mexpit – Convert log-scale numbers to probabilities

This function converts log-scale numbers to probabilities.

Usage

mexpit(...)

dmexpit(...)

Arguments

...

numeric log-scale numbers to convert to probabilities.

Value

Probabilities that add up to a number less than 1.

Details

The probabilities are calculated using the following equation:

$$p_i = \frac{e^{x_i}}{1+\sum_{j=1}^{N-1} e^{x_j}}$$

This ensures one remaining probability will add to one, that is

$$p_N = \frac{1}{1+\sum_{j=1}^{N-1} e^{x_j}}$$

For the function dmexpit(), the element-wise derivatives are calculated; that is, it returns the diagonal of the Jacobian matrix, \(d p_i / d x_i\), not the full Jacobian with off-diagonal terms.

Author

Matthew L. Fidler

Examples

m <- mlogit(0.1, 0.2, 0.3)
mexpit(m)
#> [1] 0.1 0.2 0.3

# derivatives
dmexpit(m)
#> [1] 0.09 0.16 0.21

p <- mexpit(-3, 0.5, 3)
mlogit(p)
#> [1] -3.0  0.5  3.0