Ubiquity of Benfords law and emergence of the reciprocal distribution

Ubiquit y of Benford’s la w and emergence of the recipro cal distribution J. L. F riar 1 ∗ , T. Goldman 1 † and J. P´ erez–Merc ader 2 ‡ § 1 The or etic al Di vision, L os Alamos National La b or atory, L os Alamos, NM 87545 2 Dep artment of Earth and Planetary Scienc es, Harvar d University, Cambridge, MA 02138 and Santa F e Institute, 1399 Hyde Park R o ad, Santa F e, NM 87501 (Dated: Septem b er 10, 2018 ) W e apply the Law of T otal Probabilit y to th e co nstruction of scale-in v ariant probabilit y distri- bution functions (p df ’s), and require th at probabilit y measures b e dimensionless and unitless und er a con tinuous c h ange of scales. If the scale-c hange distribution function is scale inv ariant then the constructed d istribution will also b e scale inv ariant. Rep eated app licati on of this construction on an arbitrary set of (normalizable) pdf ’s results again in scale-inv ariant distributions. The inv ari- ant funct ion of this procedu re is giv en uniquely by t he recipro cal distribution, suggesting a kin d of un iv ersalit y . W e separately demonstrate that the reciprocal distribut ion results u niquely from requiring maxim um entrop y for size-class distributions with u niform bin sizes. ∗ Electronic address: friar@lanl.gov † Electronic address: tjgoldman@post.harv ard. edu ‡ Electronic address: jperezmercader@fas.harv ard.edu § Corresp onding Author 2 INTRO DUCT ION In 1881 [1] the astrono mer and mathematician Simon Newcom b obs erv ed that the front pages of tables of lo garithms were more worn than later pa ges. I n other words ma ntissas co rresp onding to quantities that had a s maller first digit were mo r e common than for quantities with a la rger first digit. He argued tha t the distr ibutio n of “typical” mantissas was therefore logar ithmic. The physicist F rank Benford [2] rediscov ered this in 1 9 38 and provided mor e detail, for which his name is now asso ciated with this phenomenon. By now it is well do cumen ted that the frequency of first digits D in the v alues of quantities ra ndomly drawn fr om an “arbitrar y” sample follows Benford’s Law of Significant Digits, na mely , B b ( D ) = ln(1 + D ) − ln( D ) ln( b ) = Z 1+ D D dx x · ln( b ) , (1) where b is the a rbitrary base for the loga rithms and is commonly taken to be 10. W e note that the probability of first digit 1 for base 10 is log 10 (2) ∼ = . 30, far ex ceeding that for a unifor m distribution of digits. The rightmost expres sion in Eqn. (1) expres s es Newcom b’s a nd Benfor d’s lo garithmic distr ibutio n as the cum ulative distribution function (cdf ) based on the recipro cal pro babilit y distribution function (p df ), which has b een no rmalized to 1. The p df that underlies Benford’s Law is therefor e the recipro cal distr ibution, r ( x ) ≡ c/x , with normalization constant c = 1 / ln b when the rando m v aria ble x ra ng es b etw een 1 /b and 1 . W e note that Eqn. (1) is base inv ariant ( i.e. , inv aria n t under a co mmon change in the base of the v arious logarithms) a nd that the r ecipro cal pdf is scale in v ariant (a function f ( x ) is said to b e s cale inv ar ian t if f ( λ x ) = λ p · f ( x ) for any p ∈ C ). In this w ork we will concentrate on the emer gence of the recipro cal dis tribution under a v ariety of co nditio ns . The inv ariant (or fixed- point) function of a n iterative pro cedure applied to distr ibution functions that are inv ariant under a contin uous change of scales will be shown to be the recipro cal distribution. Additionally , r e q uiring maximum entropy fo r siz e-class distr ibutions with unifor mly distributed bin sizes leads to the same function. V ery relev ant to the dis cussion ab ov e is T. P . Hill’s pro of in 1 995 [3 – 6] that ra ndo m sa mples chosen from random probability distributio ns are co llectiv ely des cribed b y the re c iproca l distributio n, which is the p df for the lo garithmic or Benford distr ibutio n. In Hill’s words: “If distributions a re selected at random (in any “unbiased” way) and random sa mples are then taken from e a c h of these distributions the significant digits of the combined sample will conv erge to the logarithmic (Benford) dis tr ibution.” Beca use of this, the latter has b e e n appr opriately characterized as “the distributio n o f distributio ns ,” as Hill’s theo rem is in so me sense the obv erse (counterpart) of the Central L imit Theorem for probability distributi ons with lar ge num ber s of samples. Benford’s Law ha s bee n found to hold in an extra ordinary num b er and v a riet y o f phenomena in area s as diverse as ph ysics [7 – 12], genomics [13], engineering [14] a nd among many other s, forensic accounting [15]. Recently the num b er of examples where it applies has b een expanding ra ther rapidly . In the 196 0’s the need for unders tanding the constraints imp osed in computation b y finite word length and its impact on round-off err ors were b ehind the interest of many , including R. Hamming [16, 17], in Benfor d’s law. Impo rtan tly , Hamming argue d that rep eated application of any of the four bas ic arithmetic op erations (addition, subtraction, m ultiplication and divisio n) to num b ers leads to results whose distribution of leading floating- point digits approaches the loga rithmic (Benford) distr ibutio n. Hamming further ar g ued that if any o ne ar ithmetic op eration inv olves a quantit y already dis tr ibuted a ccording to the recipro cal distribution, r ( x ), then the res ult of this and all subsequent op erations will r esult in quantities whose p df for the leading floating-p oint digits is the r ecipro cal distribution. Hamming calle d this prop erty the “ p ersis tence of the r e c iproca l distribution” altho ug h a b etter word might be con tagiousness , since c on tact with the recipro cal distribution a t a n y p oint in a calcula tional chain mo difies the remaining chain irr ev o cably . In this pape r we us e elementary metho ds to explor e the connection b etw een Benford’s law, Hill’s theo r em and the “contagiousness” pro perty of the r ecipro cal distribution. W e will demons trate this by constructing a simple but comprehensive class of proba bilit y distr ibutions that dep ends on a sing le ra ndom v a riable that is dimensionles s and unitless under a contin uo us change o f scale s. This class dep ends on an underlying p df that is ar bitrary , and whic h can b e sampled in a manner consistent with Hill’s Theorem. W e further ge ner alize this into an iter a tiv e pro cedure whose in v ar ian t functions ar e shown uniquely to be the recipro cal distribution, and which demons trate Ha mming’s “contagiousness”. Uniqueness obtains beca use the arbitrary (or “rando m” in this se nse) underlying pdf eliminates any particular s olutions in the inv ariant functions and leav es only the g eneral so lution. Our pro cedure generaliz es the work o f Hamming[1 6], a nd to the b est o f our knowledge is b oth new and useful. W e show a lternatively by inv oking maximum entrop y for a size-class dis tr ibution function that the re ciproc al distr ibution ag ain obtains as the unique solution. W e conclude by sp eculating on the universalit y and a pplications o f these results, with particular emphasis on minimizing error s in co mputations of v ario us types. 3 RESUL TS In v ariance under changes i n units and the law of total probability In most sc ien tific applica tions a sto c hastic v ariable x is assig ned to the random v alues of some physical q uan tity . This quantit y carr ies either ph ysical dimensions ( e.g. , length or volume) or units (suc h as the num b er o f base pairs in a genome). How ever, b ecause it refer s to pro babilities, the probability measure F ( x ) · dx that characterizes x must be dimensionless and unitless. Hence, in order to re move units or dimensio ns from the measure it is necessar y to intro duce a parameter that results bo th in a dimensionless and unitless sto chastic v aria ble, as well as in a b ona fide pr obabilit y mea s ure. Ca lling this parameter σ , for a sp ecific v alue of σ we can resc a le the physical v a riable x into a dimensionles s and unitless ra ndom v ar iable b y just r eplacing x with z = x/σ . (W e als o assume for simplicity that x is p ositive semi-definite.) Then we can alwa ys intro duce a normaliz able function g suc h that F ( x ) = 1 σ g  x σ  , (2) and that has the cor rect prop erties exp e c ted from a proba bility measure. In other words, we can use a par ameter σ to r emo ve units or dimensions from the probability measure. F amilia r examples of distributions o f the t yp e g are the uniform distribution, g u ( z ) = θ (1 − z ), the Gaussian distribution, g G ( z ) = 2 √ π exp ( − z 2 ), and the ex p onential dis tribution, g e ( z ) = exp ( − z ), all of which sa tisfy the normalizatio n condition: R ∞ 0 dz g ( z ) = 1. Hea viside step functions can b e used for those cases where g ( z ) is only non-v a nishing in an interv al, such as z = [ a, b ], as was do ne ab ov e for the uniform dis tribution. But the units chosen to meas ure x a re, of course, a rbitrary . F or example, if x is a length, the units could b e meter, millimeter, Angs tr om, or even fathom, furlong, league, etc. In other words, the choice of units is itself arbitra ry [18] and we ca n think of σ a s a r andom v ariable with a distribution function h ( σ ). Th us the pr oblem we mu st study inv olves the combination of tw o stochastic v aria bles. W e can conv enient ly remov e the sca le and avoid the issue of units by using the Law of T otal P robability [1 9] to combine the distribution g with a distribution of scale choices to pro duce a distribution G ( x ): G ( x ) = Z ∞ 0 dσ g ( x | σ ) σ h ( σ ) , (3) where now G ( x ) and h ( σ ) are interpreted as the marginal pro ba bilities for ev ents x and σ , and g ( x | σ ) r e presen ts the conditional pr obabilit y for x given σ . This well known law captures the intuitiv ely clear statement that the pr obabilit y that even t x o ccurs is determined by summing the pro duct of the probabilities for a n y o f its anteceden ts σ to happ en, times the conditiona l pro babilit y that x happ ens, giv en that σ has already o ccurred. Conv ergence of the integral for small v a lues of σ is not a problem for x 6 = 0 if g ( z ) v anishes sufficiently rapidly for lar ge z . Normalizability of g is sufficient fo r our purp oses. The pr obabilit y distribution in Eqn. (3) is fairly gener al and will b e o ur template for studying the conditions underly ing the emerg ence of the recipr oca l distribution. The Law of T otal Probabili ty and its recursive application Let us consider a g ( x | σ ) that is inv ariant under changes in dimensions or units. That is, le t us ass ume that g ( x | σ ) ≡ g ( x/σ ) , (4) with a c o ncomitan t interpretation for g ( x/σ ) in the terms descr ibed in the preceding par agraph (N.B. the difference betw een“ | ” a nd “ / ” ). Changing the in tegr ation v ariable to z ≡ x/σ in Eqn. (4) lea ds to the conv enient form G ( x ) = Z ∞ 0 dz z g ( z ) h ( x/z ) . (5) It is imp ortant to note a pr oper t y of Eqn. (5) that is a consequence of its str uctur e: the function G ( x ) has an exceptional form if h ( σ ) is a scale- in v ar ian t (and p ow er–law) function. A scale-inv aria n t function h ( x/z ) m ust b e 4 a power of its arg umen t, o r h ( x/z ) ∝ ( x/z ) − s for a p o wer–law. Ignor ing the (for now) ir relev ant prop ortionality constant, we then hav e G ( x ) = 1 x s Z ∞ 0 dz z s − 1 g ( z ) (6) for h ( σ ) = 1 /σ s . W e no te that the integral R ∞ 0 dz z s − 1 g ( z ) = M s ( g ) is a cons tan t and the Mellin transfor m [20] of the function g ( z ). This allows one to rewrite Eq n. (6) in the more co mpact form G ( x ) = 1 x s M s ( g ) . (7) W e therefore conclude that any sca le-in v ariant h ( σ ) in the integral in Eqn. (5 ) repli cates itself in G ( x ) (up to an ov erall constant fa ctor). F urthermore, this co ns tan t is equal to one if h is the recipro cal distribution, since by definition the Mellin transfor m in Eqn. (7) equa ls unity for s = 1 if g ( z ) is no r malizable. Iterating the Law of T otal Probability and the inv ariant function of the iteration With the ab ov e result in hand w e are ready to tackle the following imp ortant q ue s tion: what would be the res ult of applying the Law o f T otal Probability , as written in E qn. (3), to a re p eated a nd independent combination of r andom quantities if h is scale inv ariant ( i.e. , a p o wer–law with exp onen t s > 0)? More specifica lly , s uppose that we hav e n random v aria ble s distributed acco r ding to the (in principle diffe ren t ) distributions g 1 ( z ) , g 2 ( z ) , · · · g n ( z ), and that h is sca le in v a riant. Defining the integral in Eqn. (3 ) or (5) to be an int egr al transfor m op erator , C , acting on h ( σ ), we can then op erate C rep eatedly on G a total of n times with (in principle) n different distributions to pr oduce the n -th iterate. Denoting the result of the ab ov e o pera tions by G n ( x n ), we can then write it mathematica lly as G n ( x n ) = C · C · · · · C | {z } n tim es { h } ≡ C n { h } = 1 x s n " n Y i =1 M s ( g i ) # , (8) where the constant in the large br ac ket is unit y for s = 1 and if each g is normaliza ble, as assumed. E q uation (8 ) follows immediately from Eqn. (6) in a natural w ay: each succeeding application of C regenerates the function h and thus r epro duces the previous a pplication, except for a n overall constant. This is of course the “contagiousness” or “p ersistence” prop erty noted by Hamming, which is the inevita ble (una voidable) result of using a s cale-inv a riant prior function h in E qn. (5). (W e note in passing that Hamming referred to s cale-inv a rian t distributions , but ac tua lly treated and discussed only the re c iproca l distribution.) W e can extend and unify our discussion by examining the fixed-p oint functions, or mor e precisely the inv ariant functions , of the iteration pro cedure in Eqn. (8). This is done by replacing G n with h , and results in h = C n { h } (9) for any n . Thus the inv ariant function o f the iter ativ e pro cedure introduced in Eqn. (8) is the recipro cal distribution, since the br a c keted cons tan t in that equation with multiple a rbitrary distributions (and therefore normalizable) g n will only equal unity for s = 1. This result should not co me a s a complete surpr ise: an inv a riant function canno t dep end on the details of the arbitrar y g n . The scale σ in the g n will couple to any scale in h ( σ ) to pr oduce ar bitrary results unles s h is scale inv a rian t, which is easy to demonstrate using a v ariety of simple distributions. F ur ther more, the r ecipro cal distribution can b e shown to b e the unique solution when g is the uniform distribution. Inv oking a set of a r bitrary p df ’s ( i.e. , the g n ) is simila r in spir it to Hill’s “r andom distributions.” In our case it rules out a n y particular solutions to E qn. (9), which will b e different for each choice of g , but allows the scale–inv ar ia n t general solutions. Consequently , the “p ersistence” or “cont agious ness” o f the recip ro cal distributi on is a prop ert y of the in v ariant-func tion s olutions of the i terativ e pro cedure in tro duced in E qn. (8). The preceding narr ativ e was predicated on developing an understanding o f the “ persis tence” of the recipro cal distribution. Nevertheless, w e emphasize that the b eha vior of Eqn. (9) for n = 1 is all that is needed in or der to sp ecify the r ecipro cal distribution as the unique s olution of h ( x ) = Z ∞ 0 dσ σ g i  x σ  h ( σ ) , (10) 5 where g i is an y mem b er of the set of arbitra ry normalizable pdf ’s { g 1 , g 2 , · · ·} . The gene ral solution is clea rly the recipro cal distribution, while any particular so lution will be determined b y the intrinsic pr oper ties of a particular pdf, and cannot b e a s olution for all of them (or even a few of them). DISCUSSION Selected applications to computing and to mini m al truncation e rror s Scale in v ariance (whic h only restricts h ( x ) to a gene r al p o wer law: x − s ) is th us seen to b e a necessary condition for the emergence of the recipr oca l pdf (uniquely s = 1) and its cum ulative distr ibutio n function that leads to Benford’s Law o f Significant Dig its . A sufficien t condition is that the re c iproca l p df emerge when in co n tact with a rbitrary mem b ers of the set o f nor ma lizable p df ’s. Alter nativ ely , requiring an inv ariant-function solution to the “ c on tagion” pro cess leads immediately and uniquely to the re c ipr oca l distribution. The reciprocal distribution and its contagiousness are “universal” consequences of the rep eated application of the Law of T otal Probabili t y W e hav e established ab o ve that when combining data distributed acco rding to a v ar iet y of pdf ’s into a common p df for the result, the pr esence of sca le inv ariance in at least one of the distributions for the da ta b eing combined leads to a sca le-in v a riant common distribution. Moreov er, if one o f the distributions is the recipr oca l, then all subsequent combined data is distributed ac cording to the rec ipr oca l distribution. It is like a mathematical par aphrasing o f the po pular 17th cent ury English proverb that “Once a p o or, alwa ys a p o or” into “Once Rec iproca l/Benford, always Recipro cal/Benford” . The rela tionship of the recipr oca l distribution to the inv ariant functions of the iterated L aw o f T ota l Proba bilit y was de r iv ed a bov e. W e wish to sugg e st that this r elationship is b ehind the abundant nu mber and the wide s pectrum of phenomena where Benford’s Law of Significant Digits has b een found to apply . Given the nature of these tightly constrained iterations, one can a lso envision how the r ecipro cal distr ibution and the as socia ted Benford’s Law of Significant Digits could b e considered universal in a sense a k in [21] to the one used in condens e d matter theor y a nd for cr itical phenomena. W e note that inv ariant-function solutions to iter ativ e pro cedures a re bo th extremely co mmon and q uite imp ortant in fields a s diverse as chaotic dyna mics [22], theor etical eco lo gy [23], and many others . In some sense, this result emerg es as a c o nsequence of the fact that the recipro cal distribution is the in v ariant function fo r iterations of the Law of T o tal Probability applied to algor ithmic co m bination of ph ysica l da ta. Of course, it follows from the preceding a nalysis, that this also will b e the case if one were to co m bine any data toge ther with data that is purely numerical and cla ssified acco rding to first digits. Size - Cl asses and Maximum Entro py In addition to the pro perties just discussed, the recipro cal distribution has a re ma rk able prop ert y that impacts data transmission for applications in co mmunication theory . It follows from the functiona l for m of the r ecipro c al distribution that the mea n of recipro cally distributed ra ndo m v ar iables is a cons tan t over a n y uni form in terv al . If we could asso ciate this mean with a probability distribution, then by g rouping Benford–distributed data into unifor mly distributed pa c k a ges (cor r espo nding to this average v alue) we could communicate those pack ages at the maximum infor ma tion rate [24]. W e can for mally implement the words ab ov e by consider ing the notion of size-classes [25]. T o that end we introduce the following constr uction for an arbitr a ry pdf, p ( x ): ∆Φ[ G [ j ]] ≡ Z G [ j ] G [ j − 1] dx x p ( x ) ≡ Φ[ G [ j ]] − Φ[ G [ j − 1 ]] , (11) where G [ j ] and G [ j − 1 ] deno te the uppe r a nd lower v a lues o f the sizes co n tained within the size-c lass indexe d by j . Since p ( x ) is the p df for x taking v alues b et ween x min and x max , we must hav e Z x max x min dx p ( x ) = 1 . (12) 6 The mean of x with resp ect to p ( x ) is then given by h x i = Z x max x min dx · x · p ( x ) . (13) The pro babilit y that the v a lues of the v ariable x fall in the c la ss j ( i.e. , b etw een G [ j ] and G [ j − 1 ]) is determined by a new p df, p ∗ ( j ): p ∗ ( j ) = Z G [ j ] G [ j − 1] dx x h x i p ( x ) = ∆Φ[ G [ j ]] h x i , (14) with the constraint fro m Eqns. (12) and (13) that j max X j =1 p ∗ ( j ) = 1 , (15) where j max is the n umber of s ize-classes into which the interv al [ x min , x max ] is par titioned. Thus for any p df, s ize-classes and a n a sso c iated pdf can b e in tro duced. It is straig htforward to show that the Benford dis tr ibution ha s the pro perty that unifor m siz e-classes a re themselves uniformly distributed. If we require p ∗ ( j ) to b e a cons ta n t (viz., α > 1) that is indep enden t o f j , then Φ[ G [ j ]] − Φ[ G [ j − 1]] = 1 α , (16) and the size-class es G [ j ] − G [ j − 1] from Eqns. (14) and (16) that a r e found for p ( x ) = a/ x must satisfy 1 α = Z G [ j ] G [ j − 1] dx x h x i p ( x ) = a h x i ( G [ j ] − G [ j − 1]) . (17) That is, the classes that sa tis fy Eqn. (17) are uniformly distr ibuted and therefor e hav e maximum entrop y; the information they contain ca n b e transmitted at the maximum achiev able rate: H = + ln( α ). F or the r ecipro cal-distribution exa mple we hav e G [ j ] − G [ j − 1] = h x i a · α = x max − x min α ≡ β , (18) where Eqn. (13) has b een used. The r ecursion rela tion in Eq n. (18) for the size-cla ss b oundary G [ j ] is so lved by G [ j ] = β j + r , (19) where r is an arbitr ary constant. The integer quantit y j max in Eqn.(15) ca n then b e shown to equal α (which must also b e in teger ). Therefore, Benfor d-distributed data g rouped into such s ize-classes will b e communicated at the maximum a c hiev a ble rate 1 . C o m bining this re sult with the co n tagion prop erty of Benford lea ds to the co nclusion that the co n tagiousnes s of the recipro cal dis tr ibution via the L TP implies that a g rouping into si ze-classes of sto c hastic v ariables, at least one of which is Benford distributed, has maximal entrop y as long as the gro uping is uniformly partitioned. These classes or grouping s of the o riginal data can then b e tr ansmitted a t the maximum achiev able rate. 1 Note that for any distribution we can alwa ys introduce the notion of an f -cla ss (corresp onding to a f unct ion f ) by constructing the equiv alent of p ∗ ( j ) i n Eqn. (14). All one needs to do is replace x h x i p ( x ) in Eqn. (14) b y f ( x ) h f ( x ) i p ( x ), where f ( x ) is a function of the stochastic v ari able x and represents some ph ysical v ariable according to which we wish to sort the system into classes. If we require that the resulting p ∗ ( j ; f ) b e constan t, then the transmiss ion of these classes wil l tak e place at the maximum r ate . It is particularly int eresting that f or the Zipf distribution, p ( x ) ∝ 1 /x 2 , the “distortion-classes” that result from ch o osing f ( x ) ∝ x 2 ac hieve maximum en tropy in the same w ay as the size-classes do for the Benford distribution. 7 Round-off error and the Re ciprocal Dis tribution. Applications to computing and to evolutionary biol ogy The a bov e obser v ation can b e turned into a principle of desig n that ca n b e applied if one is in terested in a c hieving maximum precision (equiv a len t to minimizing resulting erro rs by alwa ys choosing the sma llest intermediate er r ors) in calculational situations where erro rs are unav oidable, as is the case for round- off error s o c curring in automa tic digital computation. This is what was sugges ted by Hamming’s res ult mentioned in the introduction to this pap er. In fact, the contagiousness prop erty o f the recipro cal and the asso ciated prep onderance of mantissas sta rting with smaller dig its indicates that dis tributing quantities acco rding to the recipro cal distribution will pro duce maximal reduction in round-off err ors dictated by the necessar y truncation of r esults of op erations due to the fixed and finite word length of the machine on which the op eration takes pla ce. This contagiousness pr oper t y of the r ecipro cal p df, combined with its relationship to the Benford Law of Significant Digits, was then used by Hamming to argue that the mantissas of the r o und-off error s that result fro m the a r ithmetical combination of rando m quantities are a lways sma ller tha n what would res ult if they w ere uniformly distr ibuted. Hence the ar ithmetica l combination of quantities of which a t least one is r ecipro cally distr ibuted leads to smaller error s than those resulting from the combination of a ll-uniformly distr ibuted qua n tities. This implies for round-o ff er ror that it “pays” to design the data to b e r ecipro cally distributed, since when c o m bined it leads to minimal error s. This leads to a maximum r eduction of the unav oida ble round-o ff erro rs inherent [2 6] to any calculation done with digital computers that, of course, ar e ultimately due to the fixed nature o f their word length [2 7, 28]. Indeed this was discusse d b y Schatte [29] a s the basis for a strategy designed to reduce the accumulation of erro rs in the op eratio n of dig ital machines (cf. also [30]). W e combine this pr oper t y with the ab ov e result that maximum en trop y is achieved by grouping B enford-distributed data in to uniformly sized classes. This demonstrates that those groups will autom atically b e the fastes t transmitted pac k ets with the small est p ossible errors i n b oth their transmiss i on and con tents. In this context we note that the fundamental op erations in information handling by the geno mes of all for ms of life are co n trolled by the so-calle d cDNA frac tion of the genome. This fr action comprises the fundamenta l genes in the living s y stem, and ar e cont ained in ob jects called Ope n Rea ding F r ames (ORFs). F or a ll living systems the ORFs a r e distributed acc ording to a recipro cal distribution of the full genome size [13], no t just of only the cDNA frac tio n. This suggests to us that Life, by means of the tria l a nd er ror pro cess e s of evolution, may hav e “ discov ered” the mo st r obust and low est-er r or strategy for stor ing and tra nsmitting information. The key ingredient for distributing quantities with maximal fidelity using the Law of T o tal P r obabilit y (equiv alent to low est p ossible error) is to incorp ora te the recipro cal distribution. Subseq ue ntly the pro cess e s dep ending on these fundamental genes will b e the mo st robust and w ill therefore hav e a comp etitiv e adv ant ag e as a base to p ersist, survive and not b ecome extinct. REFERENCES [1] Newcom b S., Amer. J. Math. 4 , 39-40 (1881). [2] Benford F., Pro c. of t h e American Ph il. S oc. 78 , 551-572 (1938). [3] Hill T. P ., P . Am. Math. So c. 123 , 887-895 (1995). [4] Hill T. P ., Stat. Sci. 10 , 354-363 (1995). [5] Berger A. & H ill T. P ., The Mathematical Intelligencer 33 , 85-91 (2011). [6] Berger A. & H ill T. 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[28] Bernstein J., The A nalytical Engine: Comput ers-Past, Presen t and F u ture (R andom House, N ew Y ork, 1963). [29] Schatte P ., J. In f. Pro cess. Cyb ern. EIK 24 , 443-455 (1988). [30] Knuth D., The Art of Computer Programming, V olume 2, Seminumerical A lgori thms (A ddison-W esley , R eading, Mass., 1997). ACKNO WLEDGMENT S One of us (JP-M) would like to thank the everis F oundatio n and Repso l for generous supp ort, and the Theo retical Division of Los Alamos National Lab ora to ry fo r its hospitality .