## Truncated multivariate normal document.SUBSCRIPTION_OPTIONS = { "thing": "thread", "subscribed": false, "url": "subscribe", "icon": { "css": "fa fa-envelope-o" } };

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2011-09-07
2012-09-01
• Joris Menten - 2011-09-07

I would like to use a truncated bivariate normal to model random intercepts
and slopes, with intercept constrained to be positive and the slope to be
negative.

In OpenBugs I used:

beta ~ dmnorm(mu.beta, R)I(lim1,lim2)

with beta the random intercepts, and beta the random slopes. The limits lim 1
and lim2 are given as data.

Using the same, but replacing the I by T (as this is an a priori truncated
distribution), in jags does not seem to work.

(runtime error "Distribution cannot be bounded in distribution dmnorm").

Does this mean it is not possible to bound the multivariate normal - or am I
just doing something evident wrong?

Joris

• Aziz - 2011-09-08

If you are modelling bivariate data, I think you could by-pass the
multivariate node and bind two univariate models like this:

beta ~ dnorm(mu.beta,tau1)T(low1,up1)

beta ~ dnorm(mu.beta + lam*(beta-mu.beta),tau.iid)T(low2,up2)

Basically, you use the correlated part of the error in beta inside the mean
function of beta. Tau.iid is then the precision associated to the independent
part of the error term in beta. Since you write this as 2 univariate
distribution, it is straightforward to add truncation constraints.

You can then use the following "vague" priors/parameter transformation:

lam ~ dnorm(0,0.0001)

var1 <- 1/tau1

var2 <- 1/tau.iid + (lam^2)*var1

tau1 <- pow(sigma1,-2)

tau.iid <- pow(sigma.iid,-2)

sigma1 ~ dunif(0,1000)

sigma.iid ~ dunif(0,1000)

And the correlation between beta and beta is given by:

rho <- lamvar1/sqrt(var1var2)

Please make some trials first. I think it should work but I'm no expert in
Bayesian stat. Maybe Martyn can confirm this.

Hope it helps

• Joris Menten - 2011-09-08

Thanks - will have a try. I did sort of knew this trick from generating
bivariate normals - should indeed work. A bit more complicated, but then I can
also avoid those pesky Wishart distributions as priors. Joris