Computational Statistics

Foundations of Computational Probability and Simulation

Random Number Generation and Transformation

• Master the generation of pseudo-random numbers using linear congruential generators and Mersenne Twister algorithms, understanding their limitations in high-dimensional spaces.
• Implement inverse transform sampling to generate values from continuous distributions by applying the inverse cumulative distribution function to uniform random variables.
• Utilize the rejection sampling method to draw samples from complex, non-standard probability distributions by bounding the target density with a scaled proposal density.

Monte Carlo Integration

• Apply Monte Carlo methods to solve high-dimensional integrals that are mathematically intractable or impossible to solve with standard numerical quadrature.
• Reduce variance in simulations using techniques such as antithetic variates, control variates, and importance sampling to achieve greater precision with fewer iterations.

Markov Chain Monte Carlo (MCMC) Methods

Core MCMC Mechanics

• Construct Markov chains that converge to a target posterior distribution, ensuring the properties of ergodicity, irreducibility, and aperiodicity are met.
• Execute the Metropolis-Hastings algorithm to sample from probability distributions where only the proportional density is known, managing proposal distribution widths for optimal acceptance rates.
• Apply Gibbs sampling to multi-dimensional problems by iteratively sampling from conditional distributions, effectively breaking complex joint distributions into manageable univariate segments.

Advanced Sampling Techniques

• Implement Hamiltonian Monte Carlo (HMC) to traverse high-dimensional parameter spaces efficiently by simulating physical particle dynamics and gradient information.
• Utilize the No-U-Turn Sampler (NUTS) to automate the selection of trajectory lengths, eliminating the need for manual tuning in complex Bayesian hierarchical models.

Optimization and Stochastic Search

Deterministic vs. Stochastic Optimization

• Master gradient-based optimization techniques, including Newton-Raphson and Quasi-Newton methods like BFGS, for finding maximum likelihood estimates.
• Apply the Expectation-Maximization (EM) algorithm to find maximum likelihood estimates of parameters in statistical models that depend on unobserved latent variables.

Global Optimization

• Implement Simulated Annealing to avoid local optima by allowing probabilistic moves toward higher energy states during the early stages of the search process.
• Utilize stochastic approximation methods, such as the Robbins-Monro algorithm, to find roots of functions when the function evaluations are subject to random noise.

Computational Inference and Resampling

Bootstrap and Jackknife Methods

• Quantify uncertainty in point estimators by using empirical distribution functions to generate bootstrap replicates of complex statistics.
• Apply the jackknife technique for bias reduction and variance estimation, systematically leaving out individual observations from the dataset to calculate influence functions.
• Use permutation tests to perform exact hypothesis testing without relying on large-sample asymptotic assumptions, allowing for robust statistical inference on small or skewed datasets.

Density Estimation and Smoothing

• Perform non-parametric density estimation using kernel methods, selecting appropriate bandwidths through cross-validation to balance bias and variance.
• Implement local polynomial regression and smoothing splines to identify underlying trends in noisy data without assuming a rigid parametric functional form.

Numerical Linear Algebra for Statistics

Matrix Decompositions

• Utilize Cholesky, LU, and QR decompositions to solve large-scale linear systems efficiently, reducing the computational cost of regression and multivariate analysis.
• Apply Singular Value Decomposition (SVD) and Principal Component Analysis (PCA) to perform dimensionality reduction and noise filtering in high-dimensional feature spaces.
• Master sparse matrix operations to manage high-dimensional datasets where most entries are zero, significantly improving memory efficiency and processing speed.