estimation using half as many samples, and verify negative correlations.

Explore how simulated annealing escapes local maxima to find a global maximum by balancing exploration through temperature and probabilistic moves.

Applies genetic algorithms to the traveling salesman problem by evolving random paths with euclidean distance as fitness, using mutation, a 1000-path population, 35% rate, and 200 generations, without crossover.

Explore the monte carlo em algorithm for missing data, iterating the expected complete data log-likelihood and its maximization using observed data to estimate theta.

Explore discrete Markov chains and Markov chain Monte Carlo by examining a two-state system, transition probabilities and a transition matrix, and simulating 2000 steps to observe state changes and autocorrelation.

Explore a three-state Markov chain with a transition matrix to illustrate convergence to a limiting distribution, stationary distribution, irreducibility, and reversibility in Markov chain Monte Carlo.

Learn how the Metropolis-Hastings algorithm samples from a target density using a proposal distribution with an acceptance probability, illustrating independent Metropolis-Hastings through gamma and beta examples.

Apply Bayesian logistic regression to the o-ring dataset using a non-informative prior and a metropolis-hastings posterior, yielding 95 percent credible intervals for alpha and beta.

Explore Bayesian change point analysis on a time series of coal mine disasters (1851–1962) using a Gibbs sampler to infer a two-rate Poisson model and the change point.