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Variational Bayes is a method in Bayesian inference that approximates probability distributions through optimization, allowing for efficient computation in complex models. It transforms the problem of integration in Bayesian statistics into one of optimization, providing a scalable alternative to traditional methods like Markov Chain Monte Carlo.
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Bayesian inference is a statistical method that updates the probability of a hypothesis as more evidence or information becomes available, utilizing Bayes' Theorem to combine prior beliefs with new data. It provides a flexible framework for modeling uncertainty and making predictions in complex systems, often outperforming traditional methods in scenarios with limited data or evolving conditions.
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Approximate inference is a set of techniques used to estimate the probability distributions in complex probabilistic models where exact inference is computationally infeasible. It is crucial for making predictions and understanding data in fields such as machine learning and statistics, where dealing with large datasets and high-dimensional spaces is common.
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Variational inference is a technique in Bayesian statistics that approximates complex posterior distributions through optimization, offering a scalable alternative to traditional sampling methods like Markov Chain Monte Carlo. It transforms the inference problem into an optimization problem by introducing a family of simpler distributions and finding the closest match to the true posterior using the Kullback-Leibler divergence.
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Optimization is the process of making a system, design, or decision as effective or functional as possible by adjusting variables to find the best possible solution within given constraints. It is widely used across various fields such as mathematics, engineering, economics, and computer science to enhance performance and efficiency.
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Probability distributions describe how the values of a random variable are distributed, providing a mathematical function that assigns probabilities to each possible outcome. They are essential in statistics and data analysis for modeling uncertainty and making predictions about future events or data patterns.
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Markov Chain Monte Carlo (MCMC) is a class of algorithms used to sample from probability distributions by constructing a Markov Chain that has the desired distribution as its equilibrium distribution. It is particularly useful in Bayesian statistics and computational physics for approximating complex integrals and distributions that are difficult to compute directly.
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The Evidence Lower Bound (ELBO) is a crucial component in variational inference, serving as a tractable objective function that approximates the true posterior distribution by maximizing a Lower Bound on the model's log marginal likelihood. By optimizing the ELBO, one can effectively perform approximate Bayesian inference in complex probabilistic models where exact inference is computationally infeasible.
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Latent Variable Models are statistical models that aim to explain observed variables through the inclusion of unobserved, or 'latent', variables. These models are essential for uncovering hidden structures in data, facilitating dimensionality reduction, and improving inference in complex datasets.
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Expectation-Maximization (EM) is an iterative algorithm used for finding maximum likelihood estimates of parameters in statistical models, particularly when the data is incomplete or has hidden variables. It alternates between evaluating the expected value of the log-likelihood (E-step) and maximizing this expectation (M-step), improving parameter estimates iteratively until convergence.
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Kullback-Leibler Divergence is a measure of how one probability distribution diverges from a second, expected probability distribution. It is often used in statistics and machine learning to quantify the difference between two distributions, with applications in areas like information theory, Bayesian inference, and model evaluation.
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