Bayesian operational modal analysis: Difference between revisions

Revision as of 07:57, 27 July 2020

Revision as of 07:59, 27 July 2020

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</ref> and Section 9.6 of <ref name="bayomabook" />. For example,

*Assuming a uniform prior, the most probable value (MPV) of parameters in a Bayesian method is equal to the ___location where the likelihood function is maximized, which is the estimate in Maximum Likelihood Method

*Under a Gaussian approximation of the posterior distribution of parameters, their covariance matrix is equal to the inverse of Hessian of the negative log of likelihood function at the MPV. Generally, this covariance depends on data. However, if one assumes (hypothetically; non-Bayesian) that the data is indeed distributed as the likelihood function, then for large data size it can be shown that the covariance matrix is asymptotically equal to the inverse of the [[Fisher Information Matrix]] Matrix (FIM) of parameters (which has a non-Bayesian origin). This coincides with the [[Cramer-Rao bound]] in classical statistics, which gives the lower bound (in the sense of matrix inequality) of the ensemble variance of any unbiased estimator. Such lower bound can be reached by maximum-likelihood estimator for large data size.

*In the above context, for large data size the asymptotic covariance matrix of modal parameters depends on the 'true' parameter values (a non-Bayesian concept), often in an implicit manner. It turns out that by applying further assumptions such as small damping and high signal-to-noise ratio, the covariance matrix has mathematically manageable asymptotic form, which provides insights on the achievable precision limit of OMA and can be used to guide ambient vibration test planning. This is collectively referred as 'uncertainty law'<ref name="ulaw2018"/>.