Rejection sampling
$\require{mathtools}
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% the following cause issues with real LaTeX tho :/ maybe consider naming it \fhi instead?
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Suppose we have two (unnormalized) density functions $p(x)$ and $q(x)$ that can be easily evaluated on a point, and suppose that we’re able to get sample from $q$:
\[
\Pr\b{\BX = x} \propto q(x).
\]
Then, assuming that $p$ is not too much more concentrated than $q$ at any point, we can get samples from $p$ by sampling from $q$ then “tweaking the probabilities” appropriately.
Algorithm
Repeatedly:
- sample $x$ from $q$;
- with probability $c \f{p(x)}{q(x)}$, return $x$;
- otherwise, “reject” $x$ and continue.
Analysis
As long as the “probability” in step 2 is never greater than $1$, we’ll get value $x$ with probability
\[
\propto q(x) \times \p{c\f{p(x)}{q(x)}} = cp(x),
\]
so this has the right distribution. And it’s enough to set $c \ce \min_x \f{q(x)}{p(x)}$.
But the issue is that the smaller we set $c$, the longer the algorithm might take until it accepts for the first time. Indeed, assuming that $p$ and $q$ are actually probabilities (they sum to $1$), the probability of accepting at any iteration of the loop is
\[
\sum_x q(x) \times \p{c \f{p(x)}{q(x)}} = c \sum_x p(x) = c,
\]
so it would take $1/c = \max_x \f{p(x)}{q(x)}$ iterations on average until the algorithm succeeds. This quantity $\max_x \f{p(x)}{q(x)}$ is the max-divergence between $p$ and $q$ (see Power entropies and divergences).
Example: sampling from DNFs
Say we want to sample from the set of satisfying assignments of a DNF formula
\[
f = T_1 \or \cdots \or T_m.
\]
It’s easy to sample uniformly over the assignments $T_i^{-1}(1)$ that satisfy a particular term $T_i$ of the formula, since they form a hypercube. But the overlaps between the hypercubes $T_i^{-1}(1)$ for different $i$ make it nontrivial to sample uniformly over the entirety of $f^{-1}(1)$.
That said, we can still easily sample according to the following distribution, which is somewhat close to the desired distribution:
- pick a term $T_i$ at random according to how large $\abs{T_i^{-1}(1)}$ is,
- then pick a uniformly random $x$ within $T_i^{-1}(1)$.
This would be perfect if the hypercubes $T_i^{-1}(1)$ were disjoint, but in general, assignments $x$ that satisfy many terms have a higher chance of being drawn: we get density
\[
q(x) \ce \#\setco{i}{T_i(x)},
\]
whereas the distribution we wish for is just
\[
p(x) \ce \1[T_1(x) \or \cdots \or T_n(x)].
\]
The ratio $p(x)/q(x)$ is always at most $1$, so we can just set $c \ce 1$, and each iteration has at least a
\[
\f{p(x)}{q(x)} \ge \f1m
\]
chance of accepting $x$, so it takes $\le m$ iterations on average.
#to-write swap $p$ and $q$ and capitalize