What is Variational inference

Variational Inference: A Comprehensive Guide to Approximate Bayesian Inference

Introduction:

Bayesian inference is a powerful framework for probabilistic reasoning, which provides a systematic method to update the belief in a hypothesis, given observed data. It is widely used for applications such as parameter estimation, model selection, and uncertainty quantification. However, exact Bayesian inference is often intractable, as it involves the integration of the posterior distribution over the entire parameter space. In practice, we resort to some approximation methods to perform Bayesian inference in a computationally efficient manner. Variational inference is one such method, which approximates the posterior distribution with a simpler tractable distribution. In this article, we will provide a comprehensive guide to variational inference, covering its key concepts, algorithms, and applications.

Before we delve into variational inference, let us briefly recap the basics of Bayesian inference. Bayesian inference is based on Bayes theorem, which states that the posterior distribution of a hypothesis H given observed data D, is proportional to the product of the likelihood of the data given the hypothesis and the prior distribution of the hypothesis. P(H|D) ∝ P(D|H) * P(H) Here, P(H|D) is the posterior distribution, P(D|H) is the likelihood of the data given the hypothesis, and P(H) is the prior distribution of the hypothesis. The likelihood function represents the probability of observing the data D, given the values of the parameters of the model. The prior distribution represents the prior beliefs about the values of the parameters, before observing the data. Given the posterior distribution, we can make predictions, perform inference, and compute the expected utility of different decisions concerning the hypothesis. However, exact Bayesian inference is often intractable, because it involves the computation of the posterior distribution over the entire parameter space. In practice, we must resort to some approximation methods to perform Bayesian inference in a computationally efficient manner. Variational inference is one such method, which approximates the posterior distribution with a simpler tractable distribution.

The key idea behind variational inference is to approximate the posterior distribution with a simpler distribution, which belongs to a parametric family of distributions. The simpler distribution is called the variational distribution, and the parameters of the variational distribution are called the variational parameters. The goal is to find the values of the variational parameters that minimize a distance measure between the variational distribution and the true posterior distribution. This distance measure is usually some form of divergence measure such as Kullback-Leibler (KL) divergence or reverse KL divergence. The quality of the approximation depends on the complexity of the variational family, the choice of the distance measure, and the optimization algorithm used to find the optimum. In general, the closer the variational distribution is to the true posterior, the better the approximation.

At a high level, the steps involved in variational inference can be summarized as follows:

1. Specify a prior distribution over the parameters of the model.

2. Specify a likelihood function that describes the probability of observing the data, given the parameters of the model.

3. Choose a tractable family of distributions (e.g., Gaussian, Poisson, exponential) to approximate the posterior distribution.

4. Set up an optimization problem to find the values of the variational parameters that minimize the distance measure between the approximate posterior and the true posterior.

5. Use the optimized approximate posterior to compute marginal probabilities, expected values, and other statistics of interest.

There are several algorithms used for variational inference, such as mean-field variational inference, stochastic variational inference, and black-box variational inference. Let us briefly describe each of these methods. Mean-field Variational Inference: In mean-field variational inference, we assume that the variational distribution factorizes over the parameters, that is, q(θ) = ∏q(θ_i), where θ_i's are the parameters of the model. This assumption simplifies the optimization problem, as we can optimize each q(θ_i) independently. The optimization problem can be written as: q*(θ) = argmin{KL[q(θ)||p(θ|D)]} where q(θ) is the variational distribution, p(θ|D) is the true posterior distribution, and KL[q(θ)||p(θ|D)] is the Kullback-Leibler divergence between the variational distribution and the true posterior distribution. The optimization problem can be solved iteratively, by optimizing each q(θ_i) in turn, while keeping the others fixed. Stochastic Variational Inference: In stochastic variational inference, we use a Monte Carlo approximation to estimate the gradient of the objective function. This allows us to use the gradient-based optimization algorithm, such as stochastic gradient descent, to optimize the variational parameters. In this method, we randomly sample data points from the data set, and use them to compute an unbiased estimate of the gradient. This method is useful for large data sets, where computing the gradient on the entire data set is infeasible. Black-box Variational Inference: Black-box variational inference is a more general approach, which does not assume any particular form of the posterior distribution. Instead, it uses a flexible function approximator, such as a neural network, to approximate the posterior distribution. The neural network takes the observed data as input and outputs the parameters of the variational distribution. The optimization problem is then to minimize the difference between the true posterior and the output of the neural network, using some divergence measure. This method can handle complex models and is becoming increasingly popular in recent years.

Variational inference has been used in a wide range of applications, such as probabilistic modeling, Bayesian optimization, reinforcement learning, natural language processing, computer vision, and many more. Let us describe a few important applications of variational inference. Probabilistic Modeling: Probabilistic modeling is a key application area of variational inference, where it is used to estimate the parameters of the model and perform inference over the latent variables. This approach is particularly useful in Bayesian neural networks, where the activation functions are probabilistic and the outputs are probabilistic distributions. Bayesian Optimization: Bayesian optimization is a technique for optimizing expensive black-box functions, where the aim is to find the global minimum with as few function evaluations as possible. Variational inference can be used to accelerate the optimization process, by approximating the posterior distribution of the function values, given the observed evaluations. Reinforcement Learning: Variational inference has been applied to reinforcement learning, where it is used to learn a model of the environment and perform inference over the latent variables that capture the hidden state of the environment. Natural Language Processing: Variational inference has been used in natural language processing, such as topic modeling, text classification, and sentiment analysis. It can be used to infer the topics of a document, given the observed textual content. Computer Vision: Variational inference has been used in computer vision, such as image segmentation, object recognition, and scene parsing. It can be used to infer the latent variables that capture the hidden structure of the images.

Variational inference is a powerful technique for approximating the posterior distribution in Bayesian inference, which allows us to perform probabilistic reasoning in a computationally efficient manner. It provides a systematic method to choose a tractable family of distributions to approximate the posterior and optimize the variational parameters to minimize the distance measure between the approximate and true posteriors. Variational inference has found widespread applications in probabilistic modeling, Bayesian optimization, reinforcement learning, natural language processing, computer vision, and other areas. With the availability of powerful computing resources and increasing interest in probabilistic modeling, the use of variational inference is expected to grow further in the future.