An overview of robust Bayesian analysis

a b s t r a c tMost of the currently used techniques for linear system identification are based on classical estimation paradigms coming from mathematical statistics. In particular, maximum likelihood and prediction error methods represent the mainstream approaches to identification of linear dynamic systems, with a long history of theoretical and algorithmic contributions. Parallel to this, in the machine learning community alternative techniques have been developed. Until recently, there has been little contact between these two worlds. The first aim of this survey is to make accessible to the control community the key mathematical tools and concepts as well as the computational aspects underpinning these learning techniques. In particular, we focus on kernel-based regularization and its connections with reproducing kernel Hilbert spaces and Bayesian estimation of Gaussian processes. The second aim is to demonstrate that learning techniques tailored to the specific features of dynamic systems may outperform conventional parametric approaches for identification of stable linear systems.

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“…This approach is deeply connected with robust Bayesian estimation concepts, e.g. see Berger (1980Berger ( , 1994.…”

Section: Regularized Least Squares and James-stein Estimators ⋆mentioning

confidence: 99%

Kernel methods in system identification, machine learning and function estimation: A survey

et al. 2014

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“…So, when π(θ) is in G U S it seems that alternative values of θ (to θ 0 ) are not favored according to the point null hypothesis being studied. Some other justifications for using this class of priors can be found in Berger (1994), Casella and Berger (1987), Berger and Sellke (1987) and Gómez-Villegas and Sanz (1998).…”

Section: Preliminariesmentioning

confidence: 99%

Asymptotic relationships between posterior probabilities and p-values using the hazard rate

Gómez–Villegas

Maı́n

Sanz

et al. 2004

Statistics & Probability Letters

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“…In fact there are two distinct approaches of this kind, which have been compared by Walley [2]. The first approach, known as Bayesian sensitivity analysis or Bayesian robustness [3,4], assumes that there is an ideal prior distribution π 0 which could, ideally, model prior uncertainty. It is assumed that we are unable to determine π 0 accurately because of limited time or resources.…”

Section: Introductionmentioning

confidence: 99%

A model of prior ignorance for inferences in the one-parameter exponential family

Benavoli

Zaffalon

2012

Journal of Statistical Planning and Inference

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This paper proposes a model of prior ignorance about a scalar variable based on a set of distributions M. In particular, a set of minimal properties that a set M of distributions should satisfy to be a model of prior ignorance without producing vacuous inferences is defined. In the case the likelihood model corresponds to a one-parameter exponential family of distributions, it is shown that the above minimal properties are equivalent to a special choice of the domains for the parameters of the conjugate exponential prior. This makes it possible to define the largest (that is, the least-committal) set of conjugate priors M that satisfies the above properties. The obtained set M is a model of prior ignorance with respect to the functions (queries) that are commonly used for statistical inferences; it is easy to elicit and, because of conjugacy, tractable; it encompasses frequentist and the so-called objective Bayesian inferences with improper priors. An application of the model to a problem of inference with count data is presented.

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An overview of robust Bayesian analysis

Cited by 493 publications

References 108 publications

Kernel methods in system identification, machine learning and function estimation: A survey

Kernel methods in system identification, machine learning and function estimation: A survey

Asymptotic relationships between posterior probabilities and p-values using the hazard rate

A model of prior ignorance for inferences in the one-parameter exponential family

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