Uncertainty Quantification of Cardiac Properties | 83 w(θ) ∝ p(θ z) q(θ) . (2) The weights are normalized such that ∑θ∈Θ w(θ) =1 . Importance sampling is most effective when the proposal distribution q(θ) is close to the posterior distribution p(θ z) such that variance in weight of the samples is small and the effective sample size is close to the actual sample size. Since no information was available on the posterior distribution, we used adaptive importance sampling in which the proposal distribution is iteratively updated to better describe the posterior distribution.11 The computational cost of calculating the likelihood p(z θ) in cardiovascular models is relatively high compared to the cost of calculating the probability density function of the proposal distribution q(θ), so the samples from all previous iterations were included in defining the proposal distribution q(θ) to optimally recycle past simulations following the adaptive multiple importance sampling (AMIS) (see Figure 2).10 Figure 2. Visualization of adaptive multiple importance sampling In the first iteration, samples θ are drawn from a uniform distribution and stored in the sample set Θ. For each sample, the corresponding sample weight w is calculated. Then, based on all previous samples θ in the sample set Θ and corresponding sample weight w, the next proposal distribution is defined and new samples are added to the sample set Θ. This iterates niter times. Each iteration in this algorithm consists of two stages. First, samples are drawn from the proposal distribution and weights of all samples are updated. Second, the proposal distribution is updated based on the new sample weights. 5
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