## the Bayesian learning rule [One World ABC’minar, 27 April]

Posted in Books, Statistics, University life with tags , , , , , , , , , , , on April 24, 2023 by xi'an

The next One World ABC seminar is taking place (on-line, requiring pre-registration) on 27 April, 9:30am UK time, with Mohammad Emtiyaz Khan (RIKEN-AIP, Tokyo) speaking about the Bayesian learning rule:

We show that many machine-learning algorithms are specific instances of a single algorithm called the Bayesian learning rule. The rule, derived from Bayesian principles, yields a wide-range of algorithms from fields such as optimization, deep learning, and graphical models. This includes classical algorithms such as ridge regression, Newton’s method, and Kalman filter, as well as modern deep-learning algorithms such as stochastic-gradient descent, RMSprop, and Dropout. The key idea in deriving such algorithms is to approximate the posterior using candidate distributions estimated by using natural gradients. Different candidate distributions result in different algorithms and further approximations to natural gradients give rise to variants of those algorithms. Our work not only unifies, generalizes, and improves existing algorithms, but also helps us design new ones.

## ABConic mean evidence approximation

Posted in Statistics with tags , , , , , on March 7, 2023 by xi'an

Following a question on X validated about evidence approximation in ABC settings, i.e., on returning an approximation of the evidence based on the outputs of parallel ABC runs for the models under comparison, I wondered at the relevance of an harmonic mean estimator in that context.

Rather than using the original ABC algorithm that proposes a model, a parameter from that model, and a simulated dataset from that model with that parameter,  an alternate, cost-free, solution would be to run an ABC version of [harmonic mean evidence approximation à la Newton & Raftery (1994). Since

$\mathcal Z=1\Big/\int \dfrac{\pi(\theta|D)}{p(D|\theta)}\,\text d\theta$

the evidence can formally be approximated by

$\hat{\mathcal Z} =1\Big/\frac{1}{N}\sum_{i=1}^N\frac{1}{p(D|\theta_i)}\qquad\theta_i\sim\pi(\theta|D)$

and its ABC version is

$\hat{\mathcal Z} =1\Big/\frac{1}{N}\sum_{i=1}^N\frac{1}{K_\epsilon(d(D,D^\star(\theta_i)))}\qquad\theta_i\sim\pi^\epsilon(\theta|D)$

where Kε(.) is the kernel used for the ABC acceptance/rejection step and d(.,.) is the distance used to measure the discrepancy between samples. Since the kernel values are already computed for evidence, the cost is null. Obviously, an indicator kernel does not return a useful estimate but something like a Cauchy kernel could do.

However, when toying with a normal-normal model and calibrating the Cauchy scale to fit the actual posterior as in the above graph, the estimated evidence 5 10⁻⁵ proved much smaller than the actual one, 8 10⁻².

## ABC with path signatures [One World ABC seminar, 2/2/23]

Posted in Books, pictures, Running, Statistics, Travel, University life with tags , , , , , , , on January 29, 2023 by xi'an

The next One World ABC seminar is by Joel Dyer (Oxford) at 1:30pm (UK time) on 02 February.

Title: Approximate Bayesian Computation with Path Signatures

Abstract: Simulation models often lack tractable likelihood functions, making likelihood-free inference methods indispensable. Approximate Bayesian computation (ABC) generates likelihood-free posterior samples by comparing simulated and observed data through some distance measure, but existing approaches are often poorly suited to time series simulators, for example due to an independent and identically distributed data assumption. In this talk, we will discuss our work on the use of path signatures in ABC as a means to handling the sequential nature of time series data of different kinds. We will begin by discussing popular approaches to ABC and how they may be extended to time series simulators. We will then introduce path signatures, and discuss how signatures naturally lead to two instances of ABC for time series simulators. Finally, we will demonstrate that the resulting signature-based ABC procedures can produce competitive Bayesian parameter inference for simulators generating univariate, multivariate, irregularly spaced, and even non-Euclidean sequences.

Reference: J. Dyer, P. Cannon, S. M Schmon (2022). Approximate Bayesian Computation with Path Signatures. arXiv preprint 2106.12555

## Adversarial Bayesian Simulation [One World ABC’minar]

Posted in Statistics with tags , , , , , , , , , on November 15, 2022 by xi'an

The next One World ABC webinar will take place on 24 November, at 1:30 UK Time (GMT) and will be presented by Yi Yuexi Wang (University of Chicago) on “Adversarial Bayesian Simulation”, available on arXiv. [The link to the webinar is available to those who have registered.]

In the absence of explicit or tractable likelihoods, Bayesians often resort to approximate Bayesian computation (ABC) for inference. In this talk, we will cover two summary-free ABC approaches, both inspired by adversarial learning. The first one adopts a classification-based KL estimator to quantify the discrepancy between real and simulated datasets. We consider the traditional accept/reject kernel as well as an exponential weighting scheme which does not require the ABC acceptance threshold. In the second paper, we develop a Bayesian GAN (B-GAN) sampler that directly targets the posterior by solving an adversarial optimization problem. B-GAN is driven by a deterministic mapping learned on the ABC reference by conditional GANs. Once the mapping has been trained, iid posterior samples are obtained by filtering noise at a negligible additional cost. We propose two post-processing local refinements using (1) data-driven proposals with importance reweighting, and (2) variational Bayes. For both methods, we support our findings with frequentist-Bayesian theoretical results and highly competitive performance in empirical analysis. (Joint work with Veronika Rockova)

## Frontiers in Machine Learning and Economics: Methods and Applications

Posted in Statistics with tags , , , , , , , , on October 7, 2022 by xi'an