# Exponential errors in variables model with known uncertainties

I have $$N$$ data points that I am trying to fit using a function of the form

$$y_i = \prod_j {X_{i,j}}^{b_j}, \quad j=1..N$$

where $$\mathbf X$$ and $$\mathbf y$$ are measured values. The form of this function is motivated from physics. I also have a variance-covariance matrix containing the estimated uncertainties of $$\mathbf X$$ and $$\mathbf y$$, which I assume to be distributed according to a multivariate normal distribution. Finally, $$\mathbf b$$ are the exponents I would like to determine.

I am not sure how to handle the fact there are uncertainties in the $$\mathbf X$$ variables and furthermore that they are correlated with the errors in $$\mathbf y$$. If the errors in $$\mathbf X$$ did not exist then I would simply find the $$\mathbf b$$ that minimize

$$\chi^2 = \sum_i \frac{(y_i - \prod_j X_{i,j}^{b_j})^2}{\sigma_y^2}$$

What should I do in this case? Is there a similarly simple functional form that I can optimize?

It is not lost on me that I can take the log of both sides and it looks like a standard regression problem, except without an intercept. Is this a useful line to pursue?