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I am interested in clinical pharmacokinetics, where we have a given medicine's population model, with its parameters (mean and standard deviation), and our goal is take one or two blood samples from a patient and find the medicine blood concentrations. With that, we try to find individual's parameters, and we adjust the posology of the medicine for that patient.

To do so, we use nonlinear regression, finding the parameter set that minimize a penalty function. Which in fact is a MAP estimator, derived from Bayes' theorem, assuming normal distribution of the prior and posterior this is written in the following way (Sheiner et al):

$ \hat{\theta}_{MAP} = min (\sum_{i=1}^{n}\frac{(Y_{i}-f(\hat\theta|x_{i}))^2}{\sigma_{i}^2} + \sum_{j=1}^{p}\frac{(\overline\theta_{j}-\hat\theta_{j})^2}{\omega_{p}^2}) $

where $\sigma^2$ is the variance of the observation, and $\omega^2$ is the variance of the parameter.

As we only have 1-3 blood samples, and there are fewer than parameters on the model, I am struggling to find a way of computing the uncertainty of the parameters. Because I can not use Bootstrap nor MCMC. I have googled a lot, but all I could find is for calculate populations models, but in this case is more easier, because we would have a lot of observations.

How can I compute the uncertainty of the individual parameters?

Thank you.

Raúl

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