Modeling and Estimating Uncertainty in Parameter. Estimation. Center for Research in Scientic Computation. Box 8205.

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1 Modeling and Estimating Uncertainty in Parameter Estimation H.T. Banks and Kathleen L. Bihari Center for Research in Scientic Comutation Box 825 North Carolina State University Raleigh, NC Fax: Abstract In this aer we discuss uestions related to reliability or variability of estimated arameters in deterministic least suares roblems. By viewing the arameters for the inverse roblem as realizations for a random variable we are able to use standard results from robability theory to formulate a tractable robabilistic framework to treat this uncertainty. We discuss method stability and aroximate roblems and are able to show convergence of solutions of the aroximate roblems to those of the original roblem. The ecacy of our aroach is demonstrated in numerical examles involving estimation of constant arameters in dierential euations. Keywords: Parameter estimation, measures of variability, robability distributions, method stability, aroximation and comutational methods

2 Introduction A standard deterministic inverse roblem freuently encountered in both alied and theoretical literature can be abstractly stated as follows: Given a arameter deendent dynamical or algebraic system A( u()) = F() () with states u(), arameters and oerators (dierential or algebraic) A, use observations (ossibly incomlete) or data on the states to determine the best arameters in some admissible set Q so that the solution of euation () for = best describes the data. For such deterministic roblems there is a large literature based on diverse formulations (least suares, euation error, etc.). For discussions of some of these see [3]. Once one has \solved" this (by no means trivial) deterministic roblem, it is freuently imortant toknow something about the reliability ofthe estimates. One aroach entails attaching \error bars" to the estimated arameter values, much like one does in standard statistical analysis or in scientic comutational analysis (using a riori bounds) with nite discretization techniues (nite dierence, nite elements, etc) from numerical analysis. In essence we are asking for measurements of uncertainty (inherent in our methods rather than in our data collection) related to our best estimates of arameters. Thus we are led in a comletely natural way to stochastic or robabilistic asects of estimates from a deterministic roblem solved with deterministic algorithms. We oer here ideas for one aroach to treatment ofvariability in arameter estimation techniues. The aroach is based on viewing multile observations f^u j g N j= of the state in () as observations corresonding to a set of realizations f j g N j= of the arameter which isnow thought of as a secic (albeit unknown) random variable with robability distribution P on Q. The system () is accordingly reformulated in terms of the state u = u(p ) deending on the robability P distribution P. The observations f^u j g can be averaged so that ^u = N N ^u j= j is an observation for u(p ) and one can then attemt to estimate a best distribution P to t this data ^u, for examle in some tye of least suares t. Once one obtains P, its mean and variance 2 can be used as a best arameter estimate and measure of reliability, resectively. In the sections below wegivea concrete examle of this (using nonlinear arameter deendent ordinary dierential euations for the system ()). We resent a recise formulation of this concetual aroach, show that fundamental results from robability theory can be used to develo well-osedness results (existence, continuous deendence, and method stability) along with aroximation ideas that are comutationally tractable. We demonstrate feasibility of the resulting algorithms by resenting a summary of numerical ndings using an examle arising in estimation of eectiveness of vaccination olicies in a oulation of suscetibles in disease rohylactics.

3 We believe the underlying hilosohy as well as the secic formulation are alicable to and will be useful in a wide class of ractical alications. 2 Parameter Estimation in Nonlinear Systems To demonstrate our ideas, we will examine the estimation of constant arameters in a system of ordinary dierential euations. These ideas can be easily extended to many systems of interest in alications, including systems of artial dierential euations with unknown functional arameters (e.g., time and/or satially deendent coecients). Atyical estimation roblem emloys observations ^x = f ^x i g n i= for x(t i ), i = 2 ::: n, to estimate arameters 2 R m in the vector dynamical system _x(t) =f(t x(t) ): (2) Often a least suares formulation is used to nd a best arameter value in some admissible arameter set Q R m. In other words, we attemt to nd 2 Q which is a minimum for J( ^x) = i= jx(t i ) ; ^x i j 2 over 2 Q where x(t i ) isasolutionof(2)foragiven 2 Q. To introduce uncertainty, we view the arameters as realizations for a random variable and use the data to estimate the robability distribution function (PDF) for this random variable. Secically, letp(q) denote the set of robability distributions on Q and treat the data f ^x i g as observations for the exected value E[x(t i )jp ]= Z Q x(t i )dp () (3) for a given PDF P 2 P(Q). Note that if P is a discrete PDF with atoms f j g M j= Q and associated robabilities f j g, j, P M j= j =, then (3) can be written Z MX x(t i )dp ()= x(t i j ) j : Q j= Regardless of the form of P, the least suares estimation roblem can be described as nding P 2 P(Q) to minimize J(P )= i= je[x(t i )jp ] ; ^x i j 2 (4) 2

4 over P 2 P(Q). To develo theoretical and comutational results for this roblem, it is necessary to have a toology on P(Q), continuity of the function P! J(P ) in this toology, comatible comactness results, and some aroximation results leading to imlementable comutational algorithms. In order to address these issues we will introduce the Prohorov metric and summarize some results from Billingsley [5]. 3 The Prohorov Metric in the Sace of Probability Distributions Let P(Q) be the set of robability measures on the Borel subsets of Q, where Q is any comlete metric sace with metric d. For any closed subset F Q and >, we dene an -neighborhood of F by F = f 2 Q : d(~ ) < ~ 2 F g: We then dene : P(Q) P(Q)! R + by (P P 2 ) inff >:P [F ] P 2 [F ]+ F closed F Qg: The following roerties of are well-known: (a) is a metric (called the Prohorov metric) on P(Q) (b) (P(Q) ) is a comlete metric sace (c) if Q is comact, then (P(Q) ) is a comact metric sace. We would like to understand convergence of P k! P in the metric. Unfortunately, the Prohorov metric is neither intuitive nor easy to use directly. However,it is well known that if (Q d) is a comlete metric sace and (P(Q) ) is dened as above, then for P k P 2 P(Q), the following convergence statements are euivalent: (i) (P k P)! (ii) R Q fdp k()! R Q fdp() for all bounded, uniformly continuous f : Q! R (iii) P k [A]! P [A] for all Borel sets A Q with P [@A] =, denotes the boundary of A. The euivalence of (i) and (iii) reveals that convergence in the metric is euivalent to convergence in distribution. Moreover, if we consider P(Q) CB (Q) where C B (Q) denotes the sace of bounded, continuous functions on Q with the suremum norm, then (i) and (ii) imly that convergence in the toology is euivalent 3

5 to weak* convergence in P(Q). For our discussions, we will make critical use of the euivalence between (P k P)! and Z Q which in turn is the same as x(t )dp k ()! Z x(t )dp () Q E[x(t )jp k ]! E[x(t )jp ]: This convergence is needed to establish continuity in the toology of the ma P! J(P )= i= je[x(t i )jp ] ; ^x i j 2 : Continuity of this ma, along with the comactness of Q, which guarantees comactness of P(Q) inthe metric, is sucient to establish existence of a solution to the roblem of minimizing (4) over P(Q). If we assume existence uestions are answered, and turn to the task of characterizing and/or nding minimizers, we note that P(Q) with the metric is in general an innite dimensional sace because in general Q will be innite dimensional. Thus, to address comutational issues one must consider aroximation ideas. To do this we rst rove a density theorem that will be useful in establishing continuous deendence of estimates on data as well as in constructing aroximation schemes. There are numerous toologies on P(Q). If we dene a W-neighborhood of P as N (P )=fp 2 P(Q) :P (F i ) <P(F i )+ i = k F i closed F i 2 Sg for a given >and nite set ff i g k i=, this induces a toology on P(Q) which is euivalent to the toology of weak convergence, W (see [5],. 236). We can also dene a -neighborhood of P by N (P )=fp 2 P(Q) :(P P ) <gfor a given >. If Q is a searable sace, the W toology is euivalent tothetoology induced by the -neighborhoods. We will be using the euivalence of these two toologies in the roof of Theorem 3.. Here N + are the ositive integers, R are the rational numbers, and j is the Dirac measure with atom at j. Theorem 3. Let Q be a comlete, searable metric sace with metric d, S be the class of all Borel subsets of Q and P(Q) be thesace ofrobability measures on (Q S). Let Q = f j g j= beacountable, dense subset of Q. Then the set of P 2 P(Q) such that P has nite suort in Q and rational masses is dense in P(Q) in the metric. That is, P (Q) fp 2 P(Q) :P = kx j= is dense in P(Q) relative to. j j k2 N + j 2 Q j 2 R j kx j= j =g 4

6 Proof: Let> and let P 2 P(Q). Let N (P )bea-neighborhood of P. Since Q is searable, the W and toologies are euivalent. Thus there is a >such that N (P ) N (P ), where N (P )isaw-neighborhood of P of the form described above with closed sets F F k. S Let fb i g M be the artition of k F i= i= i Q generated by the closed sets F F k. We will assume each B i is non-emty and so M <. Since Q is a dense subset of Q, B i \ Q 6=, for i = M. For i = M, select a oint x i 2 B i \ Q.Ateachoint, x i, lace a mass, b i, which satises the following three conditions: i) b i 2 R, ii) b i P (B i ), and iii) jp (B i ) ; b i j < 2M. Now if[ k i=f i 6= Q, select a oint x M+ so that x M+ 2 Q \ ([ k i=f i ) C. If [ k F i= i = Q, choose x M+ so that x M+ 2 Q n (fx i g M i= ). In either case, lace at x M+ amassb M+ P P M ; b i= i. Note that P b M+ 2 R and b M+. Dene P M+ = b i= i i. Then P M+ (Q) = b i= i =,andp (A) for all A 2 S. Thus P 2 P (Q). Dene K i = fj : F i \ B j 6= j Mg. Note the set K i has at most M indices. Now suose [ k F i= i 6= Q. Then for any [ F i, [ jp (F i ) ; P (F i )j = jp ( B j ) ; P ( B j )j X j2k i X j2k i = j P (B j ) ; P (B j )j j2k X i X j2k i = j b j ; P (B j )j j2k X i j2k i = j [b j ; P (B j )]j j2k i X jb j ; P (B j )j j2k X i < 2M j2k i 2 < : Now suose [ k i= F i = Q. Ifx M+ =2 F i, the above argument shows jp (F i ) ; P (F i )j <.Ifx M+ 2 F i,then 5

7 [ [ jp (F i ) ; P (F i )j = jp ( B j ) ; P ( B j )j X j2k i X j2k i = j P (B j ) ; P (B j )j j2k X i j2k X i = j b j + b M+ ; P (B j )j j2k i j2k i MX X = j [b j ; P (B j )]+[; b j ]j j2k i j= X MX MX = j [b j ; P (B j )]+[ P (B j ) ; b j ]j j2k i j= j= X MX = j [b j ; P (B j )] + [P (B j ) ; b j ]j j2k i j= < X j2k i jb j ; P (B j )j + X j2k i : 2M + M X j= 2M MX j= jp (B j ) ; b j j Thus for all F i, P (F i ) <P(F i )+, sop 2 N (P ). Since N (P ) N (P ), P 2 N (P ). By construction P 2 P (Q), so P (Q) is dense in P(Q) relative to. Theorem 3. can be used as a basis for dening a class of aroximating sets to be used in tractable comutational methods for the inverse roblems dened in Section 2. First dene Q d = [ M= Q M (5) 6

8 where Q M = f M j g M j= M = 2, are chosen so that Q d is dense in Q. Note that Q d is countable. For each ositive integer M let P M (Q) =fp 2 P(Q) :P = If we then dene MX j= j M j M j 2 Q M j 2 R j MX j= j =g: (6) P d (Q) =[ M=P M (Q) (7) then by Theorem 3. we know P d (Q) is dense in P(Q), and so we can aroximate any element P 2 P(Q)by a seuence fp Mj g, P Mj 2 P Mj (Q), such that (P Mj P)! asm j!. 4 Stability of the Inverse Problem We now turn to the study of the inverse roblem. We return to our original roblem of nding a solution to min J(P ^x) = P2P(Q) i= jx(t i P) ; ^x i j 2 : (8) Given data ^x k and ^x such that^x k! ^x as k!and corresonding solutions P (^x k ) and P (^x) (which in general are sets because there is not necessarily a uniue minimizer of (8)), we say the roblem is continuously deendent on the data (or stable) ifdist(p (^x k ) P (^x))! ask!(see [3, 4] for detailed discussions and motivation). We now dene a series of aroximate roblems. Let P M (Q) bedenedasin (6) where Q d is a countable dense subset of Q as dened in (5) with Q M = f M j g. We dene the aroximate roblem as nding a solution to min J(P M ^x) = P M2P M (Q) i= jx(t i P M ) ; ^x i j 2 : (9) Let PM (^x) denote the set of solutions for a given ^x. The roblems are method stable (again, see [3, 4] for further discussions) if for any data ^x k and ^x such that^x k! ^x as k!we have dist(pm (^xk ) P (^x))! ask!and M!. Note that this is euivalent to reuiring dist(pm (^xk ) PM (^x))! ask!uniformly in M. Theorem 4. Let Q be acomact metric sace and assume solutions x(t ) of (2) are continuous in on Q. Let P(Q) be the set of all robability measures on Q and let Q d beacountable dense subset of Q as dened in (5) with Q M = fj MgM j=. Dene P d (Q) as in (7) where P M (Q) is dened asin(6). Suose P M (^xk ) is the set 7

9 of minimizers for J(P ) over P 2 P M (Q) corresonding to the data f^x k g and P (^x) is the set of minimizers over P 2 P(Q) corresonding to f^xg where ^x k, ^x 2 R n are the observed data such that ^x k! ^x. Then dist(p M (^xk ) P (^x))! as M!and ^x k! ^x. Thus the solutions deend continuously on the data and the aroximate roblems are method stable. Proof: Since Q is a comact, searable metric sace, Q d is dense in Q and P d (Q) is the sace of all robability measures with nite suort in Q d and rational masses, it follows from Theorem 3. that P d (Q) is a dense subset of P(Q). Since Q is comact, (P(Q) ) is comact, where is the Prohorov metric. Since! x(t ) is continuous from Q to R n, whenever P M! P in P(Q) wehave that lim J(P M) = lim M! = je[x(t i )jp M ] ; ^x i j 2 M! i= i= = J(P ): je[x(t i )jp ] ; ^x i j 2 Since P M (Q) is a closed subset of P(Q), P M (Q) is comact in the toology. Moreover, J is a continuous function on the comact set P M (Q) and thus for a given set of observations ^x k = f^x k i gn 2 i= Rn, there exists a (not necessarily uniue) minimizer PM k which is a solution to the roblem of minimizing Jk (P )over P 2 P M (Q) where J k (P ) J(P ^x k ) i= je[x(t i )jp ] ; ^x k i j 2 : Let f^x k g be a seuence that converges to some arbitrary ^x 2 R n. Let PM (^xk ) denote the set of minimizers of J k (P )over P M (Q). For k = 2 and M = 2, let fpm k g, P k M 2 P M (^xk ), be any seuence of minimizers in P(Q). By comactness there exists a convergent subseuence fp kj g such that M` in the metric. First note that for any P M` 2 P M`(Q) () lim P kj = ~ M` k j! M` P 2 P(Q) () J kj (P kj M` ) J kj (P M`): (2) 8

10 Then by denition of J kj (P kj ), (), and () M` lim k Jkj (P kj ) = lim j M`! M` = k j M`! i= i= = J( ~ P): je[x(t i )jp kj M` ] ; ^x kj i j2 je[x(t i )j ~ P] ; ^xi j 2 Let P be any element ofp(q). Since P d (Q) is dense in P(Q) we can nd a seuence fp M`g, P M` 2 P M`(Q), so that P M`! P in the metric as M`!. Then by denition of J kj (P M`), and () lim k Jkj (P M`) = lim j M`! So (2), (3), and (4) gives = k j M`! i= i= = J(P ): je[x(t i )jp M`] ; ^x kj i j2 je[x(t i )jp ] ; ^x i j 2 J( ~ P ) J(P ) for any P 2 P(Q). Hence P ~ is a minimizer of J(P )over P 2 P(Q), i.e., P ~ 2 P (^x). Thus any seuence PM k in P M (^xk ) has a subseuence P kj that converges to a M` ~P 2 P (^x). So dist(pm`(^x kj ) P (^x))! asm`!and k j!.itfollows that dist(pm (^xk ) P (^x))! asm!if ^x k! ^x. (3) (4) In order to address comutational issues, we use the family of aroximate minimization roblems dened above. If Q d, P M (Q), and P d (Q) are dened as in Theorem 4., we know from Theorem 3. we can aroximate any P 2 P(Q) by P d 2 P d (Q). Furthermore, from the results established above we also know we can aroximate any P 2 P(Q) by distributions P M 2 P M (Q). By choosing M suciently large we obtain Z Q x(t i ) dp () Thus we can aroximate J() by Z J M () MX x(t i ) Q j= i= j^x i ; 9 M j () dp () = MX j= x(t i M j ) j j 2 MX j= x(t i M j ) j :

11 where =( 2 M ) and P M j= j =, j, j 2 R, j M. If we dene X M i = (x(t i M ) x(t i M 2 ) ::: x(t i M M ))T X M =[X M XM n ] ^X = (^x ^x n ) then we can write J M () = i= j^x i ; X M i j 2 = jj ^X ; X M jj 2 2 : The aroximate minimization roblem thus reduces to a constrained otimization roblem for a uadratic cost functional. Such roblems are amenable to a number of standard algorithms. Thus we see that any solution must satisfy X M = ^X and that if X M is nonsingular, is uniuely determined and deends continuously on the data ^X. 5 Examles We resent a series of examles to illustrate a comutational algorithm arising from the discussions of the revious sections. Each examle uses the same system motivated by a roblem in the assessment of the eciency of a vaccination rogram [6, 2] by using data f^x i g for the aggregate oulation x(t i )ofvaccinated but not infected individuals at time t i.theevolution of the oulation is given by _x(t) =;G(t)x(t) x() = x (5) where G(t) reresents the known rate of exosure to infection, is the suscetibility to \environmental exosure" (a arameter to be chosen from an admissible arameter set Q) subseuent tovaccination of the oulation at time t =,andx is the known number of individuals initially vaccinated. All calculations were carried out using MATLAB routines. We letq =[ ] and dene Q M = f j; M; gm. Note Q j= d = [ Q M= M is a countable, dense subset of Q. Foragiven ositive integer M, we would like to nd a P that is a solution to (9) where P M (Q) is dened as in (6) with our given Q M. The data f^x i g is simulated data we generate. To generate simulated data we start by taking N samles, fj SgN on j= from Q. The time interval T =[ ] is discretized by t i = n i, i n, where n =:. \Data" f^x i g is generated by rst solving (5) at each oint x(t i j S )withg(t) given by G(t) = 3 8 < : t : t;: :8 : <t :9 :9 <t :

12 P Then, to average solutions from all samles, we dene x i = N N x(t j= i j S ). Relative random noise was added to the solutions so that the \data" was given by ^x i = x i [ + i ] where the i are indeendent Gaussian random variables with mean zero and variance 2. In our rst examle, the samles of are chosen from a normally distributed random variable on Q with mean :5 and standard deviation :. The following gures dislay the otimal estimated discrete robability densities reresented P by M =( M ) and the corresonding robability distributions P M = j= j j. In Figure we resent results of the otimization using data that was generated as described with % relative error. In Figure 2 the data was generated with 5% relative error, and in Figure 3 data was generated with % relative error. Note in each case as M increases the robability distributions are converging in the Prohorov metric as guaranteed by the theory, while the discrete densities do not converge in any sense on Q =[ ]. In each lot of the robability density, the \x"'s are the actual distribution of the generated data for and in each lot of the robability distribution, the dashed line is the continuous distribution associated with that the discrete distributions are attemting to aroximate. The otimal distributions are grahed with iecewise constant solid lines. In addition to testing the inverse roblem on with a Gaussian distribution, we also carried out the inverse roblem with M = 9 and % and 5% relative error for the following ( x(t i ) is again reresented by a M vector of values and the distribution for is aroximated by =( 9 )): A delta function at :5, =:5, with results resented in Figure 4 Two delta functions at :25 and :75, :25 j<n=2 (j) = :75 N=2 j<, with results in Figure 5 Two skewed delta functions at :5 and :6, :5 j<n=3 (j) = :6 N=3 j<, with results in Figure 6 A uniform distribution on [:25 :75], with results in Figure 7 A uniform distribution on [: :3] [ [:55 :75], with results in Figure 8 A bimodal Gaussian distribution with one mean at :3 and the other at :7 and standard deviation :6, with results in Figure 9 A right skewed distribution with mean and standard deviation :2, with results in Figure. In each examle the estimated robability distribution is a reasonable aroximation of the continuous distribution, both with no error and with 5% relative error on the data.

13 .2 Prob. Density rel. error=%.2 Prob. Distribution rel. error=% M= Prob. Density rel. error=%.2 Prob. Distribution rel. error=% M= Prob. Density rel. error=% Prob. Distribution rel. error=% Prob. Density rel. error=%.2 Prob. Distribution rel. error=% M= Prob. Density rel. error=% M= Prob. Distribution rel. error=% Figure : Aroximate robability densities and robability distributions for normally distributed and % error on data.

14 .2 Prob. Density rel. error=5%.2 Prob. Distribution rel. error=5% M= Prob. Density rel. error=5%.2 Prob. Distribution rel. error=5% M= Prob. Density rel. error=5% Prob. Distribution rel. error=5% Prob. Density rel. error=5%.2 Prob. Distribution rel. error=5% M= Figure 2: Aroximate robability densities and robability distributions for normally distributed and 5% error on data. 3

15 .2 Prob. Density rel. error=% M=3.2 Prob. Distribution rel. error=% Prob. Density rel. error=%.2 Prob. Distribution rel. error=% M= Prob. Density rel. error=% Prob. Distribution rel. error=% Prob. Density rel. error=%.2 Prob. Distribution rel. error=% M= Figure 3: Aroximate robability densities and robability distributions for normally distributed and % error on data. 4

16 .2 Prob. Density rel. error=%.2 Prob. Distribution rel. error=% Prob. Density rel. error=5%.2 Prob. Distribution rel. error=5% Figure 4: Aroximate robability density and robability distribution for =:5 with M = 9 and relative errors % and 5%. 5

17 .2 Prob. Density rel. error=%.2 Prob. Distribution rel. error=% Prob. Density rel. error=5%.2 Prob. Distribution rel. error=5% Figure 5: Aroximate robability density and robability distribution for two delta functions with M = 9 and relative errors % and 5%. 6

18 .2 Prob. Density rel. error=%.2 Prob. Distribution rel. error=% Prob. Density rel. error=5%.2 Prob. Distribution rel. error=5% Figure 6: Aroximate robability density and robability distribution for two skewed delta functions with M = 9 and relative errors % and 5%. 7

19 .2 Prob. Density rel. error=%.2 Prob. Distribution rel. error=% Prob. Density rel. error=5%.2 Prob. Distribution rel. error=5% Figure 7: Aroximate robability density and robability distribution for uniformly distributed on [:25 :75] with M = 9 and relative errors % and 5%. 8

20 .2 Prob. Density rel. error=%.2 Prob. Distribution rel. error=% Prob. Density rel. error=5%.2 Prob. Distribution rel. error=5% Figure 8: Aroximate robability density and robability distribution for uniformly distributed on [: :3] [ [:55 :75] with M = 9 and relative errors % and 5%. 9

21 .2 Prob. Density rel. error=%.2 Prob. Distribution rel. error=% Prob. Density rel. error=5%.2 Prob. Distribution rel. error=5% Figure 9: Aroximate robability density and robability distribution for binormally distributed with mean :3 and :7 and standard deviation :6 for each with M = 9 and relative errors % and 5%. 2

22 .2 Prob. Density rel. error=%.2 Prob. Distribution rel. error=% Prob. Density rel. error=5%.2 Prob. Distribution rel. error=5% Figure : Aroximate robability density and robability distribution for right skew distributed with mean and standard deviation:2 with M = 9 and relative errors % and 5%. 2

23 6 Concluding Remarks In the discussions above we have resented one aroach to the uantication and comutational treatment of uncertainty in inverse roblems of a least suares formulation. By treating the estimated arameter as a random variable with unknown distribution, we concetually reformulate the deterministic arameter estimation roblem into a roblem of estimation of a random variable using samled data from a dynamical system which deends on the arameter. We use owerful but basic results from robability theory to develo a theoretical basis for these new roblems. Aroximation results along with continuous deendence of estimates on data and method stability are discussed. To test the ideas we resent a series of numerical examles based on a single model arising in vaccination and suscetibility roblems. The comutational results resented suort the ecacy of our aroach and illustrate well the theoretical convergence results given in the aer. Acknowledgments This research was suorted in art by the U.S. Air Force Oce of Scientic Research under grants AFOSR F , AFOSR F , and AFOSR F References [] H.T. Banks, Remarks on Uncertainty Assessment and Management in Modeling and Comutation, CRSC-TR98-39, November, 998. [2] H.T. Banks, B.G. Fitzatrick, and Y. Zhang, \Estimation of Distributed Individual Rates from Aggregate Poulation Data", CRSC-TR94-3, Set. 994, in Dierential Euation and Alications to Biology and to Industry, ed.by M. Martelli et al, World Scientic Press, 996, [3] H.T. Banks and K. Kunisch. Estimation Techniues for Distributed Parameter Systems, Birkhauser, Boston, 989. [4] H.T. Banks, R.C. Smith and Y. Wang. Smart Material Structures Modeling, Estimation and Control, Masson/J. Wiley, Paris/Chichester, 996. [5] P. Billingsley. Convergence of Probability Measures, Wiley, New York, 968. [6] R.C. Brunet, C.J. Struchiner and M.E. Halloran, \On the Distribution of Vaccine Protection Under Heterogeneous Resonse", Math. Biosc., 6 (993),