# How to simulate a multivariate Logistic-Normal distribution in Python

I'm trying to generate a text document using reverse "Correlated Topic Models", which is an advanced version of LDA (Latent Dirichlet Allocation). In this version the topics are generated over a multinomial distribution with a Logistic-Normal prior. This allows adding correlation between topics through the covariance matrix.

Can anyone please help me simulate values from a multivariate Logistic-Normal distirubtion, i.e. given input parameters myu, sigma (mean vector, and covariance matrix respectively), the result is a distribution over topics.

Unfortunately, scipy.stats doesn't provide the logistic normal distribution. However, you could draw random samples from a multivariate normal distribution (e.g. using numpy) and transform them with a logistic transformation to simulate samples drawn from the logistic-normal distribution.

Let's assume your probability vectors are $D=3$ dimensional.

import numpy as np

# draw from multivariate random distribution of dimension D-1 = 2
mean = (1, 2)
cov = [[1, 0], [0, 1]]
y = np.random.multivariate_normal(mean, cov)


Now (as you can read here), you can transform your normally distributed sample $y \in \mathcal{S}^{D-1}$ to a logistic-normally distributed sample $x \in \mathcal{S}^{D}$:

$$\mathbf{y} = \left[ \log \left( \frac{ x_1 }{ x_D } \right) , \dots , \log \left( \frac{ x_{D-1} }{ x_D } \right) \right]$$

$$\mathbf{x} = \left[ \frac{ e^{ y_1 } }{ 1 + \sum_{i=1}^{D-1} e^{ y_i } } , \dots , \frac{ e^{ y_{D-1} } }{ 1 + \sum_{i=1}^{D-1} e^{ y_i } } , \frac{ 1 }{ 1 + \sum_{i=1}^{D-1} e^{ y_i } } \right]$$