Logit-Normal Distribution#

Univariate, Continuous, Asymmetric, Non-Negative, Heavy-tailed

The logit-normal distribution is a continuous probability distribution of a random variable whose logit (or log-odds) is normally distributed. Thus, if a random variable \(X\) follows a logit-normal distribution, then \( Y = \text{logit}(X) = \log\left(\frac{X}{1-X}\right)\) is normally distributed. It is defined for values of \(x\) between 0 and 1. It is characterized by two parameters: \(\mu\) and \(\sigma\), which are the mean and standard deviation of the logit-transformed variable, respectively, not the original variable.

The logit-normal distribution is useful in modeling proportions or ratios.

Key properties and parameters#

Support

\(x \in (0, 1)\)

Mean

no analytical solution

Variance

no analytical solution

Parameters:

  • \(\mu\) : (float) The mean of the logit-transformed variable.

  • \(\sigma\) : (float) The standard deviation of the logit-transformed variable.

  • \(\tau\) : (float) The precision of the logit-transformed variable, \(\tau = \frac{1}{\sigma^2}\).

Alternative parametrization

The logit-normal distribution can be parametrized in terms of \(\mu\) and \(\sigma\) or in terms of \(\mu\) and \(\tau\).

The link between the two parametrizations is given by:

\[ \tau = \frac{1}{\sigma^2} \]

Probability Density Function (PDF)#

\[ f(x \mid \mu, \sigma) = \frac{1}{x(1-x)\sigma\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{\text{logit}(x)-\mu}{\sigma}\right)^2\right) \]
../../_images/logit_normal_0_0.png
../../_images/logit_normal_1_0.png

Cumulative Distribution Function (CDF)#

\[ F(x \mid \mu, \sigma) = \frac{1}{2} + \frac{1}{2}\text{erf}\left(\frac{\text{logit}(x)-\mu}{\sigma\sqrt{2}}\right) \]

where erf is the error function.

../../_images/logit_normal_2_0.png
../../_images/logit_normal_3_0.png

See also

Related Distributions:

  • Normal Distribution - The logit-normal distribution is directly related to the normal distribution since if a variable is logit-normally distributed, its logit follows a normal distribution. This relationship is crucial for understanding the logit-normal’s properties and applications.

References#