# Multivariate lognormal distribution in Stan

Hi,

Currently, I am trying to fit a lognormal model for one of the parameters in the Stan program. For one-dimension case, the easy way is just using the built-in Stan function as:

\beta \sim lognormal(\mu, \sigma)

However, currently my parameter \beta is a vector of length 3, where \beta = (\beta_1, \beta_2, \beta_3) and I would like to jointly model the three components as multivariate lognormal distribution with some mean vector and covariance matrix (maybe treat them as hyperparameter in the model). May I ask that does Stan provide such built-in functions for multivariate lognormal distribution?

Thanks so much for your help!

The easiest way to do this would be to take the logarithm of your data in the transformed data block. Tutorial here.

Usually when we include multivariate random effects in a model, we are modeling covariances between multiple vectors, not multiple elements of a single vector. If \beta_1, \beta_2, and \beta_3 are all elements of a single random-effect vector, then there is just the variance and no additional covariance. If, on the other hand, \beta_1, \beta_2, and \beta_3 all represent single-element random effect vectors, then how do you expect to be able to estimate the underlying means, variances, and covariances? If the means, variances, and covariances are tightly constrained a priori then this could be a useful approach, but that’s not what you seem to be driving at when you write

We don’t have a multivariate lognormal built in, but it’s easy to code by inverse transforming a multivariate normal.

data {
cov_matrix Sigma;
}
transformed data {
cholesky_factor_cov L_Sigma = cholesky_decompose(Sigma);
}
parameters {
vector log_beta;
}
transformed parameters {
vector beta = exp(log_beta);
}
model {
log_beta ~ multi_normal_cholesky(mu, L_Sigma);
}


This implies that beta has a multi_lognormal(mu, Sigma) distribution, where Sigma = L_Sigma * L_Sigma'. With this code, you can use beta just as you would had it been given a multivariate lognormal directly.

Multivariate logistic normal works the same way, only with inv_logit() in place of exp().

[edit: added he covariance matrix as data and its cholesky factor as a transform—you could also take this to be a parameter, for which you might consider the LKJ prior on a cholesky factor of correlation matrices scaled by a vector of scales with their own independent prior]

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Hi Bob,

Thanks so much for your help!