Is there a way to generate a random variate from a non-standard distribution without computing CDF?

Question

I'm trying to write a Monte Carlo simulation. In my simulation I need to generate many random variates from a discrete probability distribution.

I do have a closed-form solution for the distribution and it has finite support; however, it is not a standard distribution. I am aware that I could draw a uniform[0,1) random variate and compare it to the CDF get a random variate from my distribution, but the parameters in the distributions are always changing. Using this method is too slow.

So I guess my question has two parts:

Is there a method/algorithm to quickly generate finite, discrete random variates without using the CDF?
Is there a Python module and/or a C++ library which already has this functionality?

Try [`std::discrete_distribution`](http://en.cppreference.com/w/cpp/numeric/random/discrete_distribution). — Kerrek SB, Feb 01 '13 at 22:02
if you have a closed-form solution, is there any chance you could invert it? — ev-br, Feb 02 '13 at 01:14

score 0 · Accepted Answer · answered Feb 01 '13 at 22:08

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Acceptance\Rejection: Find a function that is always higher than the pdf. Generate 2 Random variates. The first one you scale to calculate the value, the second you use to decide whether to accept or reject the choice. Rinse and repeat until you accept a value. Sorry I can't be more specific, but I haven't done it for a while.. Its a standard algorithm, but I'd personally implement it from scratch, so I'm not aware of any implementations.

answered Feb 01 '13 at 22:08

user2034068

16

It's difficult to explain the rejection method with such brevity. It may be worth it to [point to a reference on the method](https://en.wikipedia.org/wiki/Rejection_sampling) – loopbackbee Feb 01 '13 at 22:55

score 0 · Answer 2 · answered Feb 02 '13 at 00:38

Indeed acceptance/rejection is the way to go if you know analytically your pdf. Let's call it f(x). Find a pdf g(x) such that there exist a constant c, such that c.g(x) > f(x), and such that you know how to simulate a variable with pdf g(x) - For example, as you work with a distribution with a finite support, a uniform will do: g(x) = 1/(size of your domain) over the domain.

Then draw a couple (G, U) such that G is simulated with pdf g(x), and U is uniform on [0, c.g(G)]. Then, if U < f(G), accept U as your variable. Otherwise draw again. The U you will finally accept will have f as a pdf.

Note that the constant c determines the efficiency of the method. The smaller c, the most efficient you will be - basically you will need on average c drawings to get the right variable. Better get a function g simple enough (don't forget you need to draw variables using g as a pdf) but will the smallest possible c.

Quartz · Answer 3 · 2013-09-19T09:18:07.623

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If acceptance rejection is also too inefficient you could also try some Markov Chain MC method, they generate a sequence of samples each one dependent on the previous one, so by skipping blocks of them one can subsample obtaining a more or less independent set. They only need the PDF, or even just a multiple of it. Usually they work with fixed distributions, but can also be adapted to slowly changing ones.

edited Sep 19 '13 at 09:18

answered Sep 19 '13 at 09:08

Quartz

141
6

Can you give more detail on the distribution? How many samples are needed for each parameter set? – Quartz Sep 19 '13 at 09:08

Is there a way to generate a random variate from a non-standard distribution without computing CDF?

3 Answers3