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Can philosophy be of any use in counting statistics?
Nuclear Instruments and Methods in Physics Research A309 (1991) 555-559 North-Holland
NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH Section A
Can philosophy be of any use in counting statistics? Jörg W. Müller Bureau International des Poids et Mesures, Pactllon de Breteuil, F-92312 Sètres Cedex, France Received 25 June 1991
"Philosophy has been defined as "an unusually obstinate attempt to think clearly" ; I should define it rather as "an unusually ingenious attempt to think fallaciously". . . . The more profound the philosopher, the more intricate and subtle must his fallacies be in order to produce m him the desired state of intellectual acquiescence . That is why philosophy is obscure." (B . Russell [1]) . On the basis of some examples discussed in detail, we examine some general statements, put forward by philosophically-minded physicists, to see if they are applicable to practical problems met in counting statistics and are of help m solving them . The outcome of this comparison, although admittedly based on a restricted sample, indicates that thought alone, even if it appears to be general, is nearly always too narrow in scope. The complex, and usually incompletely known, structure of a physical situation is too easily misconceived by a seemingly straightforward generalization . If an essential, but perhaps hidden, aspect has been overlooked, the model is inappropriate and deductions based on it are of no value. Physicists therefore seem well advised to mistrust arguments advanced with the claim that they are based on general reasoning. Philosophical conclusions - if one cannot resist drawing them should be the outcome of serious physical investigations, both experimental and theoretical, rather than their starting point
1. Introduction Considering that philosophy, by general agreement,
describe (and thereby try to "understand") the random processes with which they deal .
is regarded as the most respectable form of human
thinking, and since all attempts in science to improve
knowledge make use, in a decisive way, of rational procedures, we are led to associate philosophical reasoning with scientific progress . One may therefore wonder how this partnership works in reality. For this
purpose we shall examine their possible interrelation in the particular field of counting statistics . Perhaps a word should first be said on what is meant by counting statistics . For many persons the
word statistics has rather a negative connotation. This is the result of its daily misuse, which can be summa-
rized by the provocative title of the well-known humor-
ous booklet "How to lie with statistics", by D. Huff,
2. What is a random start? Suppose that we have a material bar of length L. If it is split into two pieces, their average length will be
L/2,
whatever the process of division . Let us now assume that the length varies according to a certain statistical law, which may be known or unknown. If the expectation (or mean) value is denoted by
L,
and if
each piece - whether large or small - is likewise subdivided into two parts, adherents of any philosophical school will no doubt still accept that the mean
which draws a legion of striking examples from politics, business and medicine. In what follows, statistics is
_ length for the parts resulting from this process must be L/2. Nothing in all this looks doubtful or suspicious .
tions in a compact form, but in particular also to
interval, obtained by a process analogous to that used for the bar, should be T/2. Let us transpose this step by step to a simple experiment. We choose the inter-
essentially considered as a branch of mathematics, and counting statistics is understood not only to be an objective attempt to summarize the results of observa-
* An abridged version of this article was presented on 28 May 1991 at the International Symposium on Radionuclide Metrology in Madrid (Spain). It is published here with the kind permission of the Organizing Committee Elsevier Science Publishers B.V . All rights reserved
We now replace our material bar of length L by a time interval of length T. What is true for one quantity must also hold for the other, so the average time
vals to be the distance in time between two events and the events to be pulses originating from a long-lived radionuclide . It is well known that such pulses follow either exactly or in good approximation - the Poisson law . This implies that the time interval between con-
55 6
J W. Miller / Can philosophy be of any use in counting statistics?
secutive pulses is described by the exponential probability density f(t) =p e- '",
for t < 0,
(1a)
and that its mean value is T=
ftf(t) dt= 1/p . 0
(1b)
Let us now choose the starting point for the time interval at random ; it may therefore be anywhere between two registered events . Since there is no preferred region, our philosophical intuition tells us that, on the average, such an interval with random start will have half the length of an interval between counts . Hence, the mean "waiting time" to the arrival of the first event is expected to be 112p . This can be readily checked experimentally but, alas, our expectation is not confirmed by the measurements ; the mean waiting time is still 1 /p . This result, known for long as the "Poisson paradox", may be embarrassing at first sight; for a statistician, it presents a challenge to find a convincing explanation . The solution is both simple and subtle . What do we really mean by "random start"? Apparently, we mean that any infinitesimal time element dt has the same chance of being chosen as origin . But this, in turn, means that a long interval is more likely to be selected than a small one, the probability being proportional to the length t. In consequence, the density of the intervals chosen "randomly" in this way is not given by f(t) but by g(t) =
4,
residual interval (from the random start to the next arrival), although only half of the average, is still exactly the same as for an interval which begins with a registered event. This solves the paradox which, as we now see, was created by a superficial analogy. There are practical situations where the "anomaly" of the long interval before the time origin can be a nuisance . An example occurs in the selective sampling method [3] where one observes the arrival of events preceding a pulse which has "survived" an extendable dead time and is taken as time origin . The problem also arises when we deal with undistorted Poisson events . Let us observe their arrivals before a time zero, marked by a registered event. What is the total density of pulses in this region? Let us denote by x the time -t, that is those times which precede the origin t = 0. The arrival density for event number k then corresponds to the convolution gk(x)=g(x)*lf(x))~(k-1) = g(x)
* fk-1(x) = fk+1(x) ,
with fk (x) explicitly given by eq . (3). Therefore, the total density is described by G(x) = Y_
k-1
gk(x)
_ Y_
k-2
fk(x)
MO = MO)
k
= P(Pt)k-1 e-Pt . (k- 1)!
For the density g(t), with "random start", we therefore obtain to -pr g(t) = e =P 2 t e-°` =f2(t) t e - o` dt
pf
This is a surprising relation . It tells us that the pulse, which we thought was chosen at random, is in fact preceded by a time interval which, on the average, is twice as long as it should be . This explains why the
(b)
For x < 1/p this density shows a marked deviation from the limiting value p. The effect can be attenuated by taking only event number K (instead of the first) as time origin . This can be shown to lead to the density
G(x)-p 1 - ( Px)K-1 tf(t) , (2) (7) e-' tf( t ) dt (K- 1)!
where the denominator ensures the correct normalization [2]. Can we specify how g(t) differs from f(t)? This is easy for a Poisson process. We first remember that multiple intervals between events correspond to sums of independent random variables. Hence, the k-fold density is given by the convolution (k = 1, 2. . . . )
=P(1 - el") .
K= 1, 2, . . . .
These distributions have been fully confirmed experimentally by direct accumulation of the arrival times on a multichannel analyzer in time mode [2]. As expected, the deviation from a uniform density diminishes markedly for K >> 1, but it never disappears completely, and this may be disturbing . A satisfactory solution to this problem was found only recently . In the new arrangement, the arriving pulses are counted separately, e.g . by a scale of ten. Since the corresponding output pulses are not correlated with the length of intervals, they can be used as time origin for repeated sampling . However, no measurements to confirm the exact flatness of the density have yet been performed . Another unexpected and important application concerns the usual method of measuring the lifetime of an excited nuclear state, in which the arrival times of pulses are recorded relative to a start event which signals the excitation . This gives rise to an exponential distribution, apparently superimposed on a constant "background" which is measured before the start by an electronic delay. However, since this is just the time
557
J.W. Miner /Can philosophy be of any use to counting statistics?
region with diminished pulse arrivals, the limiting tail of the curve differs from this "background" and the lifetime obtained by such an adjustment is systematically wrong. This is an example of an error which has survived several decades. The first measurement avoiding this trap was performed recently for the 304 keV level of 7-5As [4] .
also includes a presentation of my own approach to weights. Let us consider two results x1 and x,, with sample variances s 2 and s2 and a sample covariance SL2 = S2,1' We may assume that these results have been obtained on the basis of m measurements of x 1 and x,, performed pairwise, and by using the expression 1
m-1 1
k,
m
1
i .i
k,,
1
m
m
i .,
m
i
(10)
3. Negative statistical weights A question occasionally discussed among experimentalists concerns the assignment of statistical weights to numerical results. This is a problem of great practical importance as it turns up in all fields of measurement . Elementary textbooks on statistics seem to inform us fully on this point . They show, relying on an argument of minimal least squares, that for a series of measurements with results
where j,k = 1 or 2, and the variances have been obtained by the identification s, ., = s, . By definition, the statistical weights w 1 and w, must be such that the mean value of x 1 and x, is W I x 1 + w2x2 (11) x= wi + w, Application of the propagation law for uncertainties yields the relation (w1
where s, is the (estimated) standard derivation of x,, the "best" choice for assigning the statistical weights w, is to put However, such a prescription is at least incomplete . I Apart from the obvious fact that the variances s, should have a reliable and uniform basis for a given set of measurements, two additional requirements are essential for a meaningful weighting by means of relation (9), namely that s, must not be a function of x, alone, and that the results x, must be pairwise independent of each other. The first condition normally causes little trouble, except when the xi data follow Poisson statistics so that the mean and variance have the same expectation value. In this case one should resist the temptation to use 1/x, as statistical weights, as this would introduce a bias by favouring small xi values [5]. More serious - and apparently little known - is the requirement of independence . It may not be immediately obvious how a correlation between the results of measurements affects their weights. There is nevertheless one point on which both "common sense" and deep philosophical reasoning seem to agree, namely that weights must surely have non-negative values . After all, they are some kind of generalization for the effective numbers of measurements performed, and these are clearly positive quantities . Analogy with mass, which is even positive for antiparticles (as we are told), strengthens our confidence . Yet, another application of simple reasoning may be instructive . For an excellent summary of problems concerning experimental uncertainties see e.g . [6], which
+ w,) 2 S 2 (x)
= W2S
+ w2s2 +2w 1w2s 1,, .
Since additivity of the weights demands that
w(x) = w 1 + w,, with w(x) =
1/S2(z),
we obtain
w 1 +w 2 =w ;s z1 +w2s2+2w 1w2 s12 .
(12)
Symmetry in the decomposition requires that 2 2
=w iS1
+WIw2S1,2,
w2 = w2SZ
+ W Iw2s1,2 .
w1
From this, after a simple rearrangement, we find that S Z -S1,z
s1
2 -
2
si z
,
wz =
S zi -S1,z > z 2 sisz -s 1,2
(13)
Relation (13) shows that our philosophical prediction was wrong since w 1 is negative for s1,2 > S2, i.e . for a sufficiently strong (positive) correlation, and similarly for w z . It is surprising that this basic result is not better known, as it is no more than a special case illustrating the general relation W = V`, which requires that the weight matrix W is the inverse of the variance-covariance matrix V. Ignorance of how to account for correlations occasionally causes people to suggest formulae for which there is no basis. In the general case of n quantities x,, the elements wj, k of the weight matrix W can be used to form w, = E wl,k = E wk,, , k=1
k=1
(14)
thereby yielding the required statistical weights for correlated measurements .
55 8
J. W. Miller / Can philosophy be of any use in counting statistics?
It is disappointing to realize that, once more, philosophical advice has been of little help . The realization that negative statistical weights exist should have serious repercussions on the very concept of probability since it may imply that conditional probabilities can no longer be assumed to be positive definite [7]. 4. Coincidences without coincidences In this last example we take a cursory look at measurements of coincidences, a basic technique originally developed in nuclear physics, but nowadays also used in other fields . When counting particles, a certain scatter in the time measurements has to be accepted . Strict simultaneity, therefore, must be replaced by "coincidence within a finite resolving time" . Hence, measured coincidence rates always include "random" coincidences . Indeed, there is no way by which pulses originating from the same nuclear disintegration could be distinguished from those due to different decays which, by chance, happen to occur within the resolving time . One concludes, therefore, that a true coincidence rate can never be obtained directly, but only after correction for the random contribution . This is a situation to which experimentalists have become fully accustomed - even those with a tendency to philosophical contemplation . Nevertheless, it appears that a technique recently developed, called "parity method" [41, allows determination of the average number of true coincidences directly . While this may put an additional shadow on the applicability of abstract thinking, it can be regarded as a success of physical imagination, so long as it remains in close touch with reality. We limit ourselves to an outline of the principle on which the approach is based. Registered nuclear pulses, which correspond to emitted betas and gammas, are classified as "singles" or "pairs", depending on whether one or both particles from a given two-step decay have been observed . Such a generic subdivision always exists, even if we cannot perform it in practice . For a given time interval t one writes for the observed betas (B) and gammas (G) B=b+p,
G=g+p,
where b and g are the numbers of singles and p denotes the number of pairs. We also measure the sum S=B+G=b+g+2p .
(15)
For the sake of simplicity dead-time losses are neglected, thus all the quantities b, g and p, as well as sums formed of them, follow Poisson statistics . Let us now measure S by modulo 2 counting, a technique in which one determines only whether the observed number S is even or odd. This yields, after a
sufficient number of measuring cycles of length t, a quantity which may be written IT, - prob(S is odd) . Since in eq . (15) the contribution of p to S is always even, the parity of S depends only on b + g . This is so because a modulo 2 counter is blind to pairs. One may easily show that for a Poisson process with mean u II1, = i (1 - e -zw) (16) and since now It corresponds to b + g, the average sum of the singles, i.e . b_ + g, can be obtained directly from the measured value II, By an appropriate choice of t, the mean values can be kept small enough that II, is well below its limiting value of 1/2. From this it follows that the count rate C =p/t for true coincidences is readily found by the relation C=
2t
~B+G-(b+g)~,
(17)
where the mean values B and G are obtained by direct counting . In a practical application, dead times have to be taken into account, this causes some numerical complications, but it does not affect the principle. As resolving time is involved, random coincidences do not exist. Conclusion What can we learn from our short random walk in counting statistics? The examples given certainly do not allow us to "prove" anything, but they remind us of some principles on which a scientific argument should be based. Perhaps the main ingredient is an open-minded and critical attitude, which takes nothing for granted merely because it has been set in print. It may happen that what we read in the literature is superficial, or even simply wrong. We must also learn to look at a problem from different perspectives, to modify it and to find analogies for which a solution is known [8]. In the examples sketched above, the philosophical outlook has not been very successful ; in some cases - if taken seriously - it might even have prevented us from reconsidering a problem from scratch. Of course this could be due to malicious choice, but this is too simplistic an escape . Rather, there appears to be a problem of method . When we look for a possible common cause for the failures, we find that the conclusions are largely based on generalizations grounded on insufficient evidence . Lack of real insight cannot be replaced by a view from afar or above. Apparently nature always knows how to outdistance our imagination . To discover a hidden trail we must try to free ourselves from prejudices . This largely disqualifies philosophical ad-
J. W. Miller / Can philosophy be of any use in counting statistics?
vice "' which - in spite of its formal presentation in handbooks - is, I suspect, invariably an embellished presentation of subjective opinions . I note with pleasure that this view does not stand alone, as may be illustrated by the following quotation: "Philosophy, as opposed to science, springs from a kind of self-assertion : a belief that our purposes have an important relation to the purposes of the universe, and that, in the long run, the course of events is bound to be, on the whole, such as we should wish" [1]. This warning, however, has also a positive side : it is an encouragement for those who are convinced that a fresh and independent look is always worth the effort, whatever the field of interest . * 'After reading a draft version, a friend of mine noted that what I understand here by "philosophy" could perhaps more appropriately be called "common sense" . While my generous attitude towards terminology should find no objection to such a change, I doubt if the new conclusion, namely that common sense appears to be of little use, would not be considered more heretical than the original .
559
Dedication This article is dedicated to J.G .V . Taylor on the occasion of his retirement from AECL (Chalk River) . I have always admired John's deep understanding of all essential developments in radioactivity measurements, a field to which he has me lasting contributions . His involvement with the BIPM has extended over some 25 years, the later ones as Chairman of Section II of CCEMRI . References
[1] B. Russell, Philosophy's ulterior motives, in : Unpopular Essays (Unwin, London, 1950). [2] J.W . Müller, Sur le choix `au hasard' d'une impulsion, rapport BIPM-85/4 (1985). [3] J.W . Müller, Nucl . Instr. and Meth . 189 (1981) 449. [4] BIPM Proc.-Verb . Coin . Int . Poid s et Mesures 58 (1990), BIPM, F-92312 Sèvres Cedex. [5] J.W . Müller, A propos d'un abus de poids statistiques, rapport BIPM-84/7 (1984) . [6] W. Bambynek, Proc . first la Râbida summer school eds. M. Garcia-Leon and G. Madurga (World Scientific, Singapore, 1988) p. 3. [7] W. Mückenheim, Physics Rep. 133 (1986) 337. [8] G. Pôlya, How to Solve It (Princeton University Press, Princeton, 1945) .
NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH Section A
Can philosophy be of any use in counting statistics? Jörg W. Müller Bureau International des Poids et Mesures, Pactllon de Breteuil, F-92312 Sètres Cedex, France Received 25 June 1991
"Philosophy has been defined as "an unusually obstinate attempt to think clearly" ; I should define it rather as "an unusually ingenious attempt to think fallaciously". . . . The more profound the philosopher, the more intricate and subtle must his fallacies be in order to produce m him the desired state of intellectual acquiescence . That is why philosophy is obscure." (B . Russell [1]) . On the basis of some examples discussed in detail, we examine some general statements, put forward by philosophically-minded physicists, to see if they are applicable to practical problems met in counting statistics and are of help m solving them . The outcome of this comparison, although admittedly based on a restricted sample, indicates that thought alone, even if it appears to be general, is nearly always too narrow in scope. The complex, and usually incompletely known, structure of a physical situation is too easily misconceived by a seemingly straightforward generalization . If an essential, but perhaps hidden, aspect has been overlooked, the model is inappropriate and deductions based on it are of no value. Physicists therefore seem well advised to mistrust arguments advanced with the claim that they are based on general reasoning. Philosophical conclusions - if one cannot resist drawing them should be the outcome of serious physical investigations, both experimental and theoretical, rather than their starting point
1. Introduction Considering that philosophy, by general agreement,
describe (and thereby try to "understand") the random processes with which they deal .
is regarded as the most respectable form of human
thinking, and since all attempts in science to improve
knowledge make use, in a decisive way, of rational procedures, we are led to associate philosophical reasoning with scientific progress . One may therefore wonder how this partnership works in reality. For this
purpose we shall examine their possible interrelation in the particular field of counting statistics . Perhaps a word should first be said on what is meant by counting statistics . For many persons the
word statistics has rather a negative connotation. This is the result of its daily misuse, which can be summa-
rized by the provocative title of the well-known humor-
ous booklet "How to lie with statistics", by D. Huff,
2. What is a random start? Suppose that we have a material bar of length L. If it is split into two pieces, their average length will be
L/2,
whatever the process of division . Let us now assume that the length varies according to a certain statistical law, which may be known or unknown. If the expectation (or mean) value is denoted by
L,
and if
each piece - whether large or small - is likewise subdivided into two parts, adherents of any philosophical school will no doubt still accept that the mean
which draws a legion of striking examples from politics, business and medicine. In what follows, statistics is
_ length for the parts resulting from this process must be L/2. Nothing in all this looks doubtful or suspicious .
tions in a compact form, but in particular also to
interval, obtained by a process analogous to that used for the bar, should be T/2. Let us transpose this step by step to a simple experiment. We choose the inter-
essentially considered as a branch of mathematics, and counting statistics is understood not only to be an objective attempt to summarize the results of observa-
* An abridged version of this article was presented on 28 May 1991 at the International Symposium on Radionuclide Metrology in Madrid (Spain). It is published here with the kind permission of the Organizing Committee Elsevier Science Publishers B.V . All rights reserved
We now replace our material bar of length L by a time interval of length T. What is true for one quantity must also hold for the other, so the average time
vals to be the distance in time between two events and the events to be pulses originating from a long-lived radionuclide . It is well known that such pulses follow either exactly or in good approximation - the Poisson law . This implies that the time interval between con-
55 6
J W. Miller / Can philosophy be of any use in counting statistics?
secutive pulses is described by the exponential probability density f(t) =p e- '",
for t < 0,
(1a)
and that its mean value is T=
ftf(t) dt= 1/p . 0
(1b)
Let us now choose the starting point for the time interval at random ; it may therefore be anywhere between two registered events . Since there is no preferred region, our philosophical intuition tells us that, on the average, such an interval with random start will have half the length of an interval between counts . Hence, the mean "waiting time" to the arrival of the first event is expected to be 112p . This can be readily checked experimentally but, alas, our expectation is not confirmed by the measurements ; the mean waiting time is still 1 /p . This result, known for long as the "Poisson paradox", may be embarrassing at first sight; for a statistician, it presents a challenge to find a convincing explanation . The solution is both simple and subtle . What do we really mean by "random start"? Apparently, we mean that any infinitesimal time element dt has the same chance of being chosen as origin . But this, in turn, means that a long interval is more likely to be selected than a small one, the probability being proportional to the length t. In consequence, the density of the intervals chosen "randomly" in this way is not given by f(t) but by g(t) =
4,
residual interval (from the random start to the next arrival), although only half of the average, is still exactly the same as for an interval which begins with a registered event. This solves the paradox which, as we now see, was created by a superficial analogy. There are practical situations where the "anomaly" of the long interval before the time origin can be a nuisance . An example occurs in the selective sampling method [3] where one observes the arrival of events preceding a pulse which has "survived" an extendable dead time and is taken as time origin . The problem also arises when we deal with undistorted Poisson events . Let us observe their arrivals before a time zero, marked by a registered event. What is the total density of pulses in this region? Let us denote by x the time -t, that is those times which precede the origin t = 0. The arrival density for event number k then corresponds to the convolution gk(x)=g(x)*lf(x))~(k-1) = g(x)
* fk-1(x) = fk+1(x) ,
with fk (x) explicitly given by eq . (3). Therefore, the total density is described by G(x) = Y_
k-1
gk(x)
_ Y_
k-2
fk(x)
MO = MO)
k
= P(Pt)k-1 e-Pt . (k- 1)!
For the density g(t), with "random start", we therefore obtain to -pr g(t) = e =P 2 t e-°` =f2(t) t e - o` dt
pf
This is a surprising relation . It tells us that the pulse, which we thought was chosen at random, is in fact preceded by a time interval which, on the average, is twice as long as it should be . This explains why the
(b)
For x < 1/p this density shows a marked deviation from the limiting value p. The effect can be attenuated by taking only event number K (instead of the first) as time origin . This can be shown to lead to the density
G(x)-p 1 - ( Px)K-1 tf(t) , (2) (7) e-' tf( t ) dt (K- 1)!
where the denominator ensures the correct normalization [2]. Can we specify how g(t) differs from f(t)? This is easy for a Poisson process. We first remember that multiple intervals between events correspond to sums of independent random variables. Hence, the k-fold density is given by the convolution (k = 1, 2. . . . )
=P(1 - el") .
K= 1, 2, . . . .
These distributions have been fully confirmed experimentally by direct accumulation of the arrival times on a multichannel analyzer in time mode [2]. As expected, the deviation from a uniform density diminishes markedly for K >> 1, but it never disappears completely, and this may be disturbing . A satisfactory solution to this problem was found only recently . In the new arrangement, the arriving pulses are counted separately, e.g . by a scale of ten. Since the corresponding output pulses are not correlated with the length of intervals, they can be used as time origin for repeated sampling . However, no measurements to confirm the exact flatness of the density have yet been performed . Another unexpected and important application concerns the usual method of measuring the lifetime of an excited nuclear state, in which the arrival times of pulses are recorded relative to a start event which signals the excitation . This gives rise to an exponential distribution, apparently superimposed on a constant "background" which is measured before the start by an electronic delay. However, since this is just the time
557
J.W. Miner /Can philosophy be of any use to counting statistics?
region with diminished pulse arrivals, the limiting tail of the curve differs from this "background" and the lifetime obtained by such an adjustment is systematically wrong. This is an example of an error which has survived several decades. The first measurement avoiding this trap was performed recently for the 304 keV level of 7-5As [4] .
also includes a presentation of my own approach to weights. Let us consider two results x1 and x,, with sample variances s 2 and s2 and a sample covariance SL2 = S2,1' We may assume that these results have been obtained on the basis of m measurements of x 1 and x,, performed pairwise, and by using the expression 1
m-1 1
k,
m
1
i .i
k,,
1
m
m
i .,
m
i
(10)
3. Negative statistical weights A question occasionally discussed among experimentalists concerns the assignment of statistical weights to numerical results. This is a problem of great practical importance as it turns up in all fields of measurement . Elementary textbooks on statistics seem to inform us fully on this point . They show, relying on an argument of minimal least squares, that for a series of measurements with results
where j,k = 1 or 2, and the variances have been obtained by the identification s, ., = s, . By definition, the statistical weights w 1 and w, must be such that the mean value of x 1 and x, is W I x 1 + w2x2 (11) x= wi + w, Application of the propagation law for uncertainties yields the relation (w1
where s, is the (estimated) standard derivation of x,, the "best" choice for assigning the statistical weights w, is to put However, such a prescription is at least incomplete . I Apart from the obvious fact that the variances s, should have a reliable and uniform basis for a given set of measurements, two additional requirements are essential for a meaningful weighting by means of relation (9), namely that s, must not be a function of x, alone, and that the results x, must be pairwise independent of each other. The first condition normally causes little trouble, except when the xi data follow Poisson statistics so that the mean and variance have the same expectation value. In this case one should resist the temptation to use 1/x, as statistical weights, as this would introduce a bias by favouring small xi values [5]. More serious - and apparently little known - is the requirement of independence . It may not be immediately obvious how a correlation between the results of measurements affects their weights. There is nevertheless one point on which both "common sense" and deep philosophical reasoning seem to agree, namely that weights must surely have non-negative values . After all, they are some kind of generalization for the effective numbers of measurements performed, and these are clearly positive quantities . Analogy with mass, which is even positive for antiparticles (as we are told), strengthens our confidence . Yet, another application of simple reasoning may be instructive . For an excellent summary of problems concerning experimental uncertainties see e.g . [6], which
+ w,) 2 S 2 (x)
= W2S
+ w2s2 +2w 1w2s 1,, .
Since additivity of the weights demands that
w(x) = w 1 + w,, with w(x) =
1/S2(z),
we obtain
w 1 +w 2 =w ;s z1 +w2s2+2w 1w2 s12 .
(12)
Symmetry in the decomposition requires that 2 2
=w iS1
+WIw2S1,2,
w2 = w2SZ
+ W Iw2s1,2 .
w1
From this, after a simple rearrangement, we find that S Z -S1,z
s1
2 -
2
si z
,
wz =
S zi -S1,z > z 2 sisz -s 1,2
(13)
Relation (13) shows that our philosophical prediction was wrong since w 1 is negative for s1,2 > S2, i.e . for a sufficiently strong (positive) correlation, and similarly for w z . It is surprising that this basic result is not better known, as it is no more than a special case illustrating the general relation W = V`, which requires that the weight matrix W is the inverse of the variance-covariance matrix V. Ignorance of how to account for correlations occasionally causes people to suggest formulae for which there is no basis. In the general case of n quantities x,, the elements wj, k of the weight matrix W can be used to form w, = E wl,k = E wk,, , k=1
k=1
(14)
thereby yielding the required statistical weights for correlated measurements .
55 8
J. W. Miller / Can philosophy be of any use in counting statistics?
It is disappointing to realize that, once more, philosophical advice has been of little help . The realization that negative statistical weights exist should have serious repercussions on the very concept of probability since it may imply that conditional probabilities can no longer be assumed to be positive definite [7]. 4. Coincidences without coincidences In this last example we take a cursory look at measurements of coincidences, a basic technique originally developed in nuclear physics, but nowadays also used in other fields . When counting particles, a certain scatter in the time measurements has to be accepted . Strict simultaneity, therefore, must be replaced by "coincidence within a finite resolving time" . Hence, measured coincidence rates always include "random" coincidences . Indeed, there is no way by which pulses originating from the same nuclear disintegration could be distinguished from those due to different decays which, by chance, happen to occur within the resolving time . One concludes, therefore, that a true coincidence rate can never be obtained directly, but only after correction for the random contribution . This is a situation to which experimentalists have become fully accustomed - even those with a tendency to philosophical contemplation . Nevertheless, it appears that a technique recently developed, called "parity method" [41, allows determination of the average number of true coincidences directly . While this may put an additional shadow on the applicability of abstract thinking, it can be regarded as a success of physical imagination, so long as it remains in close touch with reality. We limit ourselves to an outline of the principle on which the approach is based. Registered nuclear pulses, which correspond to emitted betas and gammas, are classified as "singles" or "pairs", depending on whether one or both particles from a given two-step decay have been observed . Such a generic subdivision always exists, even if we cannot perform it in practice . For a given time interval t one writes for the observed betas (B) and gammas (G) B=b+p,
G=g+p,
where b and g are the numbers of singles and p denotes the number of pairs. We also measure the sum S=B+G=b+g+2p .
(15)
For the sake of simplicity dead-time losses are neglected, thus all the quantities b, g and p, as well as sums formed of them, follow Poisson statistics . Let us now measure S by modulo 2 counting, a technique in which one determines only whether the observed number S is even or odd. This yields, after a
sufficient number of measuring cycles of length t, a quantity which may be written IT, - prob(S is odd) . Since in eq . (15) the contribution of p to S is always even, the parity of S depends only on b + g . This is so because a modulo 2 counter is blind to pairs. One may easily show that for a Poisson process with mean u II1, = i (1 - e -zw) (16) and since now It corresponds to b + g, the average sum of the singles, i.e . b_ + g, can be obtained directly from the measured value II, By an appropriate choice of t, the mean values can be kept small enough that II, is well below its limiting value of 1/2. From this it follows that the count rate C =p/t for true coincidences is readily found by the relation C=
2t
~B+G-(b+g)~,
(17)
where the mean values B and G are obtained by direct counting . In a practical application, dead times have to be taken into account, this causes some numerical complications, but it does not affect the principle. As resolving time is involved, random coincidences do not exist. Conclusion What can we learn from our short random walk in counting statistics? The examples given certainly do not allow us to "prove" anything, but they remind us of some principles on which a scientific argument should be based. Perhaps the main ingredient is an open-minded and critical attitude, which takes nothing for granted merely because it has been set in print. It may happen that what we read in the literature is superficial, or even simply wrong. We must also learn to look at a problem from different perspectives, to modify it and to find analogies for which a solution is known [8]. In the examples sketched above, the philosophical outlook has not been very successful ; in some cases - if taken seriously - it might even have prevented us from reconsidering a problem from scratch. Of course this could be due to malicious choice, but this is too simplistic an escape . Rather, there appears to be a problem of method . When we look for a possible common cause for the failures, we find that the conclusions are largely based on generalizations grounded on insufficient evidence . Lack of real insight cannot be replaced by a view from afar or above. Apparently nature always knows how to outdistance our imagination . To discover a hidden trail we must try to free ourselves from prejudices . This largely disqualifies philosophical ad-
J. W. Miller / Can philosophy be of any use in counting statistics?
vice "' which - in spite of its formal presentation in handbooks - is, I suspect, invariably an embellished presentation of subjective opinions . I note with pleasure that this view does not stand alone, as may be illustrated by the following quotation: "Philosophy, as opposed to science, springs from a kind of self-assertion : a belief that our purposes have an important relation to the purposes of the universe, and that, in the long run, the course of events is bound to be, on the whole, such as we should wish" [1]. This warning, however, has also a positive side : it is an encouragement for those who are convinced that a fresh and independent look is always worth the effort, whatever the field of interest . * 'After reading a draft version, a friend of mine noted that what I understand here by "philosophy" could perhaps more appropriately be called "common sense" . While my generous attitude towards terminology should find no objection to such a change, I doubt if the new conclusion, namely that common sense appears to be of little use, would not be considered more heretical than the original .
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Dedication This article is dedicated to J.G .V . Taylor on the occasion of his retirement from AECL (Chalk River) . I have always admired John's deep understanding of all essential developments in radioactivity measurements, a field to which he has me lasting contributions . His involvement with the BIPM has extended over some 25 years, the later ones as Chairman of Section II of CCEMRI . References
[1] B. Russell, Philosophy's ulterior motives, in : Unpopular Essays (Unwin, London, 1950). [2] J.W . Müller, Sur le choix `au hasard' d'une impulsion, rapport BIPM-85/4 (1985). [3] J.W . Müller, Nucl . Instr. and Meth . 189 (1981) 449. [4] BIPM Proc.-Verb . Coin . Int . Poid s et Mesures 58 (1990), BIPM, F-92312 Sèvres Cedex. [5] J.W . Müller, A propos d'un abus de poids statistiques, rapport BIPM-84/7 (1984) . [6] W. Bambynek, Proc . first la Râbida summer school eds. M. Garcia-Leon and G. Madurga (World Scientific, Singapore, 1988) p. 3. [7] W. Mückenheim, Physics Rep. 133 (1986) 337. [8] G. Pôlya, How to Solve It (Princeton University Press, Princeton, 1945) .