# logit: The logit and inverse-logit functions In LaplacesDemon: Complete Environment for Bayesian Inference

## Description

The logit and inverse-logit (also called the logistic function) are provided.

## Usage

 ```1 2``` ```invlogit(x) logit(p) ```

## Arguments

 `x` This object contains real values that will be transformed to the interval [0,1]. `p` This object contains of probabilities p in the interval [0,1] that will be transformed to the real line.

## Details

The `logit` function is the inverse of the sigmoid or logistic function, and transforms a continuous value (usually probability p) in the interval [0,1] to the real line (where it is usually the logarithm of the odds). The `logit` function is log(p / (1 - p)).

The `invlogit` function (called either the inverse logit or the logistic function) transforms a real number (usually the logarithm of the odds) to a value (usually probability p) in the interval [0,1]. The `invlogit` function is 1 / (1 + exp(-x)).

If p is a probability, then p/(1-p) is the corresponding odds, while the `logit` of p is the logarithm of the odds. The difference between the logits of two probabilities is the logarithm of the odds ratio. The derivative of probability p in a logistic function (such as `invlogit`) is: (d / dx) = p * (1 - p).

In the LaplacesDemon package, it is common to re-parameterize a model so that a parameter that should be in an interval can be updated from the real line by using the `logit` and `invlogit` functions, though the `interval` function provides an alternative. For example, consider a parameter theta that must be in the interval [0,1]. The algorithms in `IterativeQuadrature`, `LaplaceApproximation`, `LaplacesDemon`, `PMC`, and `VariationalBayes` are unaware of the desired interval, and may attempt theta outside of this interval. One solution is to have the algorithms update `logit(theta)` rather than `theta`. After `logit(theta)` is manipulated by the algorithm, it is transformed via `invlogit(theta)` in the model specification function, where theta in [0,1].

## Value

`invlogit` returns probability `p`, and `logit` returns `x`.

`interval`, `IterativeQuadrature`, `LaplaceApproximation`, `LaplacesDemon`, `plogis`, `PMC`, `qlogis`, and `VariationalBayes`.
 ```1 2 3 4``` ```library(LaplacesDemon) x <- -5:5 p <- invlogit(x) x <- logit(p) ```