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I want to optimize the parameters that minimize a particular function. These parameters are typically lower and upper bounded (i.e. some can only lie between 0 and 1, some only between 4 and 6, etc.). Some algorithms allow you to set some boundaries, but ideally, I'd like to transform the parameters into a continuous infinite space, so that when I transform back, they'll be bounded. I came across some of these transformations such as $y = x/(1-x)$ (for $x\in [0,1)$ maps into $[0,\infty)$. I also seem to remember a rather convoluted one with trigonometric functions but I can't quite remember it. Ideally, I'd like to transform my parameters so that they go from a closed interval to $(-\infty, \infty)$.

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    $\begingroup$ Clearly $(x - a)/(b - a)$ maps $x$ in $[a, b]$ to $[0,1]$. $\text{logit}(x)$ maps $(0,1)$ to $(-\infty, \infty)$. $\endgroup$
    – Nick Cox
    Feb 20, 2014 at 22:45
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    $\begingroup$ @NickCox +1 -- indeed the inverse of any continuous-strictly-monotonic-cdf-on-$(-\infty,\infty)$ maps $(0,1)$ to $(-\infty,\infty)$. There's uncountable infinities of choices. $\endgroup$
    – Glen_b
    Feb 20, 2014 at 22:51

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Partially answered in comments:

Clearly $(x−a)/(b−a)$ maps $x \in [a,b]$ to $[0,1]$. logit(x) maps (0,1) to (−∞,∞). – Nick Cox

indeed the inverse of any continuous-strictly-monotonic-cdf-on-(−∞,∞) maps (0,1) to (−∞,∞). There's uncountable infinities of choices. – Glen_b

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