We present a Bayesian framework based on a new exponential likelihood function driven by the quadratic Wasserstien metric. Compared to conventional Bayesian models based on Gaussian likelihood functions driven by the least-squares norm ($L_2$ norm), the new framework features several advantages. First, the new framework does not rely on the likelihood of the measurement noise and hence can treat complicated noise structures such as combined additive and multiplicative noise. Secondly, unlike the normal likelihood function, the Wasserstein-based exponential likelihood function does not usually generate multiple local extrema. As a result, the new framework features better convergence to correct posteriors when a Markov Chain Monte Carlo sampling algorithm is employed. Thirdly, in the particular case of signal processing problems, while a normal likelihood function measures only the amplitude differences between the observed and simulated signals, the new likelihood function can capture both the amplitude and the phase differences. We apply the new framework to a class of signal processing problems, that is, the inverse uncertainty quantification of waveforms, and demonstrate its advantages compared to Bayesian models with normal likelihood functions.

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Wasserstein 度量驱动的贝叶斯反演在信号处理中的应用

我们提出了一个基于由二次 Wasserstien 度量驱动的新指数似然函数的贝叶斯框架。与基于由最小二乘范数（$L_2$ 范数）驱动的高斯似然函数的传统贝叶斯模型相比，新框架具有几个优点。首先，新框架不依赖于测量噪声的可能性，因此可以处理复杂的噪声结构，例如组合加性和乘性噪声。其次，与正常似然函数不同，基于 Wasserstein 的指数似然函数通常不会产生多个局部极值。因此，当采用马尔可夫链蒙特卡罗采样算法时，新框架具有更好的收敛性以校正后验。第三，在信号处理问题的特殊情况下，正常似然函数仅测量观测信号和模拟信号之间的幅度差异，而新的似然函数可以同时捕获幅度和相位差异。我们将新框架应用于一类信号处理问题，即波形的逆不确定性量化，并展示了与具有正态似然函数的贝叶斯模型相比的优势。