We propose a general formalism of iterated random functions with semigroup property, under which exact and approximate Bayesian posterior updates can be viewed as specific instances. A convergence theory for iterated random functions is presented. As an application of the general theory we analyze convergence behaviors of exact and approximate message-passing algorithms that arise in a sequential change point detection problem formulated via a latent variable directed graphical model. The sequential inference algorithm and its supporting theory are illustrated by simulated examples.

Most existing literature on supervised machine learning assumes that the training dataset is drawn from an i.i.d. sample. However, many real-world problems exhibit temporal dependence and strong correlations between the marginal distributions of the data-generating process, suggesting that the i.i.d. assumption is often unrealistic. In such cases, models naturally include time-series processes with mixing properties, as well as irreducible and aperiodic ergodic Markov chains. Moreover, the learning rates typically obtained in these settings are independent of the data distribution, which can lead to restrictive choices of hypothesis classes and suboptimal sample complexities for the learning algorithm. In this article, we consider the case where the training dataset is generated by an iterated random function (i.e., an iteratively defined time-homogeneous Markov chain) that is not necessarily irreducible or aperiodic. Under the assumption that the governing function is contractive with respect to its first argument and subject to certain regularity conditions on the hypothesis class, we first establish a uniform convergence result for the corresponding sample error. We then demonstrate the learnability of the approximate empirical risk minimization algorithm and derive its learning rate bound. Both rates are data-distribution dependent, expressed in terms of the Rademacher complexities of the underlying hypothesis class, allowing them to more accurately reflect the properties of the data-generating distribution.

There has been much recent interest in two-sided markets and dynamics thereof. In a rather a general discrete-time feedback model, which we show conditions that assure that for each agent, there exists the limit of a long-run average allocation of a resource to the agent, which is independent of any initial conditions. We call this property the unique ergodicity. Our model encompasses two-sided markets and more complicated interconnections of workers and customers, such as in a supply chain. It allows for non-linearity of the response functions of market participants. Finally, it allows for uncertainty in the response of market participants by considering a set of the possible responses to either price or other signals and a measure to sample from these.

We propose a general formalism of iterated random functions with semigroup property, under which exact and approximate Bayesian posterior updates can be viewed as specific instances. A convergence theory for iterated random functions is presented. As an application of the general theory we analyze convergence behaviors of exact and approximate message-passing algorithms that arise in a sequential change point detection problem formulated via a latent variable directed graphical model. The sequential inference algorithm and its supporting theory are illustrated by simulated examples.

We propose a general formalism of iterated random functions with semigroup property, under which exact and approximate Bayesian posterior updates can be viewed as specific instances. A convergence theory for iterated random functions is presented. As an application of the general theory we analyze convergence behaviors of exact and approximate message-passing algorithms that arise in a sequential change point detection problem formulated via a latent variable directed graphical model. The sequential inference algorithm and its supporting theory are illustrated by simulated examples.

We propose a general formalism of iterated random functions with semigroup property, under which exact and approximate Bayesian posterior updates can be viewed as specific instances. A convergence theory for iterated random functions is presented. As an application of the general theory we analyze convergence behaviors of exact and approximate message-passing algorithms that arise in a sequential change point detection problem formulated via a latent variable directed graphical model. The sequential inference algorithm and its supporting theory are illustrated by simulated examples. Papers published at the Neural Information Processing Systems Conference.

We propose a general formalism of iterated random functions with semigroup property, under which exact and approximate Bayesian posterior updates can be viewed as specific instances. A convergence theory for iterated random functions is presented. As an application of the general theory we analyze convergence behaviors of exact and approximate message-passing algorithms that arise in a sequential change point detection problem formulated via a latent variable directed graphical model. The sequential inference algorithm and its supporting theory are illustrated by simulated examples.

The sequential posterior updates play a central role in many Bayesian inference procedures. As an example, in Bayesian inference one is interested in the posterior probability of variables of interest given the data observed sequentially up to a given time point. As a more specific example which provides the motivation for this work, in a sequential change point detection problem [1], the key quantity is the posterior probability that a change has occurred given the data observed up to present time. When the underlying probability model is complex, e.g., a large-scale graphical model, the calculation of such quantities in a fast and online manner is a formidable challenge. In such situations approximate inference methods are required - for graphical models, message-passing variational inference algorithms present a viable option [2, 3].