Sketches on how the treatment of nuisance parameters can cause systematic errors on the target parameter posteriors. For both panels, we assume a 2D space where the parameter θ = (θ_Target, θ_Nuisance). Left: bias due to underfitting a nuisance effect. In a 2D model, let 0 be the real value of the parameter θ_Target, despite the true value of θ_Nuisance. When θ_Nuisance is a free parameter of the 2D Gaussian model presented in the panel, the true value of the target parameter is recovered with the 2D Maximum a Posteriori (2D MAP) corresponding to the red mark. With a simpler nuisance model that fixes θ_Nuisance = 0 and thus underfits the nuisance, the 1D MAP (blue mark) of θ_Target is biased towards negative values as a result. Right: prior volume effect. In this toy illustration, we show the 2D and 1D marginalized posteriors of a distribution with the 1σ and 2σ confidence contours with blue and light blue respectively. Such posteriors can, for example, result from a model that is highly non-Gaussian in the full posterior space. The black dashed line represents the fiducial value, and the intersection of them is inside the 1σ contour. If we only have access to the 1D marginalized posterior of θ_Target, we would hardly estimate the fiducial value with our point estimators. In other words, even if the posterior in the entire parameter space is centered around the correct parameters, the posterior marginalized over the nuisance parameters can be off of the correct target parameters. This bias error is called the prior volume effect, and in this paper, we propose a pipeline to overcome it by constraining systematically the priors on the nuisance parameters.