VIII.
DETECTION AND ESTIMATION THEORY
Academic and Research Staff
Prof. H. L. Van Trees
Prof. A. B. Baggeroer
Prof. D. L. Snyder
Graduate Students
I. L. Bruyland
A. E. Eckberg
A.
T. P. McGarty
J. R. Morrissey
J. E. Evans
LIKELIHOOD RATIOS AND MUTUAL INFORMATION FOR
POISSON PROCESSES WITH STOCHASTIC INTENSITY
FUNCTIONS
1.
Introduction
In this report we formally obtain some results
regarding likelihood ratios
and
mutual information for (conditional) Poisson processes with stochastic intensity functions. Our results for likelihood ratios are an extension of results obtained recently
by Snyderl ' Z by a somewhat different approach (Snyder considers the case where the
stochastic intensity function is Markovian while our results include the case of nonMarkovian stochastic intensities). To the best of our knowledge, the results for mutual
information are new.
2.
Likelihood Ratios
Given the observations {N(u), u-- u -<t of a condetermine so as to minimize an expected risk
We consider the following problem.
ditional Poisson counting process,
function, which of the following hypotheses is true.
3
ko +X(t)
H 1:
Xr(t) =
H or
Xr(t) = X,o
(1)
(2)
where Xr (t) is the (stochastic) intensity function of N(t). X might represent the rate
2
o
of arrival of photons from background radiation. Following Snyder, by a conditional
Poisson counting process, we mean that if {(r(t), t >0O
is given, then {N(t), t >0} is
an
integer valued process with independent increments, and for t > 0,
This work was supported in part by the Joint Services Electronics Programs (U. S.
Army, U. S. Navy, and U. S. Air Force) under Contract DA 28-043-AMC-02536(E).
QPR No. 98
103
DETECTION AND ESTIMATION THEORY)
(VIII.
Pr [N(t + At)-N(t)=l xr(t)] = Xr(t) At + o(At)
(3)
Pr [N(t + At)-N(t)=0 Xr(t)] = 1 - XrAt + o(At).
The method by which we obtain the likelihood ratio for the detection problem is
4-6
to obtain results for additive white Gaussian
quite similar to that used by Duncan
We first consider the likelihood ratio conditional on knowledge of
This is equivalent to the detection problem with known intensity function and
noise channels.
X (t).
r
the form of the likelihood ratio is well known.
7-13
The unconditional likelihood ratio
is then obtained by averaging over X(t) and utilizing certain properties of stochastic
integrals.
The likelihood ratio for the detection problem above conditional on knowledge of
X(t) has been shown by Reiffen and Sherman
to be
x(u)
+
st
k(u) du +
= X(t) = exp
0
p(N I Ho)
(4)
dN(u)
in
[
where
p(Nt H 1 , X) = conditional probability density of the observed counting process
Nt = {N(u), 0 < u
O
p(N
Ho
=
<t}, given X(t), under the assumption that H 1 is true.
conditional probability density of N , under the assumption that H
0
is true.
The second integral on the right-hand side of (4) (and all other integrals involving dN(u))
will be interpreted in the sense introduced by Ito (see Doobl).
Before averaging (3) over possible paths of X(t), we shall put (3) into a more convenient form by use of the Ito differential rule.
Snyder
2
has shown that if z t is a
scalar Markov process satisfying the stochastic differential equation,
(5)
dzt = at dt + bt dN t ,
where at and bt are nonanticipative functionals of the Poisson counting process
and
Nt
,
t(zt) is a differentiable function of zt and t, then
(zt) +
dqt(zt) =
Applying (6)
to (4),
(zt) a
we find that X(t)
dt + [t(zt+bt)-t(zt)]dNt.
satisfies the following stochastic
equation:
QPR No. 98
104
(6)
differential
DETECTION AND ESTIMATION THEORY)
(VIII.
dx(t) = -X(t) X(t) dt +
Lx
Xo x(t)
- X(t)
X(t) e I n
dN(t)
(7)
X(t) X(t) dN(t),
= -x(t) x(t) dt + -
or equivalently
X(t) = X(o) -
X(u) %(u) du + 0
with respect to all possible paths of
We now take the expectation of both sides of (7)
X(t).
From (4),
(8)
X(u) X(u) dN(u).
o0
we see that X(t) is related to the unconditional likelihood ratio, A(t),
A(t) = EX[X(t) ]
(9)
,
where EX[. ] denotes averaging over all sample paths of X(t).
A(t) = A(o) - EX
X(u) X(u) du
0
+
Thus
X(u)X(u) dN(u)
E
0
(10)
= A(o) -
EX[X(u)X(u)] du +
0
In Eqs.
fixed,
it
8-10,
is
o 0
EX[X(u)X(u)
dN(u).
]
important to keep in mind that N(u) is the observation and thus
as far as the expectation in (9) and (10) is concerned.
The stochastic differential equation corresponding to (10)
dA(t) = -EX[X(t)X(t)] dt +
is
Ex[X(t)X(t)] dN(t)
o
=-
EX[(t)X(t) ]
A(t) dt
E[X(t)]
1
o
EX[X(t)X(t)
(11)
]
A(t) dN(t).
EX[X(t)]
But
EX[x(t)k(t)]
EiXL(t)p(Nt I , H)]
Hi]
-X(t) = E x(t)i N
EX[X(t)]
E,[
t Ip
X, H1
= conditional expectation of X(t),
t
given No, under the assumption that
H1 is true.
least-squares realizable estimate of X(t), under the assumption that
H 1 is true.
QPR No. 98
105
(VIII.
DETECTION AND ESTIMATION THEORY)
Thus,
we find that the likelihood ratio,
A(t), is the solution to the stochastic dif-
ferential equation
A
dA(t) = -xt)
A(t) dt +
^
1
k(t) A(t) dN(t).
-
(13)
o
The solution to (13) is
A(t) = expL
(u) du +
In
dN(u) .
(14)
A
Our result (14)
can be verified by application of the Ito differential rule for Poisson
counting processes to (14)
and noting that we obtain (13).
Furthermore,
(14)
clearly
has the proper value at t = 0.
Comparing (14)
functions is
with (4),
we see that the likelihood ratio for stochastic intensity
equivalent to that for known intensity functions,
causal minimum mean-square estimate of the intensity is
tor
in
the
same
way as
if
it
were
known.
in the sense that
the
incorporated in the detec-
An analogous
relationship
has
been
obtained for stochastic signals on additive white Gaussian noise channels by Duncan. 4 ' 5
The form above for the likelihood ratio is identical to that obtained by Snyder
for Markovian X(t).
Snyderl also has some results concerning means of obtaining
causal minimum mean-square estimates of
3.
2
k(t).
Mutual Information Results
We shall now apply our results for likelihood ratios to the problem of computing the
mutual information between the message process
m and the conditional Poisson counting
process {N(t), t >0} when the stochastic intensity function of N(t) is X + X(m, t).
Our
6
15
o
approach is quite similar to that of Duncan and Zakai
who obtained mutual information results for additive white Gaussian noise channels:
of calculating certain probabilities
We convert the problem
into a detection problem by introducing a dummy
hypothesis corresponding to Ho of the previous
section.
By using appropriate like-
lihood ratio results and some properties of stochastic integrals,
we are able to obtain
results in terms of certain causal filtering estimates.
Let us define
N t = channel output in time interval [u, t]
Sr(t) =
= counting process
k(t, m) + ko = intensity function of N(t)
m = message.
The mutual information between N t and m
0
QPR No. 98
is given (see Gelfand and Yaglom
106
15
by
(VIII.
I[N
DETECTION AND
ESTIMATION THEORY)
(15)
= E log.
m
The likelihood ratio appearing in the expression for the mutual
written
p(Ntm
information can be
p(NtIm
(16)
where
likelihood ratio for detection between the hypothesis that X (t)
X(t, m) + Xo with the particular m and the hypothesis that Xr(t) =
-(Nt
p(Nt )
. = likelihood ratio for detection between the hypothesis
m random and the hypothesis that k =
(Nt
r
k = X(t, m) + X with
o
Now, from section 2,
p(Nt
m
)
= exp
rt
X(u, rn) du +
0
FX+(u, m)
t
t In
0
(u m)
dN(u)
(17)
p(N t )u)
-
exp
oro
X(u) du + YIn
X.
A
where X is the conditional mean of X(u, m), given N(u), o - u -<t.
X(t)= EIX(t' m) I Nt, X(t)
=
X(t, m)
+X].
Thus we have
QPR No. 98
107
(18)
dN(u),
That is,
(19)
(VIII.
DETECTION AND ESTIMATION THEORY)
I(Nto , m)
t [-X(u, m)+ At(u)] du +
= E
X + k(u, m)
In
dN(u)
A
.
(20)
OX+ (u)
The first integral can be shown to be zero by utilizing the properties of a conditional
expectation:
E
E[ (u,m)INto
=E
[(u, m)-X (u)]du}
-X(u)
du
= 0.
(21)
To evaluate the second integral, we write
+ X(u,)
t in
n
dN(u)
0+
dN(u)
0O
=
I
n
X + X(u, m)
[dN(u) -k(u, m)du] +
A
X0 + X(U)
5O
In
k(u, m) du.
o
0
Now
t
X + X(u, m)
o
In
E
]- [dN(u)-k(u, m)du]
(22)
= 0
from the definition of the stochastic integral, since N(u)14
).
(Doob
fOu k(u, m) du is a martingale
Thus we have
[t
ILN ,m
t
=
E
tX
In
X + X(u, m)
o
w n+
k(u, m)
(23)
dt.
(u)
Next, we note that
IN 0, m
Y
E{ln [o
+ X(u, m)]
+
(u, m)-ln [X
X(u)]
X(u, m)} du.
t
Again taking expectations conditional on No we obtain
I[N, m]
QPR No. 98
n
=
(u)]
n[
E In [Xo+ k(u)] k(u) -In
u
u]
[Xo + k(u)] k(u)
0
108
du,
(24)
(VIII.
DETECTION AND ESTIMATION
THEORY)
where
In [o + k(u)]
k(u)
= realizable least-squares estimate of X(u,m) In [ko +(u,m)],
given the observations {N(s), O - s -- u}, under the assumption
that
(t) = X(t, m) + X
One can readily verify that I[N , m]
is
a monotone increasing function of t.
In the case of large background intensity, that is,
X >>X(u, m)
A
X0 >>X(u),
we can relate IIN t,m to a realizable least-squares filtering error by using the expansion
In [Xo+X] = In X + X/Xo
to obtain
I[ No'm
=
E[
(u)-k(u)] du =
(25)
E{p(u)} du,
A
where p(u) is the conditional variance of the estimate X(u).
A somewhat more interesting
result can be obtained by using this expansion
on Eq. 23 to obtain
I[Nto, m]
(26)
E{[X(u, m)-(u)] X(u, m)} du.
=
A
A
The error, X(u, m) - X(u), is seen to be orthogonal to X(u) as follows:
E{[X(u, m)-X(u)](u)} = E(E [(u, m)-X(u)]X(u) Nt
so that (12) is
I N
m
= 0,
equivalent to
=
o0
5
E ([X(t, m)-X(u)]2)
du.
(27)
Equation 27 is our desired result for the relationship between the mutual information
and the realizable filtering error for the estimation of X(t).
A result quite similar to (27) was obtained for the mutual information on the white
Gaussian channel with (or without) feedback by Duncan and Zakai.
They consider
the problem
y(t) =
QPR No. 98
(28)
s, ys, m) ds + w(t),
0
109
(VIII.
DETECTION AND ESTIMATION
THEORY)
where m denotes the message, y(t) denotes the channel output at time
path y(u), o - u < s, w(t) is a Brownian motion, and m is the channel
that they obtain is that the amount of information between the message
path yo I
m , is related to a causal mean-square filtering error
['
Iym
= 1
E
(s,yo
= E [s,
yo, m Iy ].
Y,
Yo,
_-
yo
2
t, yS denotes the
input. The result
m and the output
by
ds,
(29)
where
4.
Summary
We have formally derived some results regarding likelihood ratios and mutual information for (conditional) Poisson processes with stochastic intensity functions. The
results emphasize the role of realizable least-squares estimates of the stochastic intensity. Several comments on extensions and utilization of the results presented here
are appropriate.
1.
The derivation here is formal; for example, we have assumed that several
interchanges of integration and expectation are valid without giving sufficient conditions. It would be of interest to rigorously establish these results by using appropriate Radon-Nikodym derivative results, 17 , 18 Borel fields, and so forth.
2.
We can readily extend our result to cover the case of feedback channels, by
making a minor extension of the likelihood ratio results. Vector channels can also
be readily considered, by minor modifications to the likelihood-ratio results.
3. These results may be of interest in optical communication, 9 biomedical
processing, 1 and studies of auditory psychophysics.19
data
J. E. Evans
[Mr. James E. Evans is an M. I. T. Lincoln Laboratory Staff Associate.]
References
1.
D. Snyder, "Estimation of Stochastic Intensity Functions of Conditional Poisson Processes," Monograph No. 128, Washington University Biomedical Computer Laboratory, St. Louis, Missouri, April 1970.
2.
3.
D. Snyder "Detection of Nonhomogeneous Poisson Processes Having Stochastic
Intensity Functions," Monograph No. 129, Washington University Biomedical Computer Laboratory, St. Louis, Missouri, April 1970.
The likelihood ratio when X = X + X(t) on hypothesis H can be obtained by using
4.
the chain rule for likelihood ratios. See Snyder 2 for details.
T. E. Duncan, "Evaluation of Likelihood Functions," Inform. Contr. 13, 62-74 (1968).
QPR No. 98
r
oo
110
(VIII.
DETECTION AND ESTIMATION THEORY)
Noise"
T. E. Duncan, "Likelihood Functions for Stochastic Signals
(unpublished paper, dated 1969).
6.
T. E.
tion).
7.
B. Reiffen and H. Sherman, "An Optimum Demodulator for Poisson Processes:
Photon Source Detectors," Proc. IEEE 51, 1316-1320 (1963).
8.
J. W. Goodman,
tum Electronics,
9.
R. M. Gagliardi and S. Karp, "N-ary Poisson Detection and Optical Communications," IEEE Trans., Vol. COM-17, No. 2, pp. 208-216, April 1969.
Duncan,
in
White
5.
"On the Calculation of Mutual Information" (submitted for publica-
"Optimum Processing of Photoelectron
Vol. QE-1, pp. 180-181, July 1965.
Counts," IEEE J.
Quan-
10.
G. L. Fillmore and G. Lachs, "Information Rates for Photocount Detection Systems," IEEE Trans., Vol. IT-15, No. 4, pp. 465-468, July 1969.
11.
J. Bar-David, "Communication under Poisson Regime," IEEE Trans., Vol. IT-15,
No. 1, pp. 31-37, January 1969.
12.
R. W. Chang, "Photon Detection for an Optical Pulse Code Modulation System,"
IEEE Trans., Vol. IT-15, pp. 725-728, November 1969.
13.
K. Abend, "Optimum Photon Detection," IEEE Trans.,
January 1966.
14.
J.
15.
M. Zakai, "On the Mutual Information of the White Gaussian Channel with or
without Feedback" (submitted for publication).
16.
I. M. Gelfand and A. M. Yaglom, "Calculation of the Amount of Information
about a Random Function Contained in Another Such Function," Am. Math. Soc.
Translations (Series 2) 12, 199-246 (1959).
17.
A. V. Skorokhod, "On the Differentiability of Measures Which Correspond to
Stochastic Processes I. Processes with Independent Increments," in Theory of
Probability and Its Applications, Vol. 2, No. 4, pp. 407-432, 1957.
18.
A. V. Skorokhod, "Absolute Continuity of Infinitely Divisible Distributions under
Translations," in Theory of Probability and Its Applications, Vol. 10, No. 4,
pp. 465-472, 1965.
19.
W. M. Siebert, "Stimulus Transformations in the Peripheral Auditory System,"
in P. A. Kolers and M. Eden (eds.), Recognizing Patterns in Living and Automatic
Systems (The M. I. T. Press, Cambridge, Mass. , 1968), pp. 104-133.
L. Doob,
QPR No. 98
Vol. IT-12,
Stochastic Processes (John Wiley and Sons,
111
Inc.,
pp.
64-65,
New York,
1953).