84 | Chapter 5 Draw samples and calculate sample weights At the start of each iteration πi(θ) , 100 samples are drawn from the current proposal distribution πi(θ). Samples are drawn without statistical dependencies between parameters, which may result in non-physiological combinations of parameters. For example, the model is not parameterized for a low contractile heart to be able to supply a high cardiac output (CO) and is therefore likely to become numerically instable. To circumvent this, only a small uniform distribution around the reference is used as initial proposal distribution q0(θ). AMIS will increase and decrease the search area of the proposal distribution and will move this to the area of interest in which physiological samples will be drawn close to the desired posterior distribution. Each iteration, the weights are updated based on the proposal function and likelihood (Equation 2). The probability density function of all previous proposal distributions is given by the sum of all individual proposal distributions qi(θ) = 1 Nsamples niter−1 ∑ i=0 nsamples, i ⋅ πi(θ), (3) with nsamples, i the number of samples in iteration nsamples, i and Nsamples = niter−1 ∑ i=0 nsamples, i the total number of samples. Samples drawn from poorly performing proposal distributions are eliminated through the erosion of their low weights.10 The likelihood function is defined based on the normalized dimensionless summed squared error Χ(θ)2. This Χ(θ)2 is problem dependent and the Χ2 used in this study is described in the section ‘Problem description’. We assumed a non-informative uniform prior and neglected all interactions between individual errors. Furthermore, annealed adaptive importance sampling12 was used to prevent the algorithm from premature convergence13,14, resulting in a likelihood p(z θ, Ti) ∝e −Χ(θ) 2 T i , (4) in which Ti ≥1 ≥ 1 in each iteration nsamples, i and represents the annealing temperature. This method is included to control convergence rate, thereby improving global search capabilities and limiting premature convergence towards local minima. The initial temperature is set to Tmax =10, and decreases each iteration nsamples, i such that Ti+1 = min(10, Ti +ΔΧ2 opt) if Χ2 is improved max(1, 0 .8⋅ Ti) else (5) with ΔΧ2 opt the difference between the old and new Χ2 of the best sample. Update proposal distribution Each iteration, the proposal distribution is updated based on all drawn samples in the sample set Θ and its corresponding weight w. In the updated proposal distributions, samples were drawn along the principal component axes of the weighted sample set Θ.
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