82 | Chapter 5 Figure 1. Non-invasive measurements were used as input for a fully automatic automated uncertainty quantification algorithm This algorithm produced a digital twin based on estimated parameters with accompanying uncertainty. This digital twin can be used to get more insight in the estimated tissue properties. RVfw: right ventricle free wall; LVfw: left ventricle free wall; IVS: inter ventricular septum; HR: heart rate; EDV: end-diastolic volume; EF: ejection fraction; RVD: right ventricular diameter. Mathematical basis of adaptive multiple importance sampling We consider an nθ- dimensional vector as a set of parameters θ of a numerical model z =ℳ(θ). This model ℳ:ℛnθ → ℛnz maps the parameter vector to an nz-dimensional vector of modelled data z. Measurement uncertainties are included in the likelihood function p(z θ) representing the similarity between patient observation and model output. The posterior distribution p(θ z) is the probability of having parameters θ given the observation z and is given by Bayes’ rule as p(θ z) = p(z θ)p(θ) p(z) ∝p(z θ)p(θ), (1) with p(θ) the prior knowledge of the parameters and p(z) the normalizing constant. No prior knowledge of the parameters p(θ) is known, so p(θ) was assumed to be uniform. Importance sampling is an algorithm which estimates the posterior distribution p(θ z) . 11 The set of samples Θ={θ~q(θ)} drawn from the proposal distribution q(θ) form an empirical estimation of the posterior distribution p(θ z) in which each sample is weighted with the sample weight w described by
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