Bayesian Predictive Modeling: Towards Martingale Posterior Distributions for Dynamical Systems

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Bayesian inference is a principled way to quantify uncertainty over parameters. The predominant approach involves specifying a prior and a likelihood in order to compute a posterior distribution. Prediction is then achieved through computing the posterior predictive distribution. However, prediction can also be viewed as the primary task of Bayesian inference, in which specifying a predictive model comes first, and inferring the posterior distribution follows next. This approach is appealing in several ways, including that one reasons over quantities we can observe, as opposed to parameters that cannot be observed. In this talk I will introduce this Bayesian "predictive approach", and discuss a particular method called the martingale posterior distribution implemented by the predictive resampling algorithm. Next I will present preliminary work in which I show how the predictive resampling algorithm can be useful for posterior inference in the setting of non-i.i.d observations generated by a dynamical system.