#StackBounty: #bayesian-optimization #map-estimation Bayesian parameter estimation with unit observations

Bounty: 50

I am trying to do a MAP estimation of the model below, which comes from the literature in non-Bayesian form ($y = normalcdf(frac{1}{alpha} * log(A/beta)$). Because the model function has a normalcdf, the outcome of the model is in the range $0-1$. My observations are bound to the same range. However, using a Gaussian distribution for the likelihood means that the uncertainty bounds are sometimes outside the $0-1$ range (see Figure below). Statistically speaking, is this a problem, and if so, what would be a better solution?

$R_i sim mathcal{N}(mu,sigma)$

$mu_i = normalcdf(frac{1}{alpha} * log(A/beta) )$

$alpha sim mathcal{N}(0.35, 0.1) $

$betasim mathcal{N}(70, 10) $

$sigma sim mathcal{U}(0, 0.3) $

Bayesian parameter estimation

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