外文翻译(机械)(4)

measured mean force values given specified values of the force coefficients. The posterior (i.e., the new belief after updating) is proportional to the prior multiplied by the likelihood. For multiple measurements, Bayes’ rule can incorporate all data in a single calculation. The likelihood functions for each measurement are multiplied together to obtain a total likelihood function. The posterior pdf is calculated by multiplying the prior and the total likelihood function. Note that the posterior distributions must be normalized so that a unit volume under the pdf is obtained; this is the purpose of the denominator in Eq. (12).

Application to Bayesian Inference This section describes the application of MCMC to Bayesian inference. As stated in Sec. 2, Bayesian inference provides a formal way to update beliefs about the posterior distribution (the normalized product of the prior and the likelihood functions) using experimental results. In the case of updating force coefficients (Eq. (13)), the prior is a joint pdf of the force coefficients, Kt, Kn, Kte, and Kne. As a result, the posterior is also a joint pdf of the force coefficients. In Bayesian inference, the MCMC technique can be used to sample from multivariate posterior distributions. The single-component MH algorithm facilitates sampling from multivariate distributions without sensitivity to the number of variables [7]. The joint posterior pdf is the target pdf for MCMC. The posterior, or target, pdf is the product of the prior and likelihood density functions. Note that the normalizing constant of the posterior pdf is not required for sampling. Effect of the Prior Selection

In this section, the effect of the prior on the posterior distribution of force coefficients is studied. For the numerical results presented in Sec. 4.1, a uniform prior was selected. A uniform prior represents a noninformative case, where any coefficient