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In real applications, the construction of prior and acceleration of samplingfor posterior are usually two key points of Bayesian inversion algorithm forengineers. In this paper, q-analogy of Gaussian distribution, q-Gaussiandistribution, is introduced as the prior of inverse problems. And anacceleration algorithm based on spectral likelihood approximation is discussed.We mainly focus on the convergence of the posterior distribution in the senseof Kullback-Leibler divergence when approximated likelihood function andtruncated prior distribution are used. Moreover, the convergence in the senseof total variation and Hellinger metric is obtained. In the end two numericalexamples are displayed.

Q-Hermite polynomials chaos approximation of likelihood function based on q-Gaussian prior in Bayesian inversion