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Copy file name to clipboardexpand all lines: developers/transforms/bijectors/index.qmd
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@@ -28,6 +28,7 @@ A _bijection_ between two sets ([Wikipedia](https://en.wikipedia.org/wiki/Biject
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That is to say, if we have two sets $X$ and $Y$, then a bijection maps each element of $X$ to a unique element of $Y$.
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To return to our univariate example, where we transformed $x$ to $y$ using $y = \exp(x)$, the exponentiation function is a bijection because every value of $x$ maps to one unique value of $y$.
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The input set (the domain) is $(-\infty, \infty)$, and the output set (the codomain) is $(0, \infty)$.
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(Here, $(a, b)$ denotes the open interval from $a$ to $b$ but excluding $a$ and $b$ themselves.)
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Since bijections are a one-to-one mapping between elements, we can also reverse the direction of this mapping to create an inverse function.
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In the case of $y = \exp(x)$, the inverse function is $x = \log(y)$.
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However, we specified that the bijection $y = \exp(x)$ maps values of $x \in (-\infty, \infty)$ to $y \in (0, \infty)$, so the point $y = 0$ is not within the domain of the inverse function.
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:::
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Specifically, one of the primary purposes of Bijectors.jl is used to construct _bijections which map constrained distributions to unconstrained ones_.
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For example, the log-normal distribution which we saw above is constrained: its _support_, i.e. the range over which $p(x) > 0$, is $(0, \infty)$.
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Specifically, one of the primary purposes of Bijectors.jl is to construct _bijections which map constrained distributions to unconstrained ones_.
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For example, the log-normal distribution which we saw in [the previous page]({{< meta dev-transforms-distributions >}}) is constrained: its _support_, i.e. the range over which $p(x) > 0$, is $(0, \infty)$.
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However, we can transform that to an unconstrained distribution (the normal distribution) using the transformation $y = \log(x)$.
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