Description Usage Arguments Details Value Author(s) References

The softplus (and inverse softplus) transform is useful to introduce positivity constraints on parameters of a function that will be optimized (e.g. MLE of the scale parameter of a density function). The softplus has been introduced to replace the exponential which might blow up for large argument. The softplus is given by : log(1+exp(x)) and converges to x for large values of x. Some care has been taken in the implementation of the softplus function to handle some numerical issues.

1 2 | ```
softplus(x)
softplusinv(y)
``` |

`x` |
is the value of the unconstrained parameter which is optimized |

`y` |
is the value of the positively constrained parameter |

Let sigma be the scale parameter of a density for which maximumm
likelihood estimation will be performed. Then we can consider optimizing
`softplusinv`

(sigma) to ensure positivity of this parameter. Let sigma.unc
be the optimzed unconstrained parameter, then `softplus`

(sigma.unc) is the
value of the MLE.

The value of the softplus (or inverse softplus) transform.

Julie Carreau

Dugas, C., Bengio, Y., Belisle, F., Nadeau, C. and Garcia, R. (2001), A universal approximator of convex functions applied to option pricing, 13, Advances in Neural Information Processing Systems

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