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

```
invlogit(x)
logit(p)
```

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.

`invlogit`

returns probability `p`

, and
`logit`

returns `x`

.

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 \(\frac{1}{1 + \exp(-x)}\).

If \(p\) is a probability, then \(\frac{p}{1-p}\) 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: \(\frac{d}{dx}
= p(1-p)\).

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 \(\theta \in [0,1]\).

`interval`

,
`IterativeQuadrature`

,
`LaplaceApproximation`

,
`LaplacesDemon`

,
`plogis`

,
`PMC`

,
`qlogis`

, and
`VariationalBayes`

.

```
# NOT RUN {
library(LaplacesDemon)
x <- -5:5
p <- invlogit(x)
x <- logit(p)
# }
```

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