## 4 Estimation of VF Material Properties
### Bayesian Inference  
The Bayesian framework for parameter inference seeks a joint probability distribution that represents the probability of all potential values of the parameters of interest, \(\chi\). Such a density is found through Bayes equation [32]  
\[\pi(\chi|\mathbf{y})=\frac{\pi(\mathbf{y}|\chi)\pi_{\rm pri}(\chi)}{\pi( \mathbf{y})}\propto\pi(\mathbf{y}|\chi)\pi_{\rm pri}(\chi), \tag{13}\]  
where \(\pi(\chi|\mathbf{y})\) is the posterior probability density function, which contains all probabilistic information about \(\chi\) given observed measurements \(\mathbf{y}\). The density \(\pi_{\rm pri}(\chi)\) is the \"prior\" probability density, \(\pi(\mathbf{y}|\chi)\) is the \"likelihood\", and \(\pi(\mathbf{y})\) is the \"evidence\". The prior contains known or expected statistical properties of the parameters based on all knowledge available prior to obtaining the measurements. For instance, if subglottal pressure is a model parameter to be inferred, it is known ahead of time that the value cannot be negative, and is likely within a specified bound. The likelihood quantifies the probability of an observed measurement occurring given fixed parameter values; that is, given a particular model with set parameters, what is the likelihood that the measured data would be observed. Lastly, the evidence is a normalization constant that ensures the Law of Total Probability is satisfied. In the present work, the importance sampling approach is used due to the computational complexity of the model [33]. Such approaches have been successfully used previously for the study of phonation [10; 11; 34]. The fundamental premise of importance sampling is that certain values of the inputs are more important to the parameter being estimated than others. So a greater weight, which we hereafter refer to as an importance weight, is allocated to those regions in the parameter space that exhibit a better fit to the measurements. In particular, random samples of \(\chi\) are drawn from some proposal distribution, and the resulting observations are simulated using the randomly drawn parameters. The likelihood distribution is then used to probabilistically quantify the goodness of fit to the measured data, from which an importance weight is allocated. When a sufficiently large number of random draws have been computed, a new ensemble is constructed by sampling from the random draws in proportion to their computed importance weight. This can be summarized as follows:  
* Initialization: generate an ensemble \(\{\tilde{\chi}^{(\ell)}\}_{\ell=1}^{N}\) of \(N\) random samples from the proposal distribution \(\pi_{0}(\chi)\). * Update: for each of the drawn samples, calculate the relative likelihood and normalize to get the importance weight for that sample \[w^{(\ell)}=\frac{1}{\overline{W}}\pi(\mathbf{y}|\tilde{\chi}^{(\ell)}),\ \ W=\sum_{\ell=1}^{N}\pi(\mathbf{y}|\tilde{\chi}^{(\ell)}).\] (14)
* Resample: generate another ensemble \(\{\chi^{(\ell)}\}_{\ell=1}^{N}\) by sampling each \(\tilde{\chi}^{\ell}\) with probability \(w^{\ell}\). If the proposal distribution is chosen to be the prior density, the final resampled ensemble converges in distribution to the posterior [35]. As a result, the resampled ensemble can then be used to compute sample-based point and spread estimates of the posterior. Herein, the sampling ensembles consist of 50,000 random draws from the prior distributions, which treats each parameter as independent. The sample mean of each ensemble is used to estimate the material properties and the sample standard deviation serves as an uncertainty estimate. The sensitivity of the results to the number of samples is considered in Section 5.3. The above algorithm defines importance weights in terms of the likelihood density \(\pi(\mathbf{y}|\tilde{\chi})\), which is determined by the specific error model [33].