Bayesian Model Class Selection

Keywords asymptotic expansion evidence information entropy Markov Chain Monte Carlo simulation modal identification Ockham factor regression problem robustness seismic attenuation 

Bayesian Methods for Structural Dynamics and Civil Engineering Ka-Veng Yuen 2010 John Wiley Sons (Asia) Pte Ltd 

Bayesian Methods for Structural Dynamics and Civil Engineering 

In recent years, there has been a re-appreciation of the work of Jeffreys on the application of Bayesian methods [121], especially due to the expository publications of E.T. Jaynes [118,120], In particular, the Bayesian approach to model class selection has been further developed by showing that the evidence for each model class provided by the data (that is, the probability of getting the data based on the whole model class) automatically enforces a quantitative expression of a principle of model parsimony or of Ockham s razor [98,164,242], There is no need to introduce any ad-hoc penalty term as was done in some of the earlier work on this problem. 

Influenced by the mind of forward modeling problems, it is easily directed to adopt complicated model classes so as to capture various complex physical mechanisms. However, the more complicated the model class is utilized, the more uncertain parameters are normally induced unless extra mathematical constraints are imposed. In the former case, the model output may not necessarily be accurate even if the model well characterizes the physical system since the combination of the many small errors from each uncertain parameter can induce a large output error. In the latter case, it is possible that the extra constraints induce substantial errors. Therefore, it is important to use a proper model class for system identification purpose. In this chapter, the Bayesian model class selection approach is introduced and applied to select the most plausible/suitable class of mathematical models representing a static or dynamical (structural, mechanical, atmospheric.) system (from some specified model classes) by using its response measurements. This approach has been shown to be promising in several research areas, such as artificial neural networks [164,297], structural dynamics and model updating [23], damage detection [150] and fracture mechanics [151], etc. 

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In the next section, the Bayesian model class selection method is introduced for quantification and selection of model classes. It will be discussed for the globally identifiable case and the general case. The Ockham factor is introduced and it serves as the penalty for a complicated model, which appears naturally from the evidence. Computational issues will be discussed and... 

In Chapter 2, Section 2.4, parametric identification was introduced for linear and nonlinear regression problems. In this section, the Bayesian model class selection is applied to these problems. In order to smooth the presentation, some of the equations from Section 2.4 are repeated in this section. 

In order to balance the data fitting capability and robustness, a relatively simple model class is chosen by the Bayesian model class selection approach and its optimal model is given by ... 

Bayesian model class selection (or model comparison) is essentially Bayesian updating at the model class level to make comparisons between alternative candidate model classes for predicting the response of a system. It has long been recognized that comparisons between model classes should factor in not only the quality of the data fit, but also the complexity of the model. Jeffreys referred to the need for a simplicity postulate, that is, simpler models that are consistent with the data should be preferred over more complex models which offer only slight improvements in the fit to the data... 

Table 4 Bayesian model class selection results. ...

In section Structural Parametric Identification by Extended Kalman Filter, online structural parametric identification using the EKF will be briefly reviewed. In section Online Identification of Noise Parameters, an online identification algorithm for the noise parameters in the EKF is introduced. Then, in section Outlier-Resistant Extended Kalman Filter, an online outlier detection algorithm is presented, and it is embedded into the EKF. This algorithm allows for robust structural identification in the presence of possible outliers. In section Online Bayesian Model Class Selection, a recursive Bayesian model class section method is presented for non-parametric identification problems. 

Bayesian model class selection is utilized for selecting the most plausible model class from a set of Nc dynamic model class candidates C, C2, , Cvc by considering their plausibility P Cj D) conditional rai the available set of dynamic measurement D (Beck and Yuen 2004 Yuen 2010b) ... 

Next, an online Bayesian model class selection algorithm is introduced. It was first developed to model the transportation system of particulate matters (Hoi et al. 2011). The plausibility of model class C, conditional on the measured data up to the (k + )th time step D +i = zi,. .., can be rewritten into the following form... 

In order to avoid incorrect or unsuitable assumptions and thereby influencing the Bayesian updating results in an unfounded way, one can make use of the available observed data to try and estimate (characteristics of) the prediction error, for instance, using Bayesian model class selection. 

Bayesian inference can be applied at model class level to assess the plausibility of several alternative model classes based on the available observations d this is referred to as Bayesian model class selection or MCS (Beck and Yuen 2004 Yuen 2010). The set of alternative model classes commonly concern (mechanical) prediction model classes Mm-, but here the method will be used to distinguish between alternative probabi-hstic prediction error models Mjy The following, however, is elaborated for a set of general model classes Mi. 

The term p A Mi) in the numerator on the right-hand side of Eq. 9 is the evidence (sometimes also referred to as the model class likelihood) for the model class Al, provided by the data d. The evidence, hereafter denoted with e, is a very important quantity in Bayesian model class selection and can be determined based on the law of total probability as... 

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