Inverse transform sampling

Inverse transform sampling (also known as inversion sampling, the inverse probability integral transform, the inverse transformation method, or the Smirnov transform) is a basic method for pseudo-random number sampling, i.e., for generating sample numbers at random from any probability distribution given its cumulative distribution function.

between 0 and 1, interpreted as a probability, and then returns the smallest number

for the cumulative distribution function

The table below shows samples taken from the uniform distribution and their representation on the standard normal distribution.

We are randomly choosing a proportion of the area under the curve and returning the number in the domain such that exactly this proportion of the area occurs to the left of that number.

Intuitively, we are unlikely to choose a number in the far end of tails because there is very little area in them which would require choosing a number very close to zero or one.

Computationally, this method involves computing the quantile function of the distribution — in other words, computing the cumulative distribution function (CDF) of the distribution (which maps a number in the domain to a probability between 0 and 1) and then inverting that function.

Note that for a discrete distribution, computing the CDF is not in general too difficult: we simply add up the individual probabilities for the various points of the distribution.

As a result, this method may be computationally inefficient for many distributions and other methods are preferred; however, it is a useful method for building more generally applicable samplers such as those based on rejection sampling.

For the normal distribution, the lack of an analytical expression for the corresponding quantile function means that other methods (e.g. the Box–Muller transform) may be preferred computationally.

It is often the case that, even for simple distributions, the inverse transform sampling method can be improved on:[1] see, for example, the ziggurat algorithm and rejection sampling.

On the other hand, it is possible to approximate the quantile function of the normal distribution extremely accurately using moderate-degree polynomials, and in fact the method of doing this is fast enough that inversion sampling is now the default method for sampling from a normal distribution in the statistical package R.[2] For any random variable

is the generalized inverse of the cumulative distribution function

[3] For continuous random variables, the inverse probability integral transform is indeed the inverse of the probability integral transform, which states that for a continuous random variable

with cumulative distribution function

to be a continuous, strictly increasing function, which provides good intuition.

We want to see if we can find some strictly monotone transformation

The problem that the inverse transform sampling method solves is as follows: The inverse transform sampling method works as follows: Expressed differently, given a cumulative distribution function

[3] In the continuous case, a treatment of such inverse functions as objects satisfying differential equations can be given.

[4] Some such differential equations admit explicit power series solutions, despite their non-linearity.

be a cumulative distribution function, and let

be its generalized inverse function (using the infimum because CDFs are weakly monotonic and right-continuous):[6] Claim: If

is a uniform random variable on

Proof: Inverse transform sampling can be simply extended to cases of truncated distributions on the interval

without the cost of rejection sampling: the same algorithm can be followed, but instead of generating a random number

In order to obtain a large number of samples, one needs to perform the same number of inversions of the distribution.

One possible way to reduce the number of inversions while obtaining a large number of samples is the application of the so-called Stochastic Collocation Monte Carlo sampler (SCMC sampler) within a polynomial chaos expansion framework.

This allows us to generate any number of Monte Carlo samples with only a few inversions of the original distribution with independent samples of a variable for which the inversions are analytically available, for example the standard normal variable.

[7] There are software implementations available for applying the inverse sampling method by using numerical approximations of the inverse in the case that it is not available in closed form.

For example, an approximation of the inverse can be computed if the user provides some information about the distributions such as the PDF [8] or the CDF.

Inverse transform sampling for normal distribution
Graph of the inversion technique from to . On the bottom right we see the regular function and in the top left its inversion.
Schematic of the inverse transform sampling. The inverse function of can be defined by .
An animation of how inverse transform sampling generates normally distributed random values from uniformly distributed random values
Random numbers y i are generated from a uniform distribution between 0 and 1, i.e. Y ~ U(0, 1). They are sketched as colored points on the y-axis. Each of the points is mapped according to x=F −1 (y), which is shown with gray arrows for two example points. In this example, we have used an exponential distribution. Hence, for x ≥ 0, the probability density is and the cumulative distribution function is . Therefore, . We can see that using this method, many points end up close to 0 and only few points end up having high x-values - just as it is expected for an exponential distribution.