I was in the middle of writing a long answer to

Uncertainty estimation in high-dimensional inference problems without sampling?

but I was suddenly struck by doubt: since the model assumes a Gaussian likelihood and a Gaussian prior over parameters, shouldn't the posterior be a normal-inverse-gamma? However, this would be the case if the model parameters, over which we put a Gaussian prior, were actually the mean and the variance-covariance matrix of the distribution of the data we observe. They're not, because the parameters enter the model through the complex nonlinear function $\mathcal{E}=(R_1(x_1,y_1)+z_1R_2(x_1,y_1),\dots,R_1(x_p,y_p)+z_pR_2(x_p,y_p))$, where the parameter vector is $\boldsymbol{\theta}=(x_1,y_1,z_1,\dots,x_p,y_p,z_p)\in\mathbb{R}^{3p}$. In other words, the model, in the OP notation, is

$$ d= \mathbf{G}\cdot (R_1(x_1,y_1)+z_1R_2(x_1,y_1),\dots,R_1(x_p,y_p)+z_dR_2(x_p,y_p))+\epsilon,\quad \epsilon\sim \mathcal{N}(0,\sigma^2)$$



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