How do you test the quality and reliability of your random numbers for Monte Carlo?

When testing the quality and reliability of random numbers for Monte Carlo simulations, it's crucial to ensure that they exhibit the necessary statistical properties. This includes properties such as uniformity, independence, and reproducibility. One common approach is to conduct statistical tests, such as the chi-square test or the Kolmogorov-Smirnov test, to assess whether the generated random numbers conform to the expected distribution and exhibit the desired properties. Additionally, practitioners may employ visualization techniques to inspect the randomness of the generated numbers visually.

In some cases, random numbers come from unexpected places. In the field of casino gaming, there are local jurisdictions where only Bingo-type games are allowed. However, it is permissible to create an “overlay” on top of the Bingo game, giving the game the flavor of more traditional slot machines. In this case, Bingo serves as an underlying “random number generator.” One has to carefully construct a function to map outcomes in the Bingo game to outcomes in the overlay. For example, drawing B7, I21, G58, B12, and N34 in Bingo might correspond to landing 7-7-7 on a slot machine. But if done correctly, the mapping will provide a realistic experience in the new game.

While many people generate random numbers with computers, there are unusual cases where you have to do it in your head! In high-stakes poker, often the best strategy is to make a play that is randomly chosen according to some probability. For example, if you have bad cards at the end of a hand, you might bluff with a 25% probability. If you are not able to make these decisions randomly, your opponents will predict your decisions and gain an advantage. Similar decisions occur in sports. For example, in a soccer kickoff, to which side of the net should you kick? So how do you generate random numbers in your head? In Texas Hold’Em you can use the cards you are dealt. For example the probability of being dealt two red cards is roughly 25%.

We can say within the universe of RNGs there are 3 categories: - Pseudo RNG: The numbers are generated using a mathematical function in this category. The most common and simplest models are those based on the Modulo/Absolute Value Function, Linear Congruential Models, whether additive or multiplicative and other methods from Number Theory. - True RNG: Sources from nature are used to generate entropy, such as radioactive decay, DNA, and electrical impulses, for example. - Quantum RNG: We can say QRNGs are TRNGs that use the fundamental concepts of quantum physics (superposition and quantum tunneling, among others) as the basis for entropy. QRNGs are usually placed separately from TRNGs, although in principle they are also TRNGs.

In order to generate samples from a common distribution e.g. gaussian, uniform, exponential etc., there are widely available samplers, for instance in python thanks to the libraries numpy.random and scipy.stats. For more exotic distribution arising when doing e.g. bayesian inference and trying to compute the maximum a pesteriori, one needs to resort to more advanced methods such as Metropolis-Hastings, Gibbs or more generally Markov-Chain Monte Carlo (MCMC) methods. MCMC has the advantage of only needing the target probability density function up to a constant, the latter often being uncomputable. Some algorithms do not even require the knowledge of the likelihood, such as Approximate Bayesian Computing.

Yes, the two main fundamental aspects of random numbers are Independence and Uniformity. We can say within the universe of RNGs there are 3 categories: - PRNGs - TRNGs - QRNGs It's true that each category has its own distinctive aspects, and the purpose for which the random number generator will be used also influences the other aspects in addition to uniformity and independence. Therefore, when trying to add other aspects such as period size, ease of implementation, portability, disjoint sub-sequences, and others that are important, they become unnecessary when you want to approach the subject in a summarized way. In this case, if you want to know other aspects, we recommend subdividing the topics by their different RNG categories.

In my opinion, verifying the quality of random numbers is a continuous process. If the stakes are high in your Monte Carlo analysis, you may need to iterate and enhance your procedures. High-quality random numbers are the cornerstone of a trustworthy Monte Carlo simulation. You may improve the dependability of your simulation findings and make better informed judgments by selecting dependable RNGs, completing thorough checks for uniformity and distribution matching, and paying attention to aspects like autocorrelation.

In the case of MCMC sampling, if the autocorrelation decreases not too slowly, it is possible to improve independance by simply only keeping one every few samples, with the disadvantage that it is more computationally costly to obtain the same number of samples. To improve the generation process itself, it is beneficial to observe the acceptance rate and to use adaptive methods to try and keep it around 24%. If the target distribution is multimodal, a lot of classical algorithms will fail and one should consider parallel tempering or equi-energy sampling and the like.

There is a wide array of use cases to rngs: approximating integrals (e.g. averages, variances, probabilities over a set), maximum a posteriori in Bayesian inference, generative tasks (e.g. image, text generation), algorithm testing, expectation maximization...