Bayesian inverse modeling with non-identifiable parameters?
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If I have a physical model begin{equation} y = frac{1}{beta_0} (beta_1 x_1 + beta_2 x_2) end{equation} and want to estimate coefficients $beta_0$ , $beta_1$ , and $beta_2$ from given data set $(y, x_1, x_2)$ , with regression I would typically have to combine them into two coefficients $beta_1/beta_0$ and $beta_2/beta_0$ since that would be what I would be able to solve (contributions from $1/beta_0$ and $beta_1$ cannot be distinguished from each other individually). However, if I have separate prior distributions for $beta_0$ , $beta_1$ , and $beta_2$ - is it possible to obtain sensible posterior distributions for each of them using MCMC or Laplace's method without combining the variables? Edits Corrected title and variable name error based on comments and answers.
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