TTHE POISSON BINOMIAL DISTRIBUTION – OLD & NEW
WENPIN TANG AND FENGMIN TANG
Abstract.
This is an expository article on the Poisson binomial distribution. We reviewlesser known results and recent progress on this topic, including geometry of polynomials anddistribution learning. We also provide examples to illustrate the use of the Poisson binomialmachinery. Some open questions of approximating rational fractions of the Poisson binomialare presented.
Key words :
Distribution learning, geometry of polynomials, Poisson binomial distribution,Poisson/normal approximation, optimal transport, stochastic ordering, strongly Rayleighproperty. 1.
Introduction
The binomial distribution is one of the earliest examples a college student encountersin his/her first course in probability. It is a discrete probability distribution of a sum ofindependent and identically distributed (i.i.d.) Bernoulli random variables, modeling thenumber of occurrence of some events in repeated trials. An integer-valued random variable X is called binomial with parameters ( n, p ), denoted as X ∼ Bin( n, p ), if P ( X = k ) = (cid:0) nk (cid:1) p k (1 − p ) n − k , 0 ≤ k ≤ n . It is well known that if n is large, the Bin( n, p ) distribution isapproximated by the Poisson distribution for small p ’s, and is approximated by the normaldistribution for larger values of p . See e.g. [77] for an educational tour.Poisson [81] considered a more general model of independent trials, which allows hetero-geneity among these trials. Precisely, an integer-valued random variable X is called Poissonbinomial, and denoted as X ∼ PB( p , . . . , p n ) if X ( d ) = ξ + · · · + ξ n , where ξ , . . . , ξ n are independent Bernoulli random variables with parameters p , . . . , p n . Itis easily seen that the probability distribution of X is P ( X = k ) = (cid:88) A ∈ [ n ] , | A | = k (cid:32)(cid:89) i ∈ A p i (cid:89) i/ ∈ A (1 − p i ) (cid:33) , (1.1)where the sum ranges over all subset of [ n ] := { , . . . , n } of size k .The Poisson binomial distribution has a variety of applications such as reliability analysis[16, 57], survey sampling [29, 104], finance [40, 92], and engineering [44, 100]. Though thistopic has been studied for a long time, the literature is scattered. For instance, the Poissonbinomial distribution has different names in various contexts: P´olya frequency (PF) distribu-tion, strongly Rayleigh distribution, convolutions of heterogenous Bernoulli, etc. Researchersoften work on some aspects of this subject, and ignore its connections to other fields. In late Date : August 28, 2019. a r X i v : . [ m a t h . P R ] A ug WENPIN TANG AND FENGMIN TANG
Distributional properties of Poisson binomial variables
In this section, we review a few distributional properties of the Poisson binomial distribu-tion. For X ∼ PB( p , . . . , p n ), we have µ := E X = n ¯ p and σ := Var X = n ¯ p (1 − ¯ p ) − n (cid:88) i =1 ( p i − ¯ p ) , (2.1)where ¯ p := (cid:80) ni =1 p i /n . It is easily seen that by keeping E X (or ¯ p ) fixed, the variance of X isincreasing as the set of probabilities { p , . . . , p n } gets more homogeneous, and is maximizedas p = · · · = p n . There is a simple interpretation in survey sampling: taking samples fromdifferent communities ( stratified sampling ) is better than taking from the same group ( simplerandom sampling ).The above observation motivates the study of stochastic orderings for the Poisson binomialdistribution. The first result of this kind is due to Hoeffding [53], claiming that among allPoisson binomial distributions with a given mean, the binomial distribution is the mostspread-out. Theorem 2.1. [53] (Hoeffding’s inequalities) Let X ∼ PB( p , . . . , p n ) , and ¯ X ∼ Bin( n, ¯ p ) .(1) There are inequalities P ( X ≤ k ) ≤ P ( ¯ X ≤ k ) for ≤ k ≤ n ¯ p − , and P ( X ≤ k ) ≥ P ( ¯ X ≤ k ) for n ¯ p ≤ k ≤ n. (2) For any convex function g : [ n ] → R in the sense that g ( k + 2) − g ( k + 1) + g ( k ) > , ≤ k ≤ n − , we have E g ( X ) ≤ E g ( ¯ X ) , where the equality holds if and only if p = · · · = p n = ¯ p . The part (2) in Theorem 2.1 indicates that among all Poisson binomial distributions, thebinomial is the largest one in convex order. This result was extended to the multidimensional
OISSON BINOMIAL 3 setting [9], and to non-negative random variables [8, Proposition 3.2]. See also [68] forinterpretations. Next we give several applications of Hoeffding’s inequalities.
Examples 2.2. (1) Monotonicity of binomials.
Fix λ >
0. By taking ( p , . . . , p n ) = (0 , λn − , . . . , λn − ),we get for X ∼ Bin( n − , λn − ) and X (cid:48) ∼ Bin( n, λn ), P ( X ≤ k ) < P ( X (cid:48) ≤ k ) for k ≤ λ − P ( X ≤ k ) > P ( X (cid:48) ≤ k ) for k ≥ λ. Similarly, by taking ( p , . . . , p n ) = (1 , λ − n − , . . . , λ − n − ), we get for X ∼ Bin( n − , λ − n − )and X (cid:48) ∼ Bin( n, λn ), P ( X ≤ k − < P ( X (cid:48) ≤ k ) for k ≤ λ − P ( X ≤ k − > P ( X (cid:48) ≤ k ) for k ≥ λ. These inequalities were used in [3] to derive the monotonicity of error in approximat-ing the binomial distribution by a Poisson distribution. By letting X ∼ Bin( n, p ) and Y ∼ P oi ( np ), they proved P ( X ≤ k ) − P ( Y ≤ k ) is positive if k ≤ n p/ ( n + 1) and isnegative if k ≥ np . The result quantifies the error of confidence levels in hypothesistesting when approximating the binomial distribution by a Poisson distribution. (2) Darroch’s rule. It is well known that a Poisson binomial variable has either one,or two consecutive modes. By an argument in the proof of Hoeffding’s inequalities,Darroch [32, Theorem 4] showed that the mode m of the Poisson binomial distributiondiffers from its mean µ by at most 1. Precisely, he proved that m = k if k ≤ µ < k + k +2 ,k or k + 1 if k + k +2 ≤ µ ≤ k + 1 − n − k +1 ,k + 1 if k + 1 − n − k +1 < µ ≤ k + 1 . (2.2)This result was reproved in [91]. See also [60] for a similar result concerning themedian. (3) Azuma-Hoeffding inequality. By the Azuma-Hoeffding inequality [5, 54], for ξ , . . . , ξ n independent random variables such that 0 ≤ ξ i ≤ P (cid:32) n (cid:88) i =1 ξ i ≥ t (cid:33) ≤ (cid:16) µt (cid:17) t (cid:18) n − µn − t (cid:19) n − t for t > µ, (2.3)where µ := (cid:80) ni =1 E ξ i . Now we show how to derive a version of (2.3) via a Poissonbinomial trick. Given ξ , . . . , ξ n , let b i be independent Bernoulli with parameter ξ i and X ∼ Bin (cid:0) n, n (cid:80) ni =1 ξ i (cid:1) . We have P (cid:32) n (cid:88) i =1 b i ≥ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:88) i =1 ξ i ≥ t (cid:33) ≤ P ( (cid:80) ni =1 b i ≥ t ) P ( (cid:80) ni =1 ξ i ≥ t ) . (2.4)Given (cid:80) ni =1 ξ i ≥ t , (cid:80) ni =1 b i is Poisson binomial with mean greater than t . Accordingto Hoeffding’s inequality, P (cid:32) n (cid:88) i =1 b i ≥ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:88) i =1 ξ i ≥ t (cid:33) ≥ P (cid:32) X ≥ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:88) i =1 ξ i ≥ t (cid:33) ≥ c, (2.5)for some universal constant c >
0. Combining (2.4) and (2.5) yields P ( (cid:80) ni =1 ξ i ≥ t ) ≤ c P ( (cid:80) ni =1 b i ≥ t ). Note that (cid:80) ni =1 b i is Poisson binomial with mean µ . Applying WENPIN TANG AND FENGMIN TANG
Hoeffding’s inequality to (cid:80) ni =1 b i with bounds for binomial tails [71], we get P (cid:32) n (cid:88) i =1 b i ≥ t (cid:33) ≤ (cid:16) µt (cid:17) t (cid:18) n − µn − t (cid:19) n − t for t ≥ µ + 1 . (2.6)As a consequence, P ( (cid:80) ni =1 ξ i ≥ t ) ≤ c (cid:0) µt (cid:1) t (cid:16) n − µn − t (cid:17) n − t which achieves the same rateas in (2.3) up to a constant factor.The original proof of Theorem 2.1 was brute-force, and it was soon generalized by using theidea of majorization and Schur convexity . To proceed further, we need some vocabularies.Let { x (1) , . . . , x ( n ) } be the order statistics of { x , . . . , x n } . Definition 2.3.
The vector xxx is said to majorize the vector yyy , denoted as xxx (cid:23) yyy , if k (cid:88) i =1 x ( i ) ≤ k (cid:88) i =1 y ( i ) for k ≤ n − and n (cid:88) i =1 x ( i ) = n (cid:88) i =1 y ( i ) . See [66] for background and development on the theory of majorization and its applications.The following theorem gives a few lesser known variants of Hoeffding’s inequalities.
Theorem 2.4.
Let X ∼ PB( p , . . . , p n ) , X (cid:48) ∼ PB( p (cid:48) , . . . , p (cid:48) n ) and Y ∼ Bin( n, p ) .(1) [48, 104] If ( p , . . . , p n ) (cid:23) ( p (cid:48) , . . . , p (cid:48) n ) , then P ( X ≤ k ) ≤ P ( X (cid:48) ≤ k ) for ≤ k ≤ n ¯ p − , and P ( X ≤ k ) ≥ P ( X (cid:48) ≤ k ) for n ¯ p + 2 ≤ k ≤ n. Moreover,
Var( X ) ≤ Var( X (cid:48) ) .(2) [80] If ( − log p , . . . , − log p n ) (cid:23) ( − log p (cid:48) , . . . , − log p (cid:48) n ) , then X is stochastically largerthan X (cid:48) , i.e. P ( X ≥ k ) ≤ P ( X (cid:48) ≥ k ) for all k .(3) [17] X is stochastically larger than Y if and only if p ≤ ( (cid:81) ni =1 p i ) n , and X is stochas-tically smaller than Y if and only if p ≥ − ( (cid:81) ni =1 (1 − p i )) n . Consequently, if ( (cid:81) ni =1 p i ) n ≥ − ( (cid:81) ni =1 (1 − p (cid:48) i )) n then X is stochastically larger than X (cid:48) . The proof of Theorem 2.4 relies on the fact that xxx (cid:23) yyy implies the components of xxx are morespread-out than those of yyy . For example in part (1), it boils down to proving if k ≤ n ¯ p − P ( X ≤ k ) is a Schur concave function in ppp , meaning its value increases as the componentsof ppp are less dispersed. The part (3) gives a sufficient condition of stochastic orderings forthe Poisson binomial distribution. A simple necessary and sufficient condition remains open.See also [15, 16, 18, 52, 93, 107] for further results.3. Approximation of Poisson binomial distributions
In this section, we discuss various approximations of the Poisson binomial distribution.Pitman [78, Section 2] gave an excellent survey on this topic in the mid-90’s. We complementthe discussion with recent developments. In the sequel, L ( X ) denotes the distribution of arandom variable X . OISSON BINOMIAL 5
Poisson approximation . Le Cam [64] gave the first error bound for Poisson approximationof the Poisson binomial distribution. The following theorem is an improvement of Le Cam’sbound.
Theorem 3.1. [7]
Let X ∼ PB( p , . . . , p n ) and µ := (cid:80) ni =1 p i . Then
132 min (cid:18) , µ (cid:19) n (cid:88) i =1 p i ≤ d T V ( L ( X ) , Poi( µ )) ≤ − e − µ µ n (cid:88) i =1 p i , (3.1) where d T V ( · , · ) is the total variation distance. It is easily seen from (3.1) that the Poisson approximation of the Poisson binomial is goodif (cid:80) ni =1 p i (cid:28) (cid:80) ni =1 p i , or equivalently µ − σ (cid:28) µ . There are two cases: • For small µ , the upper bound in (3.1) is sharp. • For large µ , the approximation error is of order (cid:80) ni =1 p i / (cid:80) ni =1 p i .As pointed out in [59], the constant 1 /
32 in the lower bound can be improved to 1 /
14. See[6] for a book-length treatment, and [86] for sharp bounds. A powerful tool to study theapproximation of the sum of (possibly dependent) random variables is Stein’s method ofexchangeable pairs, see [26]. For instance, a simple proof of the upper bound in (3.1) wasgiven in [26, Section 3] via the Stein machinery.The Poisson approximation can be viewed as a mean-matching procedure. The failure ofthe Poisson approximation is due to a lack of control in variance. A typical example is whereall p i ’s are bounded away from 0, so that µ is large and (cid:80) ni =1 p i / (cid:80) ni =1 p i is of constantorder. To deal with these cases, R¨ollin [85] proposed a mean/variance-matching procedure.To present further results, we need the following definition. Definition 3.2.
An integer-valued random variable X is said to be translated Poisson dis-tributed with parameters ( µ, σ ) , denoted as TP( µ, σ ) , if X − µ + σ + { µ − σ } ∼ Poi( σ + { µ − σ } ) , where {·} is the fraction part of a positive number. It is easy to see that a TP( µ, σ ) random variable has mean µ , and variance σ + { µ + σ } which is between σ and σ +1. The following theorem gives an upper bound in total variationbetween a Poisson binomial variable and its translated Poisson approximation. Theorem 3.3. [85]
Let X ∼ PB( p , . . . , p n ) , and µ := (cid:80) ni =1 p i and σ := (cid:80) ni =1 p i (1 − p i ) .Then d T V ( L ( X ) , TP( µ, σ )) ≤ (cid:113)(cid:80) ni =1 p i (1 − p i ) σ , (3.2) where d T V ( · , · ) is the total variation distance. Note that if all p i ’s are bounded away from 0 and 1, the approximation error is of order1 / √ n which is optimal. See [70] for the most up-to-date results of the Poisson approximation.Now we give an application of translated Poisson approximation in observational studies. Example 3.4.
Sensitivity analysis.
In matched-pair observational studies, an sensitivityanalysis accesses the sensitivity of results to hidden bias. Here we follow a modern approachof Rosenbaum [88, Chapter 4]. Precisely, the sample consists of n matched pairs and unitsin each pair are indexed by i = 1 ,
2. Each pair k = 1 , . . . , n is matched on a set of observedcovariates xxx k = xxx k , and only one unit in each pair receives the treatment. Let Z ki be the WENPIN TANG AND FENGMIN TANG treatment assignment, so Z k + Z k = 1. Common test statistics for matched pairs are sign-score statistics of the form: T = (cid:80) nk =1 d k ( c k Z k + c k Z k ), where d k ≥ c ki ∈ { , } .For simplicity, we take d k = 1 and the statistics of interest are T = n (cid:88) k =1 ( c k Z k + c k Z k ) , (3.3)where c k Z k + c k Z k is Bernoulli distributed with parameter p k := c k π k + c k (1 − π k ) with π k := P ( Z k = 1 | Z k + Z k = 1). So T ∼ PB( p , . . . , p n ). For 1 ≤ k ≤ n , let Γ k := π k / (1 − π k ),which equals to 1 if there is no hidden bias.The goal is to make inference on T with different choices of ( π , . . . , π n ) and understandwhich choices explain away the conclusion we draw from the null hypothesis (i.e. there is nohidden bias). Thus, we are interested in the set R ( t, α ) := { ( π , . . . , π n ) : P ( T ≥ t ) ≤ α } , on the boundary of which the conclusion assuming no hidden bias is turned over. However,direct computation of R ( t, α ) seems hard. A routine way to solve this problem is to ap-proximate R ( t, α ) by a regular shape. To this end, we consider the following optimizationproblem: max Γ ,s.t. max πππ ∈ C Γ P ( T ( π , . . . , π n ) ≥ t ) ≤ α, (3.4)where C Γ is a constraint region. For instance, C Γ := { πππ : ≤ π k ≤ Γ1+Γ } corresponds tothe worst-case sensitivity analysis. By the translated Poisson approximation, the quantitymax πππ ∈ C Γ P ( T ( π , . . . , π n ) ≥ t ) can be evaluated by the following problem which is easy tosolve. min A ∈{ ,...,K } min πππ ∈ C Γ K (cid:88) k =0 λ k e − λ k ! s.t. K = t − A, λ = n (cid:88) k =1 p k − A, A ≤ n (cid:88) k =1 p k < A + 1 . (3.5) Normal approximation . The normal approximation of the Poisson binomial distributionfollows from Lyapunov or Lindeberg central limit theorem, see e.g. [11, Section 27]. Berryand Esseen independently discovered an error bound in terms of the cumulative distribu-tion function for the normal approximation of the sum of independent random variables.Subsequent improvements were obtained by [72, 75, 94, 102] via Fourier analysis, and by[27, 28, 67, 101] via Stein’s method.Let φ ( x ) := √ π exp (cid:0) − x / (cid:1) be the probability density function of the standard normal,and Φ( x ) := (cid:82) x −∞ φ ( y ) dy be its cumulative distribution function. The following theroemprovides uniform bounds for the normal approximation of Poisson binomial variables. Theorem 3.5.
Let X ∼ PB( p , . . . , p n ) , and µ := (cid:80) ni =1 p n and σ := (cid:80) ni =1 p i (1 − p i ) .(1) [79, Theorem 11.2] There is a universal constant
C > such that max ≤ k ≤ n (cid:12)(cid:12)(cid:12)(cid:12) P ( X = k ) − φ (cid:18) k − µσ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cσ . (3.6)
OISSON BINOMIAL 7 (2) [94]
We have max ≤ k ≤ n (cid:12)(cid:12)(cid:12)(cid:12) P ( X ≤ k ) − Φ (cid:18) k − µσ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ . σ . (3.7)Other than uniform bounds (3.6)-(3.7), several authors [14, 49, 84] studied error boundsfor the normal approximation in other metrics. For µ , ν two probability measures, consider • L p metric d p ( µ, ν ) := (cid:18)(cid:90) ∞−∞ | µ ( −∞ , x ] − ν ( −∞ , x ] | p dx (cid:19) p , • Wasserstein’s p metric W p ( µ, ν ) := inf π (cid:18)(cid:90) ∞−∞ (cid:90) ∞−∞ | x − y | p π ( dxdy ) (cid:19) p , where the infimum runs over all probability measures π on R × R with marginals µ and ν .Specializing these bounds to the Poisson binomial distribution, we get the following result. Theorem 3.6.
Let X ∼ PB( p , . . . , p n ) , and µ := (cid:80) ni =1 p n and σ := (cid:80) ni =1 p i (1 − p i ) .(1) [76, Chapter V] There exists a universal constant
C > such that d p ( L ( X ) , N ( µ, σ )) ≤ Cσ for all p ≥ . (3.8) (2) [14, 84] For each p ≥ , there exists a constant C p > such that W p ( L ( X ) , N ( µ, σ )) ≤ C p σ . (3.9)Goldstein [49] proved L p bound (3.8) for p = 1 with C = 1. The general case follows fromthe inequality d p ( µ, ν ) p ≤ d ∞ ( µ, ν ) p − d ( µ, ν ) together with Goldstein’s L bound and theuniform bound (3.7). By the Kantorovich-Rubinstein duality, d ( µ, ν ) = W ( µ, ν ). So thebound (3.9) holds for p = 1 with C = 1. For general p , the bound (3.9) is a consequence ofthe fact that for Z = (cid:80) ni =1 ξ i with ξ i ’s independent, E ξ i = 0 and (cid:80) ni =1 Var( ξ i ) = 1, W p ( L ( Z ) , N (0 , ≤ C p (cid:32) n (cid:88) i =1 E | Z i | p +1 (cid:33) p . This result was proved in [84] for 1 ≤ p ≤
2, and generalized to all p ≥ Binomial approximation . The binomial approximation of the Poisson binomial is lesserknown. The first result of this kind is due to Ehm [41] who proved that for X ∼ PB( p , . . . , p n ), d T V ( L ( X ) , Bin( n, µ/n )) ≤ − ( µ/n ) n +1 − (1 − µ/n ) n +1 ( n + 1)(1 − µ/n ) µ/n n (cid:88) i =1 ( p i − µ/n ) . (3.10)Elm’s approach was extended to a Krawtchouk expansion in [87]. The advantage of thebinomial approximation over the Poisson approximation is justified by the following resultdue to Choi and Xia [31]. WENPIN TANG AND FENGMIN TANG
Theorem 3.7.
Let X ∼ PB( p , . . . , p n ) , and µ := (cid:80) ni =1 p n . For m ≥ , let d m := d T V ( L ( X ) , Bin( m, µ/m )) . Then for m sufficiently large, d m < d m +1 < · · · < d T V ( L ( X ) , Poi( µ )) . (3.11)See also [6, 73] for multi-parameter binomial approximations, and [95] for the P´olya approx-imation of the Poisson binomial distribution.4. Poisson binomial distributions, polynomials with nonnegative coefficientsand optimal transport
In this section, we discuss aspects of the Poisson binomial distribution related to poly-nomials with nonnegative coefficients. For X ∼ PB( p , . . . , p n ), the probability generatingfunction (PGF) of X is f ( u ) := E X u = n (cid:89) i =1 ( p i u + 1 − p i ) . (4.1)It is easy to see that f is a polynomial with all nonnegative coefficients, and all of its rootsare real negative. The story starts with the following remarkable theorem, due to Aissen,Endrei, Schoenberg and Whitney [1, 2]. Theorem 4.1. [1, 2]
Let ( a , . . . , a n ) be a sequence of nonnegative real numbers, with asso-ciated generating polynomial f ( z ) := (cid:80) ni =0 a i z i . The following conditions are equivalent:(1) The polynomial f ( z ) has only real roots.(2) The sequence ( a /f (1) , . . . , a n /f (1)) is the probability distribution of a PB( p , . . . , p n ) distribution for some p i . The real roots of f ( z ) are − (1 − p i ) /p i for i with p i > .(3) The sequence ( a , . . . , a n ) is a P´olya frequency (PF) sequence, i.e. the Toeplitz matrix ( a j − i ) i,j is totally nonnegative. See [4] for background on total positivity. From a computational aspect, the condition (3)amounts to solving a system of n ( n − / Corollary 4.2.
A random variable X ∼ PB( p , . . . , p n ) for some p i if and only if X isstrongly Rayleigh on { , . . . , n } . In the sequel, we use the terminologies ‘Poisson binomial’ and ‘strongly Rayleigh’ inter-changeably. Call a polynomial f ( z ) = (cid:80) ni =0 a i z i with a i ≥ OISSON BINOMIAL 9
For n ≥
5, it is hopeless to get any ‘simple’ necessary and sufficient condition for f to bestrongly Rayleigh due to Abel’s impossibility theorem. A necessary condition for f to bestrong Rayleigh is the Newton’s inequality: a i ≥ a i − a i +1 (cid:18) i (cid:19) (cid:18) n − i (cid:19) , ≤ i ≤ n − , (4.2)The sequence ( a i ; 0 ≤ i ≤ n ) satisfying (4.2) is also said to be ultra-logconcave [74]. Conse-quently, ( a i ; 0 ≤ i ≤ n ) is logconcave and unimodal. A lesser known sufficient condition isgiven in [58, 63]: a i > a i − a i +1 . ≤ i ≤ n − . (4.3)See also [50, 61] for various generalizations. As observed in [62], the inequality (4.3) cannot beimproved since the sequence ( m i ; i ≥
0) defined by m i := inf (cid:110) a i a i − a i +1 ; f is strong Rayleigh (cid:111) decreases from m = 4 to its limit approximately 3 . X is a strong Rayleigh, or Poisson binomialrandom variable, how well can one approximate jX/k for each j, k ≥ j = 1was solved in that paper. Theorem 4.3. [47]
Let X be a strongly Rayleigh random variable. Then (cid:4) Xk (cid:5) is stronglyRayleigh for each k ≥ , where (cid:98) x (cid:99) is the integer part of x . The key to the proof of Theorem 4.3 is [47, Theorem 4.3]: For f a polynomial of degree n and k ≥
1, write f ( z ) = (cid:80) k − j =0 x j g j ( z k ), with g j a polynomial of degree (cid:98) n − jk (cid:99) . The theoremasserts that if f is strongly Rayleigh, then so are g i ’s with interlacing roots. In fact, thereal-rootedness follows from the fact that( a n ; n ≥
0) is a P´olya frequency sequence = ⇒ ( a kn + j ; n ≥
0) is a P´olya frequency sequence , for each k ≥ ≤ j < k . This result is well known, see [1, Theorem 7] or [22, Theorem3.5.4]. But the root interlacing seems less obvious by P´olya frequency sequences.A natural question is whether (cid:98) jX/k (cid:99) is strongly Rayleigh for each j, k ≥
1. It turnsout that (cid:98) X/ (cid:99) can be far away from being strongly Rayleigh. In fact, one can prove thefollowing theorem. Theorem 4.4.
Let X ∼ Bin(3 n, / , and z i be the roots of the probability generating func-tion of (cid:98) X/ (cid:99) . Then max i {(cid:61) ( z i ) } ≥ (cid:114) n − n − , (4.4) where (cid:61) ( z ) is the imaginary part of z . The reason why some roots of the PGF of (cid:98) X/ (cid:99) have large positive imaginary parts is dueto the unbalanced allocation of probability weights to even and odd numbers: P (cid:0)(cid:4) X (cid:5) = 2 k (cid:1) = (cid:0) n +13 k +1 (cid:1) while P (cid:0)(cid:4) X (cid:5) = 2 k + 1 (cid:1) = (cid:0) n k +2 (cid:1) . So the Newton’s inequality (4.2) is not satisfied. Optimal transport . For simplicity, we consider X ∼ Bin(3 n, / Y which is strongly Rayleigh on { , , . . . , n } such that sup | Y − X/ | is as smallas possible. Now we provide a formulation of this problem via optimal transport. For µ , ν two probability measures, define W ∞ ( µ, ν ) := inf γ ∈ π ( µ,ν ) { γ − ess sup | x − y |} , (4.5)where π ( µ, ν ) is the set of couplings of µ and ν . The metric W ∞ ( · , · ) is known as the ∞ -Wasserstein distance, see [103]. A coupling γ which achieves the infimum (4.5) is called anoptimal transference plan. By abuse of notation, write W ∞ ( X, Y ) for X ∼ µ , Y ∼ ν . Wewant to solve the following optimization problem:Acc (cid:18) X (cid:19) := inf (cid:26) W ∞ (cid:18) X , Y (cid:19) ; Y is strongly Rayleigh on { , , . . . , n } (cid:27) . (4.6)Here Acc(2 X/
3) stands for the accuracy of strongly Rayleigh approximations to 2 X/
3. So thesmaller the value of Acc(2 X/
3) is, the better the approximation is. In [47], it was conjecturedthat Acc(2 X/
3) = O (1). The problem (4.6) can be divided into two stages:(1) Given the distribution of Y , find an optimal transference plan Y = φ (2 X/
3) withpossibly random φ . This is the Monge(-Kantorovich) problem.(2) Find Y among all strongly Rayleigh distributions on { , , . . . , n } which achieves theinfimum of W ∞ (2 X/ , Y ).It might be difficult to solve the problem (4.6) explicitly, but one can obtain a good upperbound by constructing a suitable transference plan. For example, the transference plan belowshows that for X ∼ Bin(9 , / X/ Y ∼ Bin(6 , / W ∞ (2 X/ , Y ) ≤
1. This implies that Acc (2 X/ ≤ X ∼ Bin(9 , / X/
3) with X ∼ Bin( n, /
2) for small n ’s. Figure 1.
A transference plan from Bin(9 , /
2) to Bin(6 , / n, p ) random variable forany p can approximate 2 X/
3. Unfortunately, the approximation is not so good as proved inthe following proposition.
OISSON BINOMIAL 11
Proposition 4.5.
Let X ∼ Bin(3 n, / , and Y ∼ Bin(2 n, p ) for ≤ p ≤ . Then thereexists C p > such that W ∞ (cid:18) X , Y (cid:19) ≥ C p n for large n. (4.7) Proof.
The extreme cases p = 0 , < p <
1. Considertransfer from 2 X/ { Y = 0 } with probability mass (1 − p ) n . By definition of W ∞ , W ∞ (cid:18) X , Y (cid:19) ≥ inf (cid:40) k ; (1 − p ) n ≤ n k (cid:88) i =0 (cid:18) ni (cid:19)(cid:41) . It is well known that for any λ < / (cid:80) λni = o (cid:0) ni (cid:1) = 2 nH ( λ )+ o ( n ) , where H ( λ ) := − λ log ( λ ) − (1 − λ ) log (1 − λ ). It follows from standard analysis that for p < − / √ W ∞ (cid:0) X , Y (cid:1) ≥ λ p n , where λ p is the unique solution on [0 , /
2) to the equation H ( λ ) = log (1 − p ) + 1.Similarly by considering transfer from 2 X/ { Y = 2 n } with probability mass p n , weget for p > / √ W ∞ (cid:0) X , Y (cid:1) ≥ λ p n , where λ p is the unique solution on [0 , /
2) to theequation H ( λ ) = log ( p ) + 1. We take C p to be 3 λ p for p ≥ /
2, and 3 λ p for p < / (cid:3) The problem requires finding ( p , . . . , p n ) ∈ [0 , n such that W ∞ (2 X/ , PB( p , . . . , p n ))is small. By Proposition 4.5, the values of p , . . . , p n cannot be all too small or too large.Precisely, there exist i ∈ [2 n ] such that p i > / √
8, and j ∈ [2 n ] such that p j < − / √ p i = i n +1 for i ∈ [2 n ]. By letting Y ∼ PB(1 / (2 n + 1) , . . . , n/ (2 n + 1)), we get E (2 X/
3) = E Y = 2 n and Var (cid:0) X (cid:1) ∼ Var Y ∼ n/ W ∞ (cid:18) X, Y (cid:19) ≥ inf (cid:40) k ; n (cid:89) i =1 i n + 1 ≤ (cid:80) ki =1 (cid:0) ni (cid:1) n (cid:41) ≥ inf (cid:40) k ; (cid:18) e (cid:19) n ≤ k (cid:88) i =1 (cid:18) ni (cid:19)(cid:41) = 3 λ eq n, where λ eq ≈ . , /
2) to the equation H ( λ ) = 1 − log ( e ).Still the approximation is not good, but much better than the Bin(2 n, p ) approximation. Open problem 4.6.
Is there a random variable Y ∼ PB( p , . . . , p n ) such that W ∞ (2 X/ , Y ) is of order o ( n ) ? What is the lower bound of Acc(2 X/ ? Coefficients of Poisson binomial PGF . For simplicity, we take X ∼ Bin(3 n − , / (cid:98) X/ (cid:99) does not satisfy the Newton’s inequality.It is interesting to ask the following: can we find ( a , . . . , a n − ) ∈ R n + such that a k + a k +1 = (cid:18) n − k (cid:19) + (cid:18) n − k + 1 (cid:19) + (cid:18) n − k + 2 (cid:19) for k ∈ [ n − , (4.8)and the polynomial P ( x ) := (cid:80) n − k =1 a k x k has all real roots ? If we are able to find such( a , . . . , a n − ), then Acc(2 X/ ≤ / a , . . . , a n − ) must satisfy the Newton’s inequality and thus is unimodal. See also [90] forhigher order Newton’s inequalities. According to (4.8), a + a = Θ( n ), meaning that a + a ∼ Cn for some C >
0. If a = Θ( n ), then the condition a ≥ a a implies that a = O ( n ). Further the condition a ≥ a a gives that a = O ( n ). Consequently, a + a = O ( n ) which contradicts thefact that a + a = Θ( n ). So we have a = o ( n ) and a = Θ( n ). A similar argumentshows that for any fixed k , a k = o ( n k +2 ) and a k +1 = Θ( n k +2 ). It can be shown that a k = Θ( n k ) for any fixed k . But the choice for the bulk terms such as a n − , a n is a moresubtle issue since the terms (cid:0) n (cid:98) n/ (cid:99)− (cid:1) , (cid:0) n (cid:98) n/ (cid:99) (cid:1) and (cid:0) n (cid:98) n/ (cid:99) +1 (cid:1) are comparable.In Appendix A, we see that Acc(2 X/
3) = 1 / n = 1, and Acc(2 X/
3) = 2 / n = 2.Further we get, • n = 3: Acc(2 X/
3) = 2 /
3, achieved by a strongly Rayleigh variable with PGF12 (3 + 34 x + 91 x + 91 x + 34 x + 3 x ) . • n = 4: Acc(2 X/
3) = 2 /
3, achieved by a strongly Rayleigh variable with PGF12 (4 + 63 x + 310 x + 647 x + 647 x + 310 x + 63 x + 4 x ) . • n = 5: Acc(2 X/
3) = 2 /
3, achieved by a strongly Rayleigh variable with PGF12 (4 + 102 x + 760 . x + 2606 . x + 4719 x + 4719 x + 2606 . x + 760 . x + 102 x + 4 x ) . From small n cases, we speculate there is a strongly Rayleigh polynomial P ( x ) whosecoefficients satisfy (4.8) and the symmetric/self-reciprocal condition: a k = a n − − k for k ∈ [ n − . (4.9)Such polynomials are instances of Λ -polynomials [23], whose coefficients are symmetric andunimodal. In general, for each n ≥ n − Q k ∈ Z [ a , · · · , a n ] such that the polynomial with real coefficients P ( x ) has only real roots if andonly if Q k ≥ k . These Q k ’s can be constructed as the leading coefficients of theSturm’s sequence of P , see e.g. [99, Section 1.3]. They are also the subresultants of theSylvester matrix of P and P (cid:48) up to sign changes. In other words, we try to find whether theset S := { ( a , . . . , a n − ) ∈ R + : (4.8) , (4.9) hold and Q k ≥ k } is empty or not. The set S is semi-algebraic. According to Stengle’s Positivstellensatz [98],the non-emptiness of S is equivalent to − / ∈ C ( Q , . . . , Q n − )+ I (cid:18) a k + a k +1 − (cid:18) n − k (cid:19) − (cid:18) n − k + 1 (cid:19) − (cid:18) n − k + 2 (cid:19) , a k − a n − − k (cid:19) , where C is the cone and I is the ideal. However, the size of the polynomials Q k growsvery fast, and hence exact computations become impossible. See also [69, 83] for relateddiscussions. Hurwitz stability . Recently, Liggett [65] proved an interesting result of (cid:98) X/ (cid:99) for X astrongly Rayleigh variable. Theorem 4.7. [65]
Let X be a strongly Rayleigh random variable. Then the PGF of (cid:98) X/ (cid:99) is Hurwitz stable. That is, all its roots have negative real parts. OISSON BINOMIAL 13
The idea is to write the PGF of (cid:98) X/ (cid:99) as g ( x )+ xg ( x ), where g and g have interlacingroots. By the Hermite-Biehler theorem [10, 51], such polynomials are Hurwitz stable. Thismeans that the PGF of (cid:98) X/ (cid:99) can be factorized into polynomials with positive coefficientsof degrees no greater than 2. Thus, (cid:98) X/ (cid:99) is a Poisson multinomial variable, that is thesum of independent random variables with values in { , , } . In general, it can be shownthat (cid:98) jX/k (cid:99) is expressed as g ( x j ) + xg ( x j ) + · · · + x j − g j − ( x j ) , (4.10)where g . . . g j − have simple interlacing roots. We conjecture the following. Conjecture 4.8.
Let X be a strong Rayleigh random variable. Then (cid:98) jX/k (cid:99) is the sum ofindependent random variables with values in { , , . . . , j } . Equivalently, the PGF of (cid:98) jX/k (cid:99) can be factorized into polynomials with positive coefficients of degrees no greater than j . Let P j be the set of polynomials with positive coefficients which can be factorized intopolynomials with positive coefficients of degrees no greater than j , and Q j be the set ofpolynomials which satisfies (4.10). From the above discussion, P = Q and P = Q . Butneither implication between P and Q is true, as the following examples in [65] show: • Let f ( z ) = z + z + z + 2 z + z + . The roots of f are z , ¯ z , z , ¯ z and w with values z = 0 .
725 + 0 . i , z = 0 .
435 + 1 . i and w = 0 . z − z )( z − ¯ z )( z − w ) = 0 .
623 + 1 . z − . z + z , so f / ∈ P . But the roots of h , h , h are − , − , − f ∈ Q . • Let f ( z ) = (1 + z + 2 z )(25 + z + 2 z ) = 25 + 25 z + 51 z + 3 z + 4 z + 4 z , which isin P , However, f / ∈ Q since the roots of h , h , h are − , − , − respectively.See also [24, 106, 108] for discussion of positive factorizations of small degree polynomials.5. Computations of Poisson binomial distributions
In this section we discuss a few computational issues of learning and computing the Poissonbinomial distribution.
Learning the Poisson binomial distribution . Distribution learning is an active domainin both statistics and computer science. Following [36], given access to independent samplesfrom an unknown distribution P , an error control (cid:15) > δ >
0, alearning algorithm outputs an estimation (cid:98) P such that P ( d T V ( (cid:98) P , P ) ≤ (cid:15) ) ≥ − δ . The perfor-mance of a learning algorithm is measured by its sample complexity and its computationalcomplexity.For X ∼ PB( p , . . . , p n ), this amounts to finding a vector ( (cid:98) p , . . . , (cid:98) p n ) defining (cid:98) X ∼ PB( (cid:98) p , . . . , (cid:98) p n ) such that d T V ( (cid:98) X, X ) is small with high probability. This is often calledproper learning of Poisson binomial distributions. Building upon previous work [12, 35, 85],Daskalakis, Diakonikolas and Servedio [34] established the following result for proper learningof Poisson binomial distributions.
Theorem 5.1. [34]
Let X ∼ PB( p , . . . , p n ) with unknown p i ’s. There is an algorithm suchthat given (cid:15), δ > , it requires • (sample complexity) O (1 /(cid:15) ) · log(1 /δ ) independent samples from X , • (computational complexity) (1 /(cid:15) ) O (log (1 /(cid:15) )) · O (log n · log(1 /δ )) operations, to construct a vector ( (cid:98) p , . . . , (cid:98) p n ) satisfying P ( d T V ( (cid:98) X, X ) ≤ (cid:15) ) ≥ − δ for (cid:98) X ∼ PB( (cid:98) p , . . . , (cid:98) p n ) . The key to the algorithm is to find subsets covering all Poisson binomial distributions, andeach of these subsets is either ‘sparse’ or ‘heavy’. Applying Birg´e’s algorithm [12] to sparsesubsets, and the translated Poisson approximation (Theorem 3.3) to heavy subsets give thedesired algorithm. Note that the sample complexity in Theorem 5.1 is nearly optimal, sinceΘ(1 /(cid:15) ) samples are required to distinguish Bin( n, /
2) from Bin( n, / (cid:15)/ √ n ) which differby Θ( (cid:15) ) in total variation. See also [39] for further results on learning the Poisson binomialdistribution, and [33, 37, 38] for the integer-valued distribution. Computing the Poisson binomial distribution . Recall the probability distribution of X ∼ PB( p , . . . , p n ) from (1.1). A brute-force computation of this distribution is expensivefor large n . Approximations in Section 3 are often used to estimate the probability distribu-tion/CDF of the Poisson binomial distribution. Here we focus on the efficient algorithms tocompute exactly these distribution functions. There are two general approaches: recursiveformulas and discrete Fourier analysis.In [29], the authors presented several recursive algorithms to compute (1.1). For B ⊂ [ n ],define R ( k, B ) := (cid:88) A ⊂ B, | A | = k (cid:32)(cid:89) i ∈ A p i − p i (cid:33) . So P ( X = k ) = R ( k, [ n ]) · (cid:81) ni =1 (1 − p i ). Now the problem is to find efficient ways to compute R ( k, B ). Two recursive algorithms are proposed: • [30, 97] For B ⊂ [ n ], by letting T ( i, B ) := (cid:80) j ∈ B (cid:16) p j − p j (cid:17) i , R ( k, B ) = 1 k k (cid:88) i =1 ( − i +1 T ( i, B ) R ( k − i, B ) , (5.1) • [45] For B ⊂ [ n ], R ( k, B ) = R ( k, B \ { k } ) + p k − p k R ( k − , B \ { k } ) . (5.2)In another direction, [43, 56] used a Fourier approach to evaluate the probability distribu-tion/CDF of Poisson binomial distributions. They provided the following explicit formulas: P ( X = k ) = 1 n + 1 n (cid:88) j =0 exp( − iωkj ) x j , (5.3)and P ( X ≤ k ) = 1 n + 1 n (cid:88) j =0 − exp( − iω ( k + 1) j )1 − exp( − iωj ) x j , (5.4)where ω := πn +1 and x j := (cid:81) nk =1 (1 − p k + p k exp( iωj )). In particular, the r.h.s of (5.3) isthe discrete Fourier transform of { x , . . . , x n } which can be easily computed by Fast FourierTransform. See also [13] for a related approach. OISSON BINOMIAL 15
Appendix A. Accuracy of X/ for small n Recall the definition of Acc( · ) from (4.6). We compute the values of Acc(2 X/
3) with X ∼ Bin( n, /
2) for 1 ≤ n ≤ • n = 1: Let Y ∼ Ber(1 / p ) is a Bernoulli variable with parameter p . Itis easy to see that Acc(2 X/
3) = W ∞ (2 X/ , Y ) = 1 / . That is, the weight P (2 X/ / { Y = 0 } , and the weight P (2 X/ /
3) = 1 / { Y = 1 } . • n = 2: Let Y ∼ Ber(3 / X/
3) = W ∞ (2 X/ , Y ) = 1 / . So the weight P (2 X/ / { Y = 0 } , and the weight P (2 X/ ∈{ / , / } ) = 3 / { Y = 1 } . • n = 3: suppose that W ∞ (2 X/ , Y ) = 1 / Y . Thenthe weight P (2 X/ / { Y = 0 } , the weight P (2 X/ ∈{ / , / } ) = 3 / { Y = 1 } , and the weight P (2 X/ / { Y = 2 } . The PGF of Y is 1 / x/ x /
8, which has two distinctreal roots − ± √
8. Thus,Acc(2 X/
3) = W ∞ (cid:18) X/ , PB (cid:18)
14 + √ , − √ (cid:19)(cid:19) = 1 / . • n = 4: if W ∞ (2 X/ , Y ) = 1 / Y , then the PGF of Y is 1 / x/
16 + 4 x /
16 + x /
16. This PGF has one real root and two imaginary roots, so Y cannot be strongly Rayleigh. There are many ways to construct a strongly Rayleighvariable Y such that W ∞ (2 X/ , Y ) = 2 /
3. For instance, the weight P (2 X/ /
16 is transferred to { Y = 0 } , the weight P (2 X/ ∈ { / , / } ) = 10 /
16 istransferred to { Y = 1 } and the weight P (2 X/ ∈ { , / } ) = 5 /
16 is transferred to { Y = 2 } . SoAcc(2 X/
3) = W ∞ (cid:18) X/ , PB (cid:18)
12 + 2 / √ , − / √ (cid:19)(cid:19) = 2 / . In fact, we can find all strongly Rayleigh Y such that W ∞ (2 X/ , Y ) = 2 /
3. Thereare two cases:(1) The range of Y is { , , } . Suppose θ /
16 with θ ≤ P (2 X/ /
3) istransferred to { Y = 1 } , and θ /
16 with θ ≤ P (2 X/ /
3) is transferredto { Y = 1 } . Then the PGF of Y is5 − θ
16 + θ + θ x + 11 − θ x . So Y is strongly Rayleigh if and only if ( θ + θ ) ≥ − θ )(11 − θ ). Figure2 (Left) shows the valid region of ( θ , θ ).(2) The range of Y is { , , , } . Assume the same as in (1), and in addition θ / θ ≤ P (2 X/ /
3) is transferred to { Y = 3 } . Then the PGF of Y is5 − θ
16 + θ + θ x + 11 − θ − θ x + θ x . The discriminant of the cubic equation ax + bx + cx + d = 0 is ∆ := 18 abcd − b d + b c − ac − a d . According to a well known result of Cardano, thecubic equation has three real roots if and only if ∆ ≥ − θ )( θ + θ )(11 − θ − θ ) θ − − θ − θ ) (5 − θ )+ (11 − θ − θ ) ( θ + θ ) − θ + θ ) θ − − θ ) θ ≥ . Figure 2 (Right) shows the valid region of ( θ , θ , θ ). Figure 2.
Left: Valid region of ( θ , θ ). Right: Valid region of ( θ , θ , θ ). • n = 5: a similar argument as in the case n = 4 shows that W ∞ (2 X/ , Y ) (cid:54) = 1 / Y . Again there are many ways to construct astrongly Rayleigh variable Y such that W ∞ (2 X/ , Y ) = 2 /
3. For instance, the weight P (2 X/ /
32 is transferred to { Y = 0 } , the weight P (2 X/ ∈ { / , / } ) =15 /
32 is transferred to { Y = 1 } , the weight P (2 X/ ∈ { , / } ) = 15 /
32 is transferredto { Y = 2 } , and the weight P (2 X/ /
3) = 1 /
32 is transferred to { Y = 3 } .The PGF of Y is then 1 /
32 + 15 x/
32 + 15 x /
32 + x /
32. It is easily seen thatthe coefficients of the above PGF satisfy the Hutchinson-Kurtz condition (4.3). SoAcc(2 X/
3) = 2 /
3. It is more difficult to find all strongly Rayleigh variables Y suchthat W ∞ (2 X/ , Y ) = 2 /
3, since the conditions for a quartic function to have all realroots are more complicated [82]. • n = 6: consider the transference plan in Figure 3. It is easy to see that the PGF of Y is 1 / x ) , so Y ∼ Bin(4 , /
2) and Acc(2 X/
3) = 2 / Acknowledgment:
We thank Tom Liggett, Jim Pitman and Terry Tao for helpful discus-sions. We thank Yuting Ye for providing Example 3.4, and Tom Liggett for showing us themanuscript [65].
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