We propose a unifying framework for methods that perform Bayesian online learning in non-stationary environments. We call the framework BONE, which stands for (B)ayesian (O)nline learning in (N)on-stationary (E)nvironments. BONE provides a common structure to tackle a variety of problems, including online continual learning, prequential forecasting, and contextual bandits. The framework requires specifying three modelling choices: (i) a model for measurements (e.g., a neural network), (ii) an auxiliary process to model non-stationarity (e.g., the time since the last changepoint), and (iii) a conditional prior over model parameters (e.g., a multivariate Gaussian). The framework also requires two algorithmic choices, which we use to carry out approximate inference under this framework: (i) an algorithm to estimate beliefs (posterior distribution) about the model parameters given the auxiliary variable, and (ii) an algorithm to estimate beliefs about the auxiliary variable. We show how this modularity allows us to write many different existing methods as instances of BONE; we also use this framework to propose a new method. We then experimentally compare existing methods with our proposed new method on several datasets; we provide insights into the situations that make one method more suitable than another for a given task.

We derive a novel, provably robust, and closed-form Bayesian update rule for online filtering in state-space models in the presence of outliers and misspecified measurement models. Our method combines generalised Bayesian inference with filtering methods such as the extended and ensemble Kalman filter. We use the former to show robustness and the latter to ensure computational efficiency in the case of nonlinear models. Our method matches or outperforms other robust filtering methods (such as those based on variational Bayes) at a much lower computational cost. We show this empirically on a range of filtering problems with outlier measurements, such as object tracking, state estimation in high-dimensional chaotic systems, and online learning of neural networks.

In foreign exchange (FX), as in other asset classes, broker-client relationships are ubiquitous. The broker streams bid and ask quotes to her clients and the clients decide when to trade on these quotes, so the broker bears the risk of adverse selection when trading with better informed clients. These risks are borne by both liquidity providers who stream quotes to individual parties and by market participants who provide liquidity in the books of electronic exchanges. However, in contrast to electronic order books in which trading is anonymous for all participants (e.g., in Nasdaq, LSE, Euronext), in broker-client relationships the broker knows which client executed the order. This privileged information can be used by the broker to classify flow, i.e., toxic or benign, and to devise strategies that mitigate adverse selection costs. In the literature, models generally classify traders as informed or uninformed; see e.g., Bagehot (1971), Copeland and Galai (1983), Grossman and Stiglitz (1980), Amihud and Mendelson (1980), Kyle (1989), Kyle (1985), and Glosten and Milgrom (1985). In equity markets, many studies focus on informed flow (i.e., asymmetry of information) across various traded stocks, see e.g., Easley et al. (1996) who study the probability of informed trading at the stock level, while our study focuses on We thank Andrew Stewart, Alistair Sturgiss, Fayçal Drissi, Patrick Chang, Álvaro Arroyo, Sergio Calvo Ordoñez, and participants at the Oxford Victoria Seminar for comments. ChatGPT suggested the name PULSE for our algorithm.