Archive for Bernoulli distribution

MCMC for conditional Bernoullis

Posted in Books, Statistics, University life with tags , , , , , on February 22, 2021 by xi'an

Jeremy Heng, Pierre Jacob [currently in Paris!] and Nianqiao Ju are in a recent arXival considering the simulation of a conditional Bernoulli, namely generating a vector of N Bernoullis with different probabilities under the constraint that their sum is fixed as I. Rather than going for a perfect simulator, with cost O(NI), they opt for the simplest of MCMC samplers, where a 0 and a 1 entries are exchanged at random. In connection with a recent spate of MCMC works using coupling, they establish convergence in O(N log N) steps, even when the probabilities are arbitrarily close to zero and one. Including the case when they are Uniformly generated. From a mundane perspective, I wonder at the appeal of using the probabilities to select the exchange pair. I realise sorting the probabilities is already of order O(N log N) avoiding selecting highly probable 1’s and highly probable 0’s should speed up converge, unless the gain is negligible. And to link MCMC and exact simulation in this setting, what would the cost of perfect sampling by sampling from the past be? Presumably much higher since there is little chance a total ordering can be found on the starting states.

Bernoulli race particle filters

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , on March 27, 2019 by xi'an

Sebastian Schmon, Arnaud Doucet and George Deligiannidis have recently arXived an AISTATS paper with the above nice title. The motivation for the extension is facing intractable particle weights for state space models, as for instance in discretised diffusions.  In most cases, actually, the weight associated with the optimal forward proposal involves an intractable integral which is the predictive of the current observed variate given the past hidden states. And in some cases, there exist unbiased and non-negative estimators of the targets,  which can thus be substituted, volens nolens,  to the original filter. As in many pseudo-marginal derivations, this new algorithm can be interpreted as targeting an augmented distribution that involves the auxiliary random variates behind the unbiased estimators of the particle weights. A worthwhile remark since it allows for the preservation of the original target as in (8) provided the auxiliary random variates are simulated from the right conditionals. (At least ideally as I have no clue when this is feasible.)

“if Bernoulli resampling is per-formed, the variance for any Monte Carlo estimate will be the same as if the true weights were known and one applies standard multinomial resampling.”

The Bernoulli race in the title stands for a version of the Bernoulli factory problem, where an intractable and bounded component of the weight can be turned into a probability, for which a Bernoulli draw is available, hence providing a Multinomial sampling with the intractable weights since replacing the exact probability with an estimate does not modify the Bernoulli distribution, amazingly so! Even with intractable normalising constants in particle filters. The practicality of the approach may however be restricted by the possibility of some intractable terms being very small and requiring many rejections for one acceptance, as the number of attempts is a compound geometric. The intractability may add to the time request the drawback of keeping this feature hidden as well. Or force some premature interruption in the settings of a parallel implementation.

when perfect correlation just means… perfect!

Posted in Statistics with tags , , on February 6, 2018 by xi'an

When looking at an X validated question on generating two perfectly negatively correlated Bernoulli variates last week, my intuition was that one had to be the opposite of the other, which means their parameters had to sum up to one. Intuition that was plain easy to back up by solving the equation

corr(C¹,C²)=-1

in terms of the joint distribution of (C¹,C²). That perfect correlation implies strong constraints on the parameter of the Bernoulli is not highly surprising given its binary support. Although I had no time to pursue the issue, I idly wondered at the generalisation to, say, a Binomial case, i.e., whether or not this case still is the only possible one for the above to hold. But again a perfect correlation can only occur with perfect prediction, i.e., when the Binomial variates have the same number of trials and complementary probability parameters. (Of no particular relevance is the fact that the originator of the question preferred an answer that showed how to simulate two Bernoulli such that C¹+C²=1!)

a Bernoulli factory of sorts?

Posted in Books, Kids, Statistics with tags , , , , , on May 10, 2016 by xi'an

crane on Cockatoo Island, Sydney Harbour, Australia, July 15, 2012A nice question was posted on X validated as to figure out a way to simulate a Bernoulli B(q) variate when using only a Bernoulli B(p) generator. With the additional question of handling the special case q=a/b, a rational probability. This is not exactly a Bernoulli factory problem in that q does not write as f(p), but still a neat challenge. My solution would have been similar to the one posted by William Huber, namely to simulate a sequence of B(p) or B(1-p) towards zooming on q until the simulation of the underlying uniforms U allows us to conclude at the position of U wrt q. For instance, if p>q and X~B(p) is equal to zero, the underlying uniform is more than p, hence more than q, leading to returning zero for the B(q) generation. Else, a second B(p) or B(1-p) generation means breaking the interval (0,p) into two parts, one of which allows for stopping the generation, and so on. The solution posted by William Huber contains an R code that could be easily improved by choosing for each interval between p and (1-p) towards the maximal probability of stopping. I still wonder at the ultimate optimal solution that would minimise the (average or median) number of calls to the Bernoulli(p) generator.

standard distributions

Posted in Books, Kids, Statistics with tags , , , on February 5, 2016 by xi'an

Joram Soch managed to get a short note arXived about the Normal cdf Φ by exhibiting an analytical version, nothing less!!! By which he means a power series representation of that cdf. This is an analytical [if known] function in the complex calculus sense but I wonder at the point of the (re)derivation. (I do realise that something’s wrong on the Internet is not breaking news!)

Somewhat tangentially, this reminds me of a paper I read recently where the Geometric Geo(p) distribution was represented as the sum of two independent variates, namely a Binomial B(p/(1+p)) variate and a Geometric 2G(p²) variate. A formula that can be iterated for arbitrarily long, meaning that a Geometric variate is an infinite sum of [powers of two] weighted Bernoulli variates. I like this representation very much (although it may well have been know for quite a while). However I fail to see how to take advantage of it for simulation purposes. Unless the number of terms in the sum can be determined first. And even then it would be less efficient than simulating a single Geometric…

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