## 2 Materials and Methods
### Inverse Problems by Markov Chain Monte Carlo  
For simplicity in this section, we describe the Metropolis-Hastings Markov chain Monte Carlo algorithm (MH-MCMC) assuming \(a\) is the scalar appearing in Section 3.2. The extension of the derivation to the general case of vector \(\mathbf{a}\) is straightforward. Suppose that the coefficient \(\mu_{a}\) is unknown and \(\mu_{a}\) depends on a parameter \(a\). We will reconstruct \(a\) within the framework of the Bayesian inversion algorithm [33; 34], i.e., we will find the probability distribution \(\pi(a|\mathbf{U})\) of \(a\) for measured data \(\mathbf{U}\). Let \(f_{\text{prior}}(a)\) be the prior probability density and \(\pi(\mathbf{U}|a)\) be the likelihood density or the conditional probability density of \(\mathbf{U}\) for \(a\). Then the joint probability density \(\pi(a,\mathbf{U})\) of \(a,\mathbf{U}\) is given by \(\pi(a,\mathbf{U})=\pi(\mathbf{U}|a)f_{\text{prior}}(a)\). According to the Bayes formula, the conditional probability density \(\pi(a|\mathbf{U})\) is given by  
\[\pi(a|\mathbf{U})=\frac{L(\mathbf{U}|a)f_{\text{prior}}(a)}{\int_{-\infty}^{ \infty}L(\mathbf{U}|a)f_{\text{prior}}(a)\,da}, \tag{24}\]  
where \(L(\mathbf{U}|a)\) is a function proportional to \(\pi(\mathbf{U}|a)\). Assuming Gaussian noise, we put  
\[L(\mathbf{U}|a)=e^{-\frac{1}{2\sigma_{2}^{2}}\|\mathbf{U}-\mathbf{F}(a)\|_{2}^ {2}}. \tag{25}\]  
In this paper, we simply set \(f_{\text{prior}}(a)=1\), i.e., we can say \(f_{\text{prior}}(a)\propto\mathbf{1}_{[a_{\text{min}},a_{\text{max}}]}(a)\) and the interval \([a_{\text{min}},a_{\text{max}}]\) is large enough so that all \(a\)'s appearing in the Markov chain fall into this interval. Here, \(\mathbf{1}_{A}(a)\) is the indicator function defined as \(\mathbf{1}_{A}(a)=1\) for \(a\in A\) and \(=0\) for \(a\
ot\in A\). General uniform distributions can be used for \(f_{\text{prior}}(a)\) if we use the prior-reversible proposal that satisfies \(f_{\text{prior}}(a)q(a^{\prime}|a)=f_{\text{prior}}(a^{\prime})q(a|a^{\prime})\) (see below for the proposal distribution \(q(a^{\prime}|a)\)) [35]. Another possible choice of \(f_{\text{prior}}(a)\) is a Gaussian distribution, which turns out to be the Tikhonov regularization term in the cost function. Using the Metropolis-Hastings algorithm, we can evaluate \(\pi(a|\mathbf{U})\) even when the normalization factor is not known and only the relation \(\pi(a|\mathbf{U})\propto L(\mathbf{U}|a)f_{\mathrm{prior}}(a)\) is available [33; 34]. We can find \(\pi(a|\mathbf{U})\) using a sequence \(p_{1},p_{2},\dots\) as  
\[\pi(a|\mathbf{U})=\lim_{k\to\infty}p_{k}(a), \tag{26}\]  
where \(p_{k+1}(a)\) is obtained from \(p_{k}(a)\) (Markov chain) as  
\[p_{k+1}(a^{\prime})=\int_{-\infty}^{\infty}K(a^{\prime},a)\,p_{k}(a)\,da. \tag{27}\]  
For all \(a,a^{\prime}\), the transition kernel satisfies  
\[K(a^{\prime},a)\geq 0,\qquad\int_{-\infty}^{\infty}K(a^{\prime},a)\,da^{\prime }=1. \tag{28}\]  
Let us write \(K(a^{\prime},a)\) as  
\[K(a^{\prime},a)=\alpha(a^{\prime},a)q(a^{\prime}|a)+r(a)\delta(a^{\prime}-a), \tag{29}\]  
where \(q(a^{\prime}|a)\) is the proposal distribution and  
\[r(a)=1-\int_{-\infty}^{\infty}\alpha(a^{\prime},a)q(a^{\prime}|a)\,da^{\prime}. \tag{30}\]  
In the Metropolis-Hastings algorithm we set \(\alpha(a^{\prime},a)=\min\left\{1,\pi(a^{\prime}|\mathbf{U})q(a|a^{\prime})/ [\pi(a|\mathbf{U})q(a^{\prime}|a)]\right\}\). With this choice of \(\alpha(a^{\prime},a)\), the detailed balance is satisfied and \(K(a,a^{\prime})\pi(a^{\prime}|\mathbf{U})=K(a^{\prime},a)\pi(a|\mathbf{U})\). A common choice of \(q(a^{\prime}|a)\) is the normal distribution, i.e., \(q(\cdot|a)=\mathcal{N}(a,\varepsilon^{2})\) with the mean \(a\) and the standard deviation \(\varepsilon>0\). We note that \(q(a^{\prime}|a)=q(a|a^{\prime})\) in this case. We have  
\[\int_{-\infty}^{\infty}h(a)\pi(a|\mathbf{U})\,da=\lim_{k_{\max}\to\infty}\frac {1}{k_{\max}}\sum_{k=1}^{k_{\max}}h(a_{k}), \tag{31}\]  
where \(a_{k}\sim p_{k}(\cdot)\) and \(h\) is an arbitrary continuous bounded function.