What can you learn about new Bayesian inference techniques?

1 Variational inference

Variational inference is a technique that approximates the posterior distribution of the parameters of a Bayesian model with a simpler distribution that is easier to compute. The idea is to find the best fit between the two distributions by minimizing a measure of divergence, such as the Kullback-Leibler divergence. Variational inference can be much faster than traditional methods like Markov chain Monte Carlo (MCMC), but it can also introduce some bias and uncertainty in the approximation. Variational inference can be applied to a wide range of models, such as latent variable models, hierarchical models, and deep generative models.

2 Hamiltonian Monte Carlo

Hamiltonian Monte Carlo (HMC) is a technique that improves the efficiency and accuracy of MCMC by using the gradient of the posterior distribution to guide the sampling process. The idea is to simulate the dynamics of a particle with a position and a momentum that correspond to the parameters and their updates. The particle moves along the contours of the posterior distribution, avoiding areas of low probability and exploring areas of high probability. HMC can reduce the autocorrelation and dependence of the samples, leading to faster convergence and better mixing. HMC can be applied to models with high-dimensional and continuous parameters, such as regression models, neural networks, and Gaussian processes.

3 Automatic differentiation

Automatic differentiation (AD) is a technique that computes the derivatives of a function with respect to its inputs by applying the chain rule of calculus. The idea is to break down the function into elementary operations and apply the rules of differentiation to each operation. AD can be done in two modes: forward mode and reverse mode. Forward mode computes the derivatives from the inputs to the outputs, while reverse mode computes the derivatives from the outputs to the inputs. AD can be used to calculate the gradients of the posterior distribution, which are needed for variational inference and HMC.

4 Probabilistic programming

Probabilistic programming is a technique that allows you to express and manipulate Bayesian models using a programming language. The idea is to use variables, functions, and control structures to define the prior distributions, the likelihood functions, and the data. Then, you can use built-in or external inference engines to perform Bayesian inference on the models. Probabilistic programming can make Bayesian modeling more intuitive, flexible, and scalable. Probabilistic programming can be done using specialized languages, such as Stan, PyMC3, and JAGS, or using frameworks that extend general-purpose languages, such as TensorFlow Probability, Pyro, and Edward.

5 Bayesian optimization

Bayesian optimization is a technique that optimizes a black-box function that is expensive or noisy to evaluate. The idea is to use a Bayesian model, such as a Gaussian process, to capture the uncertainty and the structure of the function. Then, you can use an acquisition function, such as expected improvement, to select the next point to evaluate based on the trade-off between exploration and exploitation. Bayesian optimization can be used to tune the hyperparameters of machine learning models, such as learning rate, regularization, and architecture.

6 Bayesian neural networks

Bayesian neural networks are a technique that incorporate uncertainty and regularization into neural networks. The idea is to treat the weights and biases of the neural network as random variables with prior distributions, rather than fixed values. Then, you can use Bayesian inference to update the posterior distributions of the weights and biases based on the data. Bayesian neural networks can provide estimates of predictive uncertainty, avoid overfitting, and adapt to new data. Bayesian neural networks can be implemented using variational inference, HMC, or probabilistic programming.

7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?