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My aim is to study the correlation between 2 parameters knowing that I have measurement errors in both parameters, i.e. I have uncertainties on the independent and dependent parameters.

I want to study the correlation using a Bayesian approach, i.e. construct my posterior distribution, and then I would sample from my posterior distribution using MCMC. This is what's called the structural approach.

The first step would be to check whether my uncertainties are normally distributed in order to construct my likelihood accordingly. I did a QQ plot, and the results are:

  • the independent parameter is normally distributed
  • the dependent parameter is NOT normally distributed

My question is how can I proceed to write my likelihood?

EDIT

I am following the method suggested by Kelly 2007: (see section 3 for the assumptions)
Let the observed data be (x, y) and the true (unobserved) values be ($\xi$, $\eta$)
Assume that the data can be modeled by a straight line of equation: $$\eta_i = \alpha + \beta\xi_i + \epsilon $$

And the errors are normally distributed with known variances $\sigma_x$ and $\sigma_y$.

Constructing the likelihood $p(x, y|\theta, \psi):$ where $\theta = \alpha, \beta, \sigma^2$
(note that at this moment I am going to neglect $\psi$ for simplification)

The likelihood can be expressed hierarchically as:

$$p(x, y|\theta, \psi) = \int\int p(x, y|\xi, \eta) \; p(\eta |\xi, \theta) \; p(\xi|\psi) \; d\xi \; d\eta$$

To compute $p(\eta | \xi, \theta)$, the author assumed that the data really do come from a line and that the uncertainties where drawn from a Gaussian distribution of mean ZERO and known variance $\sigma^2$. Therefore, $p(\eta | \xi, \theta) = \frac{1}{\sqrt{2\pi \sigma^2}} \exp (-\frac{(y - \eta)^2}{2\sigma^2})$

This is where I got stuck, in my case, the errors are not normally distributed. How can I proceed? I hope my problem is clearer. And excuse me if my question is silly as this is my first attempt to solve a problem in a Bayesian approach.

kjetil b halvorsen
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aloha
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    What is your model? It is not clear what you are asking... In Bayesian approach there is no Normality assumption anywhere. Besides, even in frequentest approach normality testing is not that big concern: http://stats.stackexchange.com/questions/2492/is-normality-testing-essentially-useless – Tim Mar 17 '15 at 11:12
  • @Tim What do you mean there is no normality assumption anywhere in the Bayesian approach? – tchakravarty Mar 17 '15 at 11:16
  • He means that there is no mandatory normality approach anywhere in a Bayesian approach. – jlimahaverford Mar 17 '15 at 11:20
  • @jlimahaverford As opposed to where? – tchakravarty Mar 17 '15 at 11:53
  • TC. I'm confused by what you're asking. First, my first "approach" should be assumption. I can certainly understand the possibility that in some setting we would test for normality. For instance, if we put a normal prior on some collection of parameters, it is wise to inspect the distribution afterword to see if this was a reasonable prior. But this is not a general step in the Bayesian cookbook. po6, your trying to correlate Y and X, given {(x_i, y_i), i=1, 2, ...}. Correct? So are you you saying that you tested {x_i} and {y_i} separately. You wouldn't expected normality here. – jlimahaverford Mar 17 '15 at 12:15
  • @Tim I edited my questions, I hope it is clearer now. – aloha Mar 17 '15 at 15:11
  • @jlimahaverford I hope my question is clearer now – aloha Mar 17 '15 at 15:12
  • @TC please check the new edit of my question. – aloha Mar 17 '15 at 15:13
  • @jlimahaverford, replying to the question you asked, no I did not test $x_i$ and $y_i$ separately. What I did is that I tested whether my errors $\sigma_x$ and $\sigma_y$ come from a normal distribution. I studied them separately. – aloha Mar 18 '15 at 13:08

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