Uiversity of Texas at El Paso DigitalCommos@UTEP Departmetal Techical Reports (CS) Departmet of Computer Sciece 2-2017 (Hypothetical) Negative Probabilities Ca Speed Up Ucertaity Propagatio Algorithms Adrzej Powuk Uiversity of Texas at El Paso, ampowuk@utep.edu Vladik Kreiovich Uiversity of Texas at El Paso, vladik@utep.edu Follow this ad additioal works at: http://digitalcommos.utep.edu/cs_techrep Part of the Computer Scieces Commos Commets: Techical Report: UTEP-CS-17-08a To appear i Aboul Ella Hassaie, Mohamed Elhosey, Ahmed Farouk, ad Jausz Kacprzyk (eds.), Quatum Computig: a Eviromet for Itelliget Large Scale Real Applicatio, Spriger Verlag Recommeded Citatio Powuk, Adrzej ad Kreiovich, Vladik, "(Hypothetical) Negative Probabilities Ca Speed Up Ucertaity Propagatio Algorithms" (2017). Departmetal Techical Reports (CS). 1110. http://digitalcommos.utep.edu/cs_techrep/1110 This Article is brought to you for free ad ope access by the Departmet of Computer Sciece at DigitalCommos@UTEP. It has bee accepted for iclusio i Departmetal Techical Reports (CS) by a authorized admiistrator of DigitalCommos@UTEP. For more iformatio, please cotact lweber@utep.edu.
(Hypothetical) Negative Probabilities Ca Speed Up Ucertaity Propagatio Algorithms Adrzej Powuk ad Vladik Kreiovich Abstract Oe of the mai features of quatum physics is that, as basic objects describig ucertaity, istead of (o-egative) probabilities ad probability desity fuctios, we have complex-valued probability amplitudes ad wave fuctios. I particular, i quatum computig, egative amplitudes are actively used. I the curret quatum theories, the actual probabilities are always o-egative. However, there have bee some speculatios about the possibility of actually egative probabilities. I this paper, we show that such hypothetical egative probabilities ca lead to a drastic speed up of ucertaity propagatio algorithms. 1 Itroductio From o-egative to more geeral descriptio of ucertaity. I the traditioal (o-quatum) physics, the mai way to describe ucertaity whe we have several alteratives ad we do ot kow which oe is true is by assigig probabilities p i to differet alteratives i. The physical meaig of each probability p i is that it represets the frequecy with which the i-th alterative appears i similar situatios. As a result of this physical meaig, probabilities are always o-egative. I the cotiuous case, whe the umber of alteratives is ifiite, each possible alterative has 0 probability. However, we ca talk: about probabilities of values beig i a certai iterval ad, correspodigly, Adrzej Powuk ad Vladik Kreiovich Computatioal Sciece Program Uiversity of Texas at El Paso 500 W. Uiversity El Paso, Texas 79968, USA e-mail: ampowuk@utep.edu, vladik@utep.edu 1
2 A. Powuk ad V. Kreiovich about the probability desity ρ(x) probability per uit legth or per uit volume. The correspodig probability desity fuctio is a limit of the ratio of two oegative values: probability ad volume, ad is, thus, also always o-egative. Oe of the mai features of quatum physics is that i quatum physics, probabilities are o loger the basic objects for describig ucertaity; see, e.g., [4]. To describe a geeral ucertaity, we ow eed to describe the complex-valued probability amplitudes ψ i correspodig to differet alteratives i. I the cotiuous case: istead of a probability desity fuctio ρ(x), we have a complex-valued wave fuctio ψ(x). No-positive ad o-zero values of the probability amplitude ad of the wave fuctio are importat: e.g., egative values of the amplitudes are actively used i may quatum computig algorithms; see, e.g., [9]. Ca there be egative probabilities? I the curret quatum theories, the actual probabilities are always o-egative. For example: the probability p i of observig the i-th alterative is equal to a o-egative umber p i = ψ i 2, ad the probability desity fuctio is equal to a o-egative expressio ρ(x) = ψ(x) 2. However, there have bee some speculatios about the possibility of actually egative probabilities, speculatios actively explored by Nobel-rak physicists such as Dirac ad Feyma; see, e.g., [2] ad [3]. Because of the high caliber of these scietists, it makes sese to take these speculatios very seriously. What we do i this paper. I this paper, we show that such hypothetical egative probabilities ca lead to a drastic speed up of ucertaity propagatio algorithms. 2 Ucertaity Propagatio: Remider ad Precise Formulatio of the Problem Need for data processig. I may practical situatios, we are iterested i the value of a physical quatity y which is difficult or eve impossible to measure directly. For example, we may be iterested:
Negative Probabilities Ca Speed Up Ucertaity Propagatio 3 i tomorrow s temperature, or i a distace to a faraway star, or i the amout of oil i a give oil field. Sice we caot measure the quatity y directly, a atural idea is: to measure easier-to-measure related quatities ad the to use the kow relatio x 1,...,x, y = f (x 1,...,x ) betwee these quatities to estimate y as ỹ = f ( x 1,..., x ), where x i deotes the result of measurig the quatity x i. For example: To predict tomorrow s temperature y: we measure temperature, humidity, ad wid velocity at differet locatios, ad we use the kow partial differetial equatios describig atmosphere to estimate y. To measure a distace to a faraway star: we measure the directio to this star i two differet seasos, whe the Earth is o differet sides of the Su, ad the we use trigoometry to fid y based o the differece betwee the two measured directios. I all these cases, the algorithm f trasformig our measuremet results ito the desired estimate ỹ is a example of data processig. Need for ucertaity propagatio. Measuremets are ever absolutely accurate. The measuremet result x i is, i geeral, somewhat differet from the actual (ukow) value of the correspodig quatity x i. As a result, eve whe the relatio y = f (x 1,...,x ) is exact, the result ỹ of data processig is, i geeral, somewhat differet from the the actual values y = f (x 1,...,x ): It is therefore ecessary to estimate ỹ = f ( x 1,..., x ) y = f (x 1,...,x ).
4 A. Powuk ad V. Kreiovich how accurate is our estimatio ỹ, i.e., how big is the estimatio error y def = ỹ y. The value of y depeds o how accurate were the origial measuremets, i.e., how large were the correspodig measuremet errors x i def = x i x i. Because of this, estimatio of y is usually kow as the propagatio of ucertaity with which we kow x i through the data processig algorithm. Ucertaity propagatio: a equivalet formulatio. By defiitio of the measuremet error, we have x i = x i x i. Thus, for the desired estimatio error y, we get the followig formula: y = ỹ y = f ( x 1,..., x ) f ( x 1 x 1,..., x x ). Our goal is to trasform the available iformatio about x i ito the iformatio about the desired estimatio error y. What do we kow about x i : ideal case. Ideally, for each i, we should kow: which values of x i are possible, ad how frequetly ca we expect each of these possible values. I other words, i the ideal case, for every i, we should kow the probability distributio of the correspodig measuremet error. Ideal case: how to estimate y? I some situatios, we have aalytical expressios for estimatig y. I other situatios, sice we kow the exact probability distributios correspodig to all i, we ca use Mote-Carlo simulatios to estimate y. Namely, several times l = 1,2,...,L, we: simulate the values x (l) i accordig to the kow distributio of x i, ad estimate y (l) = ỹ f ( x 1 x (l) 1,..., x x (l) ). Sice the values x (l) i have the exact same distributio as x i, the computed values y (l) are a sample from the same distributio as y. Thus, from this sample y (1),..., y (L), we ca fid all ecessary characteristics of the correspodig y-probability distributio.
Negative Probabilities Ca Speed Up Ucertaity Propagatio 5 What if we oly have partial iformatio about the probability distributios? I practice, we rarely full full iformatio about the probabilities of differet values of the measuremet errors x i, we oly have partial iformatio about these probabilities; see, e.g., [10]. I such situatios, it is ecessary to trasform this partial iformatio ito the iformatio about y. What partial iformatio do we have? What type of iformatio ca we kow about x i? To aswer this questio, let us take ito accout that the ultimate goal of all these estimatios is to make a decisio: whe we estimate tomorrow s temperature, we make a decisio of what to wear, or, i agriculture, a decisio o whether to start platig the field; whe we estimate the amout of oil, we make a decisio whether to start drillig right ow or to wait util the oil prices will go up sice at preset, the expected amout of oil is too large eough to justify the drillig expeses. Accordig to decisio theory results (see, e.g., [5, 7, 8, 11]), a ratioal decisio maker always selects a alterative that maximizes the expected value of some objective fuctio u(x) kow as utility. From this viewpoit, it is desirable to select characteristics of the probability distributio that help us estimate this expected value ad thus, help us estimate the correspodig utility. For each quatity x i, depedig o the measuremet error x i, we have differet values of the utility u( x i ). For example: If we overestimate the temperature ad start platig the field too early, we may lose some crops ad thus, lose potetial profit. If we start drillig whe the actual amout of oil is too low or, vie versa, do ot start drillig whe there is actually eough of oil we also potetially lose moey. The measuremet errors x i are usually reasoably small. So, we ca expad the expressio for the utility u( x i ) i Taylor series ad keep oly the first few terms i this expasio: u( x i ) u(0) + u 1 x i + u 2 ( x i ) 2 +... + u k ( x i ) k, where the coefficiets u i are uiquely determied by the correspodig utility fuctio u( x i ). By takig the expected value E[ ] of both sides of the above equality, we coclude that E[u( x i )] u(0) + u 1 E[ x i ] + u 2 E[( x i ) 2 ] +... + u k E[( x i ) k ]. Thus, to compute the expected utility, it is sufficiet to kow the first few momets E[ x i ], E[( x i ) 2 ],...,E[( x i ) k ] of the correspodig distributio. From this viewpoit, a reasoable way to describe a probability distributio is via its first few momets. This is what we will cosider i this paper.
6 A. Powuk ad V. Kreiovich From the computatioal viewpoit, it is coveiet to use cumulats, ot momets themselves. From the computatioal viewpoit, i computatioal statistics, it is ofte more coveiet to use ot the momets themselves but their combiatios called cumulats; see, e.g., [13]. A geeral mathematical defiitio of the k-th order cumulat κ i of a radom variable x i is that it is a coefficiet i the Taylor expasio of the logarthm of the characteristic fuctio (where i def = 1) i terms of ω: χ i (ω) def = E[exp(i ω x i )] l(e[exp(i ω x i )]) = k=1 κ ik (i ω)k. k! It is kow that the k-th order cumulat ca be described i terms of the momets up to order k; for example: κ i1 is simply the expected value, i.e., the first momet; κ i2 is egative variace; κ i3 ad κ i4 are related to skewess ad excess, etc. The coveiet thig about cumulats (as opposed to momets) is that whe we add two idepedet radom variables, their cumulats also add: the expected value of the sum of two idepedece radom variables is equal to the sum of their expected values (actually, for this case, we do ot eve eed idepedece, i other cases we do); the variace of the sum of two idepedet radom variables is equal to the sum of their variace, etc. I additio to this importat property, k-th order cumulats have may of the same properties of the k-th order momets. For example: if we multiply a radom variable by a costat c, the both its k-th order momet ad its k-th order cumulat will multiply by c k. Usually, we kow the cumulats oly approximately. Based o the above explaatios, a coveiet way to describe each measuremet ucertaity x i is by describig the correspodig cumulats κ ik. The value of these cumulats also come from measuremets. As a result, we usually kow them oly approximately, i.e., have a approximate value κ ik ad the upper boud ik o the correspodig iaccuracy: κ ik κ ik ik. I this case, the oly iformatio that we have about the actual (ukow) values κ ik is that each of these values belogs to the correspodig iterval [κ ik,κ ik ],
Negative Probabilities Ca Speed Up Ucertaity Propagatio 7 where ad κ ik def = κ ik ik κ ik def = κ ik + ik. Thus, we arrive at the followig formulatio of the ucertaity propagatio problem. Ucertaity propagatio: formulatio of the problem. We kow: a algorithm the measuremet results f (x 1,...,x ), x 1,..., x, ad for each i from 1 to, we kow itervals [κ ik,κ ik ] = [ κ ik ik, κ ik + ik ] that cotai the actual (ukow) cumulats κ ik of the measuremet errors x i = x i x i. Based o this iformatio, we eed to compute the rage [κ k,κ k ] of possible values of the cumulats κ k correspodig to y = f ( x 1,..., x ) f (x 1,...,x ) = f ( x 1,..., x ) f ( x 1 x 1,..., x x ). 3 Existig Algorithms for Ucertaity Propagatio ad Their Limitatios Usually, measuremet errors are relatively small. As we have metioed, i most practical cases, the measuremet error is relatively small. So, we ca safely igore terms which are quadratic (or of higher order) i terms of the measuremet errors. For example: if we measure somethig with 10% accuracy, the the quadratic terms are of order 1%, which is defiitely much less tha 1%. Thus, to estimate y, we ca expad the expressio for y i Taylor series ad keep oly liear terms i this expasio. Here, by defiitio of the measuremet error, we have
8 A. Powuk ad V. Kreiovich thus x i = x i x i, y = f ( x 1,..., x ) f ( x 1 x 1,..., x x ). Expadig the right-had side i Taylor series ad keepig oly liear terms i this expasio, we coclude that y = c i x i, where c i is the value of the i-th partial derivative f x i at a poit ( x 1,..., x ): c i def = f x i ( x 1,..., x ). Let us derive explicit formulas for κ k ad κ k. Let us assume that we kow the coefficiets c i. Due to the above-metioed properties of cumulats, if κ ik is the k-th cumulat of x i, the the k-th cumulat of the product c i x i is equal to (c i ) k κ ik. I its tur, the k-th order cumulat κ k for the sum y of these products is equal to the sum of the correspodig cumulats: κ k = (c i ) k κ ik. We ca represet each (ukow) cumulat κ ik as the differece where is bouded by the kow value ik : κ ik = κ ik κ ik, κ ik def = κ ik κ ik κ ik ik. Substitutig the above expressio for κ ik ito the formula for κ k, we coclude that κ k = κ k κ k, where we deoted κ k def = (c i ) k κ k
Negative Probabilities Ca Speed Up Ucertaity Propagatio 9 ad κ k def = (c i ) k κ ik. The value κ k is well defied. The value κ k depeds o the approximatio errors κ ik. To fid the set of possible values κ k, we thus eed to fid the rage of possible values of κ k. This value is the sum of idepedet terms, idepedet i the sese that each of them depeds oly o its ow variable κ ik. So, the sum attais its largest values whe each of the terms (c i ) k κ ik is the largest. Whe (c i ) k > 0, the expressio (c i ) k κ ik is a icreasig fuctio of κ ik, so it attais its largest possible value whe κ ik attais its largest possible value ik. The resultig largest value of this term is (c i ) k ik. Whe (c i ) k < 0, the expressio (c i ) k κ ik is a decreasig fuctio of κ ik, so it attais its largest possible value whe κ ik attais its smallest possible value ik. The resultig largest value of this term is (c i ) k ik. Both cases ca be combied ito a sigle expressio (c i ) k ik if we take ito accout that: whe (c i ) k > 0, the (c i ) k = (c i ) k, ad whe (c i ) k < 0, the (c i ) k = (c i ) k. Thus, the largest possible value of κ k is equal to k def = (c i ) k ik. Similarly, we ca show that the smallest possible value of κ k is equal to k. Thus, we arrive at the followig formulas for computig the desired rage [κ k,κ k ]. Explicit formulas for κ k ad κ k. Here, κ k = κ k k ad κ k = κ k + k, where κ k = (c i ) k κ k ad k = (c i ) k ik.
10 A. Powuk ad V. Kreiovich A resultig straightforward algorithm. The above formulas ca be explicitly used to estimate the correspodig quatities. The oly remaiig questio is how to estimate the correspodig values c i of the partial derivatives. Whe f (x 1,...,x ) is a explicit expressio, we ca simply differetiate the fuctio f ad get the values of the correspodig derivatives. I more complex cases, e.g., whe the algorithm f (x 1,...,x ) is give as a proprietary black box, we ca compute all the values c i by usig umerical differetiatio: c i f ( x 1,..., x i 1, x i + ε i, x i+1,..., x ) ỹ ε i for some small ε i. Mai limitatio of the straightforward algorithm: it takes too log. Whe f (x 1,...,x ) is a simple expressio, the above straightforward algorithm is very efficiet. However, i may cases e.g., with weather predictio or oil exploratio the correspodig algorithm f (x 1,...,x ) is very complex ad time-cosumig, requirig hours of computatio o a high performace computer, while processig thousads of data values x i. I such situatios, the above algorithm requires + 1 calls to the program that implemets the algorithm f (x 1,...,x ): oe time to compute ad the times to compute values ỹ = f ( x 1,..., x ), f ( x 1,..., x i 1, x i + ε i, x i+1,..., x ) eeded to compute the correspodig partial derivatives c i. Whe each call to f takes hours, ad we eed to make thousads of such class, the resultig computatio time is i years. This makes the whole exercise mostly useless: whe it takes hours to predict the weather, o oe will wait more tha a year to check how accurate is this predictio. It is therefore ecessary to have faster methods for ucertaity propagatio. Much faster methods exist for momets (ad cumulats) of eve order k. For all k, the computatio of the value κ k = (c i ) k κ ik ca be doe much faster, by usig the followig Mote-Carlo simulatios. Several times l = 1,2,...,L, we:
Negative Probabilities Ca Speed Up Ucertaity Propagatio 11 simulate the values x (l) i accordig to some distributio of x i with the give value κ ik, ad estimate y (l) = ỹ f ( x 1 x (l) 1,..., x x (l) ). Oe ca show that i this case, the k-th cumulat of the resultig distributio for y (l) is equal to exactly the desired value κ k = (c i ) k κ ik. Thus, by computig the sample momets of the sample y (1),..., y (L), we ca fid the desired k-th order cumulat. For example, for k = 2, whe the cumulat is the variace, we ca simply use ormal distributios with a give variace. The mai advatage of the Mote-Carlo method is that its accuracy depeds oly o the umber of iteratios: its ucertaity decreases with L as 1/ L; see, e.g., [13]. Thus, for example: to get the momet with accuracy 20% (= 1/5), it is sufficiet to ru approximately 25 simulatios, i.e., approximately 25 calls to the algorithm f ; this is much much faster tha thousads of iteratios eeded to perform the straightforward algorithm. For eve k, the value (c i ) k is always o-egative, so (c i ) k = (c i ) k, ad the formula for k get a simplified form k = (c i ) k ik. This is exactly the same form as for κ k, so we ca use the same Mote-Carlo algorithm to estimate k the oly differece is that ow, we eed to use distributios of x i with the k-th cumulat equal to ik. Specifically, several times l = 1,2,...,L, we: simulate the values x (l) i accordig to some distributio of x i with the value ik of the k-th cumulat, ad estimate y (l) = ỹ f ( x 1 x (l) 1,..., x x (l) ). Oe ca show that i this case, the k-th cumulat of the resultig distributio for y (l) is equal to exactly the desired value
12 A. Powuk ad V. Kreiovich k = (c i ) k ik. Thus, by computig the sample momets of the sample y (1),..., y (L), we ca fid the desired boud k o the k-th order cumulat. Odd order momets (such as skewess) remai a computatioal problem. For odd k, we ca still use the same Mote-Carlo method to compute the value κ k. However, we ca o loger use this method to compute the boud k o the k-th cumulat, sice for odd k, we o loger have the equality (c i ) k = (c i ) k. What we pla to do. We will show that the use of (hypothetical) egative probabilities eables us to attai the same speed up for the case of odd k as we discussed above for the case of eve orders. 4 Aalysis of the Problem ad the Resultig Negative-Probability-Based Fast Algorithm for Ucertaity Quatificatio Why the Mote-Carlo method works for variaces? The possibility to use ormal distributios to aalyze the propagatio of variaces V = σ 2 comes from the fact that if we have idepedet radom variables x i with variaces V i = σi 2, the their liear combiatio y = is also ormally distributed, with variace V = c i x i (c i ) 2 V i
Negative Probabilities Ca Speed Up Ucertaity Propagatio 13 ad this is exactly how we wat to relate the variace (2-d order cumulat) of y with the variaces V i of the iputs. Suppose that we did ot kow that the ormal distributio has this property. How would we the be able to fid a distributio ρ 1 (x) that satisfies this property? Let us cosider the simplest case of this property, whe V 1 =... = V = 1. I this case, the desired property has the followig form: if idepedet radom variables x 1,..., x have exactly the same distributio, with variace 1, the their liear combiatio y = c i x i has the same distributio, but re-scaled, with variace V = (c i ) 2. Let ρ 1 (x) deote the desired probability distributio, ad let χ 1 (ω) = E[exp(i ω x 1 )] be the correspodig characteristic fuctio. The, for the product c i x i, the characteristic fuctio has the form E[exp(i ω (c i x 1 )]. By re-arragig multiplicatios, we ca represet this same expressio as E[exp(i (ω c i ) x 1 ], i.e., as χ 1 (c i ω). For the sum of several idepedet radom variables, the characteristic fuctio is equal to the product of characteristic fuctios (see, e.g., [13]); thus, the characteristic fuctio of the sum c i x i has the form χ 1 (c 1 ω)... χ 1 (c ω). We require that this sum be distributed the same way as x i, but with a larger variace. Whe we multiply a variable by c, its variable icreases by a factor of c 2. Thus, to get the distributio with variace
14 A. Powuk ad V. Kreiovich V = (c i ) 2, we eed to multiply the variable x i by a factor of c = (c i ) 2. For a variable multiplied by this factor, the characteristic fuctio has the form χ 1 (c ω). By equatig the two characteristic fuctios, we get the followig fuctioal equatio: ) χ 1 (c 1 ω)... χ 1 (c ω) = χ 1 ( (c i ) 2 ω I particular, for = 2, we coclude that ) χ 1 (c 1 ω) χ 1 (c 2 ω) = χ 1 ( (c 1 ) 2 + (c 2 ) 2 ω. This expressio ca be somewhat simplified if we take logarithms of both sides. The products tur to sums, ad for the ew fuctio we get the equatio l(c 1 ω) + l(c 2 ω) = l l(ω) def = l(χ 1 (ω)), ( ) (c 1 ) 2 + (c 2 ) 2 ω. This equatio ca be further simplified if we cosider a auxiliary fuctio for which F(ω) def = l( ω), l(x) = F(x 2 ). Substitutig the expressio for l(x) i terms of F(x) ito the above formula, we coclude that F((c 1 ) 2 ω 2 ) + F((c 2 ) 2 ω 2 ) = F(((c 1 ) 2 + (c 2 ) 2 ) ω 2 ). Oe ca easily check that for every two o-egative umbers a ad b, we ca take ad thus tur the above formula ito ω = 1, c 1 = a, ad c 2 = b,.
Negative Probabilities Ca Speed Up Ucertaity Propagatio 15 F(a) + F(b) = F(a + b). It is well kow (see, e.g., [1]) that every measurable solutio to this fuctioal equatio has the form F(a) = K a for some costat K. Thus, Here, hece l(ω) = F(ω 2 ) = K ω 2. l(ω) = l(χ 1 (ω)), χ 1 (ω) = exp(l(ω)) = exp(k ω 2 ). Based o the characteristic fuctio, we ca recostruct the origial probability desity fuctio ρ 1 (x). Ideed, from the purely mathematical viewpoit, the characteristic fuctio χ(ω) = E[exp(i ω x 1 )] = exp(i ω x 1 ) ρ 1 ( x 1 )d( x 1 ) is othig else but the Fourier trasform of the probability desity fuctio ρ 1 ( x 1 ). We ca therefore always recostruct the origial probability desity fuctio by applyig the iverse Fourier trasform to the characteristic fuctio. For χ 1 (ω) = exp(k ω 2 ), the iverse Fourier trasform leads to the usual formula of the ormal distributio, with K = σ 2. Ca we apply the same idea to odd k? Our idea us to use Mote-Carlo methods for odd k, to speed up the computatio of the value k = (c i ) k ik. What probability distributio ρ 1 (x) ca we use to do it? Similar to the above, let us cosider the simplest case whe 1k =... = k = 1. I this case, the desired property of the probability distributio takes the followig form: if idepedet radom variables
16 A. Powuk ad V. Kreiovich x 1,..., x have exactly the same distributio ρ 1 (x), with k-th cumulat equal to 1, the their liear combiatio y = c i x i has the same distributio, but re-scaled, with the k-th order cumulat equal to c i k. Let ρ 1 (x) deote the desired probability distributio, ad let χ 1 (ω) = E[exp(i ω x 1 )] be the correspodig characteristic fuctio. The, as we have show earlier, for the product c i x i, the characteristic fuctio has the form χ 1 (c i ω). For the sum the characteristic fuctio has the form c i x i, χ 1 (c 1 ω)... χ 1 (c ω). We require that this sum be distributed the same way as x i, but with a larger k-th order cumulat. As we have metioed: whe we multiply a variable by c, its k-th order cumulat icreases by a factor of c k. Thus, to get the distributio with the value c i k, we eed to multiply the variable x i by a factor of c = k c i k. For a variable multiplied by this factor, the characteristic fuctio has the form χ 1 (c ω).
Negative Probabilities Ca Speed Up Ucertaity Propagatio 17 By equatig the two characteristic fuctios, we get the followig fuctioal equatio: ( ) k χ 1 (c 1 ω)... χ 1 (c ω) = χ 1 c i k ω. I particular, for = 2, we coclude that ( ) k χ 1 (c 1 ω) χ 1 (c 2 ω) = χ 1 c 1 k + c 2 k ω. This expressio ca be somewhat simplified if we take logarithms of both sides. The products tur to sums, ad for the ew fuctio l(ω) def = l(χ 1 (ω)), we get the equatio ( ) k l(c 1 ω) + l(c 2 ω) = l ( c 1 k + c 2 k ω. This equatio ca be further simplified if we cosider a auxiliary fuctio for which F(ω) def = l( k ω), l(x) = F(x k ). Substitutig the expressio for l(x) i terms of F(x) ito the above formula, we coclude that F( c 1 k ω k ) + F( c 2 k ω k ) = F(( c 1 k + c 2 k ) ω k ). Oe ca easily check that for every two o-egative umbers a ad b, we ca take ad thus get As we have already show, this leads to for some costat K. Thus, ω = 1, c 1 = k a, ad c2 = k b F(a) + F(b) = F(a + b). F(a) = K a l(ω) = F(ω k ) = K ω k. Here, l(ω) = l(χ 1 (ω)),
18 A. Powuk ad V. Kreiovich hece χ 1 (ω) = exp(l(ω)) = exp(k ω k ). Case of k = 1 leads to a kow efficiet method. For k = 1, the above characteristic fuctio has the form exp( K ω ). By applyig the iverse Fourier trasform to this expressio, we get the Cauchy distributio, with probability desity ρ 1 (x) = 1 π K 1 1 + x2 K 2 Mote-Carlo methods based o the Cauchy distributio ideed lead to efficiet estimatio of first order ucertaity e.g., bouds o mea; see, e.g., [6]. What about larger odd values k? Alas, for k 3, we have a problem: whe we apply the iverse Fourier trasform to the characteristic fuctio exp( K ω k ), the resultig fuctio ρ 1 ( x 1 ) takes egative values for some x, ad thus, caot serve as a usual probability desity fuctio; see, e.g., [12]. However: if egative probabilities are physically possible, the we ca ideed use the same idea to speed up computatio of k for odd values k 3. If egative probabilities are physically possible, the we ca speed up ucertaity propagatio amely, computatio of k. If egative probabilities are ideed physically possible, the we ca use the followig algorithm to speed up the computatio of k. Let us assume that we are able to simulate a radom variable η whose (sometimes egative) probability desity fuctio ρ 1 (x) is the iverse Fourier trasform of the fuctio χ 1 (ω) = exp( ω k ). We will use the correspodig radom umber geerator for each variable x i ad for each iteratio l = 1,2,...,L. The correspodig value will be deoted by η (l) i. The value η (l) i will correspods to the value of the k-th cumulat equal to 1. To simulate a radom variable correspodig to parameter ik, we use ( ik ) 1/k η (l) i..
Negative Probabilities Ca Speed Up Ucertaity Propagatio 19 Thus, we arrive at the followig algorithm: Several times l = 1,2,...,L, we: simulate the values x (l) i ad estimate as ( ik ) 1/k η (l) i, y (l) = ỹ f ( x 1 x (l) 1,..., x x (l) ). Oe ca show that i this case, the resultig distributio for y (l) has the same distributio as η multiplied by the k-th root of the desired value k = (c i ) k ik. Thus, by computig the correspodig characteristic of the sample y (1),..., y (L), we ca fid the desired boud k o the k-th order cumulat. So, we ca ideed use fast Mote-Carlo methods to estimate both values κ k ad k ad thus, to speed up ucertaity propagatio. Ackowledgmets This work was supported i part: by the Natioal Sciece Foudatio grats HRD-0734825 ad HRD-1242122 (Cyber-ShARE Ceter of Excellece) ad DUE-0926721, ad by the award UTEP ad Prudetial Actuarial Sciece Academy ad Pipelie Iitiative from Prudetial Foudatio. Refereces 1. J. Aczél, Lectures o Fuctioal Equatios ad Their Applicatios, Dover, New York, 2006. 2. P. A. M. Dirac, The physical iterpretatio of quatum mechaics, Proceedigs of the Royal Society A: Mathematical, Physical ad Egieerig Scieces, 1942, Vol. 180, No. 980, pp. 1 39. 3. R. P. Feyma, Negative probability, I: F. D. Peat ad B. Hiley (eds.), Quatum Implicatios: Essays i Hoour of David Bohm, Routledge & Kega Paul Ltd., Abigdo-o-Thames, UK, 1987, pp. 235 248.
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