# Construct the likelihood with asymmetric uncertainties

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?

• You need to choose a distribution that models your measurement errors. Gaussian obviously can't capture the asymmetry. Can you put your measurement error data plotted so that we can recommend a likelihood? Apr 12, 2015 at 14:59
• @Memming, my data do NOT come from repetition of measurements. My data resembles properties of exoplanets. So, for example let's assume I have 10 different planets, each data point refers to the mass of the exoplanets. Example: mass of planet 1 = $3^{+1}_{-0.25}$. I am assuming that the errors are drawn from a normal distribution with mean 0 and variance $\sigma^+$ and $\sigma^-$. Apr 12, 2015 at 15:07
• I see. You could cut two Gaussians in half, and stitch together to make a frankenstein distribution in that case. Apr 12, 2015 at 15:20
• @Memming Can you give me please more details or references, this is the first time I come across cutting 2 gaussians in half! Thanks a lot :) Apr 12, 2015 at 15:23

You could cut two Gaussians in half, and stitch together to make a split normal distribution of your noise $\epsilon$: $$P(\epsilon) = N(\epsilon; 0, \sigma^+)I(\epsilon > 0) + N(\epsilon; 0, \sigma^-)I(\epsilon < 0)$$ where $I$ is the indicator function. Below is an example probability density function. 