I want to study the correlation between 2 parameters, this is done by fitting a straight line. I have uncertainties on both parameters.
I want to solve my problem using the Bayesian approach, i.e. I construct my likelihood, assume some prior information on the slope and $y$-intercept, and then I will get my posterior distribution. Then I would sample from my posterior distribution using MCMC.
The first step would be to construct my likelihood:
Assume the data come from a line of equation $y = mx + b$.
Then the likelihood would be:
$$P(D|\theta, I) = \frac{1}{\sqrt{2\pi \sigma^2}} \exp(-\frac{(y - mx - b)^2}{2\sigma^2})
$$
However, in my case the uncertainties are asymmetric, how can I construct my likelihood? For example, my data look something like this: $x^{\sigma +}_{\sigma -}$ and $y^{\sigma +}_{\sigma -}$
One way to solve this out is to calculate the average of the lower and upper uncertainties, so that: $$ \sigma = \frac{\sigma^+ + \sigma^-}{2} $$ Now I have symmetric uncertainties, and then I will be able to construct my likelihood.
Is there another method to solve this out?