Multivariate logistic approximation?

[Stephen Martin]

Hey all,

I'm just curious what you all think about this.
I know that the logistic distribution is approximately N(0,1.6).
I'm curious whether one could then use a multivariate normal ([zeroes],1.6^2*Sigma) as an approximation to a 'multivariate' logistic distribution? Is this mathematically sound?
The idea of allowing for correlated errors in multinomial probit models makes me wonder if my latent regression for latent categories would benefit from this. Currently, for K latent groups, each individual's multinomial logit values are estimated from covariates, then converted to a simplex using softmax.
So, for(k in 2:K, n in 1:N) { //k = 1 is fixed to 1 for scale in the softmax function
lambda[n,k] ~ logistic(X*Beta, 1)
}
for(n in 1:N){
pi[n] = softmax(lambda[n])
}

This means the categorization error is assumed independent, and allowing a covariance would loosen that assumption.
I can't wrap my head around multinomial probits, or how to convert them to probabilities of membership to latent groups, so I'm wondering if using a multivariate normal that's scaled to be like a logistic would suffice?

Thanks as always for your input.