Annealed importance sampling is a method that combines Markov chain Monte Carlo (MCMC) techniques with importance sampling to estimate expectations and normalizing constants for complex distributions. The method uses a sequence of intermediate distributions, gradually transitioning from a simple distribution to the target distribution, allowing for more accurate sampling even in high-dimensional spaces. This approach is particularly useful when dealing with isolated modes or when estimating normalizing constants, as it avoids some of the convergence and autocorrelation issues associated with traditional MCMC methods. The method is inspired by the concept of simulated annealing, where a sequence of distributions is used to move from a simple distribution to the target distribution. Each distribution in the sequence is designed to allow for easier sampling, and the final distribution is the target distribution of interest. The importance weights assigned to each sample help ensure that the estimates converge to the correct values as the number of annealing runs increases. Annealed importance sampling works by generating a sequence of points from a series of intermediate distributions, each of which is used to compute importance weights. These weights are then used to estimate the expectation of a function with respect to the target distribution. The method is related to thermodynamic integration, which is used to estimate ratios of normalizing constants. It also generalizes a variant of sequential importance sampling and can be used to estimate expectations and normalizing constants in Bayesian statistics. The accuracy of importance sampling estimates depends on the variability of the importance weights. When these weights vary widely, the estimates can be biased. Annealed importance sampling reduces this variability by using a sequence of intermediate distributions that gradually approach the target distribution, allowing for more accurate sampling. The efficiency of the method depends on the number and spacing of the intermediate distributions, as well as the quality of the Markov chain transitions used to generate samples from each distribution. Demonstrations on simple distributions and a linear regression problem show that annealed importance sampling can provide accurate estimates of expectations and normalizing constants, even in the presence of isolated modes. The method is particularly effective when the target distribution has multiple modes, as it allows for more thorough sampling across different modes. However, the method can be less efficient when the number of intermediate distributions is not optimized, leading to higher variance in the importance weights. Overall, annealed importance sampling is a valuable tool for estimating expectations and normalizing constants in complex distributions, especially when traditional MCMC methods may struggle with convergence or autocorrelation issues.Annealed importance sampling is a method that combines Markov chain Monte Carlo (MCMC) techniques with importance sampling to estimate expectations and normalizing constants for complex distributions. The method uses a sequence of intermediate distributions, gradually transitioning from a simple distribution to the target distribution, allowing for more accurate sampling even in high-dimensional spaces. This approach is particularly useful when dealing with isolated modes or when estimating normalizing constants, as it avoids some of the convergence and autocorrelation issues associated with traditional MCMC methods. The method is inspired by the concept of simulated annealing, where a sequence of distributions is used to move from a simple distribution to the target distribution. Each distribution in the sequence is designed to allow for easier sampling, and the final distribution is the target distribution of interest. The importance weights assigned to each sample help ensure that the estimates converge to the correct values as the number of annealing runs increases. Annealed importance sampling works by generating a sequence of points from a series of intermediate distributions, each of which is used to compute importance weights. These weights are then used to estimate the expectation of a function with respect to the target distribution. The method is related to thermodynamic integration, which is used to estimate ratios of normalizing constants. It also generalizes a variant of sequential importance sampling and can be used to estimate expectations and normalizing constants in Bayesian statistics. The accuracy of importance sampling estimates depends on the variability of the importance weights. When these weights vary widely, the estimates can be biased. Annealed importance sampling reduces this variability by using a sequence of intermediate distributions that gradually approach the target distribution, allowing for more accurate sampling. The efficiency of the method depends on the number and spacing of the intermediate distributions, as well as the quality of the Markov chain transitions used to generate samples from each distribution. Demonstrations on simple distributions and a linear regression problem show that annealed importance sampling can provide accurate estimates of expectations and normalizing constants, even in the presence of isolated modes. The method is particularly effective when the target distribution has multiple modes, as it allows for more thorough sampling across different modes. However, the method can be less efficient when the number of intermediate distributions is not optimized, leading to higher variance in the importance weights. Overall, annealed importance sampling is a valuable tool for estimating expectations and normalizing constants in complex distributions, especially when traditional MCMC methods may struggle with convergence or autocorrelation issues.