Rejection sampling in ${\displaystyle A_{f,r}}$

The above statement does not specify how one should perform the uniform sampling in ${\displaystyle A_{f,r}}$. However, the interest of this method is that under mild conditions on ${\displaystyle f}$ (namely that ${\displaystyle f(x_{1},x_{2},\ldots ,x_{d})^{\frac {1}{1+rd}}}$ and ${\displaystyle x_{i}f(x_{1},x_{2},\ldots ,x_{d})^{\frac {r}{1+rd}}}$ for all ${\displaystyle i}$ are bounded), ${\displaystyle A_{f,r}}$ is bounded. One can define the rectangular bounding box ${\displaystyle {\tilde {A}}_{f,r}}$ such that

$${\displaystyle A_{f,r}\subset {\tilde {A}}_{f,r}=\left[0,\sup _{x_{1},\ldots ,x_{d}}{f(x_{1},\ldots ,x_{d})^{\frac {1}{1+rd}}}\right]\times \prod _{i}\left[\inf _{x_{1},\ldots ,x_{d}}{x_{i}f(x_{1},\ldots ,x_{d})^{\frac {r}{1+rd}}},\sup _{x_{1},\ldots ,x_{d}}{x_{i}f(x_{1},\ldots ,x_{d})^{\frac {r}{1+rd}}}\right]}$$

This allows to sample uniformly the set ${\displaystyle A_{f,r}}$ by rejection sampling inside ${\displaystyle {\tilde {A}}_{f,r}}$. The parameter ${\displaystyle r}$ can be adjusted to change the shape of ${\displaystyle A_{f,r}}$ and maximize the acceptance ratio of this sampling.

Parametric description of the boundary of ${\displaystyle A_{f,r}}$

The definition of ${\displaystyle A_{f,r}}$ is already convenient for the rejection sampling step. For illustration purposes, it can be interesting to draw the set, in which case it can be useful to know the parametric description of its boundary:

$${\displaystyle u=f\left(x_{1},x_{2},\ldots ,x_{d}\right)^{\frac {1}{1+rd}}\quad {\text{and}}\quad \forall i\in [|1,n|],v_{i}=x_{i}u^{r}}$$

or for the common case where ${\displaystyle X}$ is a 1-dimensional variable, ${\displaystyle (u,v)=\left(f(x)^{\frac {1}{1+r}},x\,f(x)^{\frac {r}{1+r}}\right)}$.

Generalized ratio of uniforms

Here parameterized only with ${\displaystyle r}$, the ratio of uniforms can be described with a more general class of transformations.^[5]

In the 1-dimensional case, if ${\displaystyle g:\mathbb {R} ^{+}\rightarrow \mathbb {R} ^{+}}$ is a strictly increasing and differentiable function such that ${\displaystyle g(0)=0}$, then we can define ${\displaystyle A_{f,g}}$ such that

$${\displaystyle A_{f,g}=\left\{(u,v)\in \mathbb {R} ^{2}:0\leq u\leq g^{-1}\left[f\left({\frac {v}{g'(u)}}\right)\right]\right\}}$$

If ${\displaystyle (U,V)}$ is a random variable uniformly distributed in ${\displaystyle A_{f,g}}$, then ${\displaystyle {\frac {V}{g'(U)}}}$ is distributed with the density ${\displaystyle p}$.

The exponential distribution

Assume that we want to sample the exponential distribution, ${\displaystyle p(x)=\lambda \mathrm {e} ^{-\lambda x}}$ with the ratio of uniforms method. We will take here ${\displaystyle r=1}$.

We can start constructing the set ${\displaystyle A_{f,1}}$:

${\displaystyle A_{f,1}=\left\{(u,v)\in \mathbb {R} ^{2}:0\leq u\leq {\sqrt {p\left({\frac {v}{u}}\right)}}\right\}}$

The condition ${\displaystyle 0\leq u\leq {\sqrt {p\left({\frac {v}{u}}\right)}}}$ is equivalent, after computation, to ${\displaystyle 0\leq v\leq -{\frac {u}{\lambda }}\ln {\frac {u^{2}}{\lambda }}}$, which allows us to plot the shape of the set (see graph).

This inequality also allows us to determine the rectangular bounding box ${\displaystyle {\tilde {A}}_{f,1}}$ where ${\displaystyle A_{f,1}}$ is included. Indeed, with ${\displaystyle g(u)=-{\frac {u}{\lambda }}\ln {\frac {u^{2}}{\lambda }}}$, we have ${\displaystyle g\left({\sqrt {\lambda }}\right)=0}$ and ${\displaystyle g'\left({\frac {2}{\mathrm {e} {\sqrt {\lambda }}}}\right)=0}$, from where ${\displaystyle {\tilde {A}}_{f,1}=\left[0,{\sqrt {\lambda }}\right]\times \left[0,g\left({\frac {2}{\mathrm {e} {\sqrt {\lambda }}}}\right)\right]}$.

From here, we can draw pairs of uniform random variables ${\displaystyle U\sim \mathrm {Unif} \left(0,{\sqrt {\lambda }}\right)}$ and ${\displaystyle V\sim \mathrm {Unif} \left(0,g\left({\frac {2}{\mathrm {e} {\sqrt {\lambda }}}}\right)\right)}$ until ${\displaystyle u\leq {\sqrt {\lambda \,\mathrm {e} ^{-\lambda {\frac {v}{u}}}}}}$, and when that happens, we return ${\displaystyle {\frac {v}{u}}}$, which is exponentially distributed.

Normal mixture distribution before and after change of variables by the ratio of uniforms method. Top: graph of the mixture distribution on ${\displaystyle \mathbb {R} }$. Bottom: the set ${\displaystyle A_{f,r}}$ is represented for two different values of ${\displaystyle r}$. The solid lines on the top represent the de-transformation of the bounding boxes on the bottom. The solid lines on the bottom represent the locations of different values of ${\displaystyle x}$ in the set.

A mixture of normal distributions

Consider the mixture of two normal distributions ${\displaystyle {\mathcal {D}}=0.6\,N(\mu =-1,\sigma =2)+0.4\,N(\mu =3,\sigma =1)}$. To apply the method of the ratio of uniforms, with a given ${\displaystyle r}$, one should first determine the boundaries of the rectangular bounding box ${\displaystyle {\tilde {A}}_{f,r}}$ enclosing the set ${\displaystyle A_{f,r}}$. This can be done numerically, by computing the minimum and maximum of ${\displaystyle u(x)=f(x)^{\frac {1}{1+r}}}$ and ${\displaystyle v(x)=x\,f(x)^{\frac {r}{1+r}}}$ on a grid of values of ${\displaystyle x}$. Then, one can draw uniform samples ${\displaystyle (u,v)\in {\tilde {A}}_{f,r}}$, only keep those that fall inside the set ${\displaystyle A_{f,r}}$ and return them as ${\displaystyle {\frac {v}{u^{r}}}}$.

It is possible to optimize the acceptance ratio by adjusting the value of ${\displaystyle r}$, as seen on the graphs.