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Bayesian neural network

nounAlso: BNN, Bayesian deep learning, Bayes by Backprop

In plain wordsA neural network that keeps a whole range of plausible settings for its internal numbers instead of one best guess, so that it can say how unsure it is about each prediction.

  1. A neural networkneural network, n.: A kind of computer program that learns a task from examples by tuning thousands to billions of connections between simple number-crunching units, an arrangement loosely inspired by nerve cells in the brain. whose weights are treated as random variables with a priorprior, n.: What you believe, and how strongly, before looking at the evidence in front of you. In AI it also means built-in assumptions that steer what a system learns. distribution, so that learning means computing, or approximating, a posteriorposterior, n.: What you believe after weighing new evidence against what you believed before. It is the updated belief that Bayes' theorem produces. over weights given the data, and prediction averages the outputs of all weight settings in proportion to their posterior probability. The spread of those outputs estimates epistemic uncertainty, the uncertainty due to limited data, which supports calibrationcalibration, n.: How well a model's stated confidence matches reality: when a well-calibrated system says it is 80% sure, it turns out to be right about 80% of the time., out-of-distribution detection, active learning and cautious decisions. The exact posterior is intractable for realistic networks, so practice relies on approximations: the Laplace approximation around a trained solution, Markov chain Monte CarloMarkov chain Monte Carlo, n.: A family of methods for drawing random examples from a complicated probability distribution by taking a long, carefully designed random walk that eventually visits each region in proportion to its probability. such as Hamiltonian Monte Carlo, variational inferencevariational inference, n.: A way of approximating a hard-to-compute probability distribution by picking the closest match from a simpler family of distributions, turning a hard problem of adding up every possibility into an optimization problem. over weights (Bayes by Backprop), Monte Carlo dropoutdropout, n.: A training trick that randomly switches off a fraction of a network's neurons at each step, so the network cannot lean too heavily on any one of them and learns sturdier patterns. and weight-averaging methods; deep ensembles are often used as a strong, partly Bayesian substitute. Computational cost, choice of prior and the 'cold posterior' effect, in which artificially sharpening the posterior improves accuracy, remain open problems.

    The Bayesian neural network flagged scans from the new hospital as high-uncertainty, so they went to a radiologist.

p(y∗∣x∗,D)=∫p(y∗∣x∗,w) p(w∣D) dwp(y^{*} \mid x^{*}, \mathcal{D}) = \int p(y^{*} \mid x^{*}, \mathbf{w})\, p(\mathbf{w} \mid \mathcal{D})\, d\mathbf{w}

Origin

Early probabilistic treatments of network weights include Naftali Tishby, Esther Levin and Sara Solla (1989) and John Denker and Yann LeCun (1991). David MacKay's 1992 evidence framework, using the Laplace approximation, and Radford Neal's 1995 thesis, using Hamiltonian Monte Carlo and showing that infinitely wide networks become Gaussian processesGaussian process, n.: A way of predicting an unknown curve or surface that gives not just a best guess at each point but an honest error bar, narrow near measured points and wide far from them., established the field. Variational weight uncertainty (Charles Blundell et al., 2015) and dropout as approximate inference (Yarin Gal and Zoubin Ghahramani, 2016) revived it for deep learning.

Usage note

Not to be confused with a Bayesian networkBayesian network, n.: A diagram of causes and effects with probabilities attached — say, how rain and a sprinkler can each make grass wet — that lets a computer update its beliefs about hidden causes when it sees evidence., a graphical model of dependencies among variables that need not involve neural networks. Whether approximate BNNs beat cheaper uncertainty methods in practice is debated.

Related

See also
  • Bayesian inferenceBayesian inference, n.: Reasoning that treats unknowns as uncertain beliefs and updates them with evidence using Bayes' theorem. The result is not one answer but a range of answers, each with a probability.
  • uncertainty quantificationuncertainty quantification, n.: Measuring how unsure an AI system's predictions are, and why, so that people know when to trust an answer and when to double-check it.
  • variational inferencevariational inference, n.: A way of approximating a hard-to-compute probability distribution by picking the closest match from a simpler family of distributions, turning a hard problem of adding up every possibility into an optimization problem.
  • Markov chain Monte CarloMarkov chain Monte Carlo, n.: A family of methods for drawing random examples from a complicated probability distribution by taking a long, carefully designed random walk that eventually visits each region in proportion to its probability.
  • Gaussian processGaussian process, n.: A way of predicting an unknown curve or surface that gives not just a best guess at each point but an honest error bar, narrow near measured points and wide far from them.
  • dropoutdropout, n.: A training trick that randomly switches off a fraction of a network's neurons at each step, so the network cannot lean too heavily on any one of them and learns sturdier patterns.
  • ensemble learningensemble learning, n.: Combining the predictions of several models — by voting or averaging — so their individual mistakes partly cancel out, giving more accurate and stable results than any one model.
  • calibrationcalibration, n.: How well a model's stated confidence matches reality: when a well-calibrated system says it is 80% sure, it turns out to be right about 80% of the time.
Contrast with
  • Bayesian networkBayesian network, n.: A diagram of causes and effects with probabilities attached — say, how rain and a sprinkler can each make grass wet — that lets a computer update its beliefs about hidden causes when it sees evidence.

Sources

  1. MacKay, D. J. C. (1992). A Practical Bayesian Framework for Backpropagation Networks. Neural Computation 4(3), 448–472.
  2. Neal, R. M. (1996). Bayesian Learning for Neural Networks. Lecture Notes in Statistics 118. Springer.
  3. Blundell, C., Cornebise, J., Kavukcuoglu, K. & Wierstra, D. (2015). Weight Uncertainty in Neural Networks. Proc. ICML 2015.

Mentioned in

Bayesian networkBayesian network, n.: A diagram of causes and effects with probabilities attached — say, how rain and a sprinkler can each make grass wet — that lets a computer update its beliefs about hidden causes when it sees evidence.