Stochastic methods are computational techniques used to model and analyze systems that involve randomness and uncertainty. These methods involve generating or simulating random variables or processes to approximate or solve complex problems.
- Markov Chain Monte Carlo (MCMC) is a widely used stochastic method for sampling from complex probability distributions. It is particularly useful in Bayesian inference, where the goal is to estimate the posterior distribution of model parameters given observed data. MCMC algorithms, such as the Metropolis-Hastings algorithm and Gibbs sampling, construct a Markov chain that converges to the target distribution, allowing for efficient sampling even in high-dimensional spaces.
- Bayesian Optimization is a stochastic method used for optimizing black-box functions that are expensive to evaluate. It combines Bayesian modeling and optimization to iteratively explore and exploit the function space efficiently. By building a surrogate model of the objective function and using Bayesian inference to update the model based on observed evaluations, Bayesian Optimization guides the search toward promising regions to find the optimal or near-optimal solution.
Stochastic methods are valuable in various fields, including machine learning, optimization, physics, finance, and many other domains where uncertainty and randomness play a significant role. They offer powerful tools for tackling complex problems by simulating or approximating random processes, sampling from probability distributions, and optimizing functions under uncertainty.