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About the assumption of multivariate normal distribution and parameter correlation

Hi everyone,

I estimated my HB model with data gathered from a choice experiment whose design contained some prohibitions. Using default values for the alpha matrix, my prior distributions of the mean of the parameters was noninformative. Testing for normality of the marginal posterior distributions of the parameters at population level (as contained in the alpha file) failed. Moreover, the posterior covariance matrix shows some off-diagonal element different from zero. As expected, a test over the joint posterior distribution of the parameters revealed the presence of correlation across the attribute levels and therefore the assumption of respondents being drawn from a multivariate normal distribution does not hold anymore. Having said that, I was now wondering the following:
1) What are the implications of the violation of the assumption of multivariate normal distribution on my estimates? In other words, how good is the software at accomodating correlation across parameters?
2) Value parameters (e.g. price) are not expected to be normally distributed, but what could explain other parameters not being normally distributed? Might this be due to miss-specification of the priors? Given that I am estimating 22 parameters using 12180 observations, would the miss-specification of the priors actually affect my estimates?
2) Does it make sense to use the estimates I obtained to better shape the alpha matrix? Which other information could I use?

Thanks for sharing your thoughts with me!
asked Feb 20, 2015 by lotika (455 points)

1 Answer

0 votes
The posterior estimates of our model are explicitly non-normal. Rarely is it feasible in Bayesian estimation to have a normal posterior distribution for estimates.

The priors are normal, but the priors times the non-normal likelihood guarantee that the posterior is not normal either. So it shouldn't be worrisome that the posterior draws fail a normality test.

1) There are no real implications to the data being not multivariate normal because of the model. The software carries the correlation across parameters and it is well represented in the estimates generated.

2) Unless you set extremely informative priors, they will have almost no influence on your final results with the number of observations you have.
answered Feb 21, 2015 by Kenneth Fairchild Bronze Sawtooth Software, Inc. (3,520 points)