5
$\begingroup$

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?

$\endgroup$
4
  • $\begingroup$ 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? $\endgroup$
    – Memming
    Commented Apr 12, 2015 at 14:59
  • $\begingroup$ @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 $\sigma^-$. $\endgroup$
    – aloha
    Commented Apr 12, 2015 at 15:07
  • $\begingroup$ I see. You could cut two Gaussians in half, and stitch together to make a frankenstein distribution in that case. $\endgroup$
    – Memming
    Commented Apr 12, 2015 at 15:20
  • $\begingroup$ @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 :) $\endgroup$
    – aloha
    Commented Apr 12, 2015 at 15:23

1 Answer 1

2
$\begingroup$

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.

enter image description here

As you can see, positive noise is distributed as a guassian with a larger standard deviation than the negative noise.

$\endgroup$
6
  • $\begingroup$ what are your thoughts? $\endgroup$
    – Memming
    Commented Apr 13, 2015 at 13:11
  • $\begingroup$ weird enough, even google does not know what a frankenstein distribution is! I am not sure if I should ask you for more details or if I should figure this on my own. $\endgroup$
    – aloha
    Commented Apr 13, 2015 at 21:38
  • $\begingroup$ Can you please provide me with references to your suggested answer? $\endgroup$
    – aloha
    Commented Apr 14, 2015 at 13:33
  • 1
    $\begingroup$ There's no reference, because I came up with it. It's a logical answer to your specific problem. $\endgroup$
    – Memming
    Commented Apr 14, 2015 at 14:29
  • 1
    $\begingroup$ This distribution is known as the split-normal or two-piece normal distribution, see en.wikipedia.org/wiki/Split_normal_distribution. If the name Frankenstein distribution was made up for this answer, this should perhaps be edited to use the 'real' name. $\endgroup$ Commented Jul 5, 2015 at 11:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.