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r

SOME BAYESIAN STATISTICAL TECHNIQUES
USEFUL IN ESTIMATING
FREQUENCY AND DENSITY

UNITED STATES DEPARTMENT OF THE INTERIOR

FISH AND WILDLIFE SERVICE

Speq^aj^Scientific Report— Wildlife No. 203

Boston J':;;f-^i DocuiTior/.:'

Library of Congress Cataloging in Publication Data

Johnson, Douglas H

Some Bayesian statistical techniques useful in estimating frequency
and density.

(Special scientific report — wildlife; no. 203)

1. Animal populations — Mathematical models. 2. Bayesian
statistical decision theory. I. Title. II. Series: United States. Fish and
Wildlife Service. Special scientific report — wildlife; no. 203.
SK361.A256 no. 203 [QH352] 639'.97'90973s [598.2'5'24]
76-50075

NOTE: Use of trade names does not imply U.S. Government endorsement of commercial products.

Some Bayesian Statistical Techniques Useful in
Estimating Frequency and Density

by Douglas H. Johnson

UNITED STATES DEPARTMENT OF THE INTERIOR

i^l°' Tfie,^ FISH AND WILDLIFE SERVICE

Special Scientific Report— Wildlife No. 203
Washington, D.C. • 1977

Some Bayesian Statistical Techniques Useful in
Estimating Frequency and Density

by

Douglas H. Johnson

U.S. Fish and Wildhfe Service

Northern Prairie WildHfe Research Center

P.O. Box 1747, Jamestown, North Dakota 58401

Abstract

This paper presents some elementary applications of Bayesian statistics to problems faced
by wildlife biologists. Bayesian confidence limits for frequency of occurrence are shown to be
generally superior to classical confidence limits. Population density can be estimated from
frequency data if the species is sparsely distributed relative to the size of the sample plot. For
other situations, limits are developed based on the normal distribution and prior knowledge
that the density is non -negative, which insures that the lower confidence limit is non-negative.
Conditions are described under which Bayesian confidence limits are superior to those
calculated with classical methods; examples are also given on how prior knowledge of the
density can be used to sharpen inferences drawn from a new sample.

Biologists often require estimates of distribution
and abundance for a species of animal or plant. Many
approaches, such as mark-recapture techniques,
complete censuses, counts along prescribed transects,
and animal signs, have been used to estimate
frequency of occurrence or density (e.g., Overton
1969). These methods provide an index to abundance
which may allow comparisons between years or
between areas, but some do not permit valid estimates
of the absolute population size. Variance estimates,
which are necessary to assess the precision of a
quantity, are typically not available. Because time
and expense limit the amount of data that can be
collected, the investigator generally desires es-
timators with as much precision as possible.

This paper wdll explore methods of analyzing
counts of organisms to obtain accurate estimators
of population size and frequency of occurrence, as
well as to gauge the precision of these estimators.
The techniques described here apply the Bayesian
philosophy of statistical inference (e.g., Good 1965,
Schmitt 1969, Lindley 1971) to description of frequen-
cy and density. Other authors have discussed
applications of Bayesian theory to a variety of
problems; see DeGroot (1970) for an extensive
bibliography. The methods described here will be
illustrated by examples of censuses of breeding pairs
of birds (Stewart and Kantrud 1972), but the methods
are applicable to a broad class of populations. The

term "pair" is used for these examples, but in other
situations the word "individual" might be sub-
stituted.

Stewart and Kantrud (1972), who studied the
populations of breeding birds in North Dakota, drew
a random sample of land areas (quarter-sections, 160
acres or 64.75 ha), and counted pairs of all avian
species breeding on them. They wanted to estimate
the statewide breeding population of each species
and to measure the precision of those estimates. They
assumed that a census of a quarter-section was
essentially perfect; no pairs were missed and none
counted more than once. Under this assumption, a
census of all quarter-sections in the State would
provide the exact population total, and no statistics
would be required. Because complete censuses were
not feasible for such a large area, however, sampling
had to be employed.

A Bayesian Approach

We now consider some estimation procedures
which employ the Bayesian philosophy of statistics.
A distinguishing feature of this philosophy is the
concept of a parameter. Whereas in classical thought
a parameter is considered to be a fixed constant with
unknown value, Bayesian method treats a parameter
as a random variable, the distribution of which
depends on available knowledge about the

parameter. The parameter does not vary, but our
degree of belief in its value changes with increasing
information, and the probability function reflects the
belief. We will use this philosophy in two ways. First,
it will be convenient to combine information about
the parameter from more than one source. The
Bayesian approach provides a rigorous framework in
which information about a parameter can be applied
either from experiments performed simultaneously or
sequentially, or from knowledge available before the
experiment. Second, the concept of a confidence
interval will be replaced by a more natural statement
about the probability that a parameter is contained
within an interval. A classical 95% confidence
interval (a, b) for a parameter 6 consists of two
functionsoftheobservations,a(xi,A:2, . . . , Atn)and
6(aci, X2, . ■ ■ , s:^), such that, if the experiment were
repeated an infinite number of times under identical
conditions, 95% of the intervals obtained would
include the true value of the parameter 9. A
comparable Bayesian confidence interval is more
simply the interval (a, b) such that 95% of the
probability content lies in that interval.

To oversimplify the Bayesian method, we will
say that it combines information about a pa-
rameter 9 from two sources. One source is the
experiment (or, in the present case, the random
sample of counts). Information from the experiment
is contained in the probability function of the
observations which, considered as a function of ^ , is
called the "likelihood" function. The other source of
information is that which is available before the
experiment occurs, and is contained in the "prior"
distribution of 9 , denoted p{9)- We combine the two
sources of information by multiplying the prior and
likelihood functions; the product is termed the
"posterior" distribution function. The Bayesian
draws his inference from the posterior distribution,
but the classical statistician generally uses only the
likelihood function. Often our prior knowledge is
lacking, so we take p{9) to be noninformative. A
Bayesian employing a noninformative prior obtains
results closely paralleling classical results, although
the Bayesian may do so more directly. We will later
demonstrate how even a modest amount of prior
knowledge can sharpen the inferences drawn.

Application of the Bayesian Method
to Frequency of Occurrence

To put this concept in focus, let X be a random
variable taking the value 1 if breeding mallards
(Anas platyrhynchos) are present in a quarter-
section selected at random, and 0 if they are
absent. Then X is distributed binomially with
Pr| X = \\ - 9 and 9 , the expected value of X,

represents the frequency of occurrence of mallards
per quarter-section.

Useful prior knowledge may be available, either
from censuses conducted previously on these quarter-
sections or from a knowledge of how well the habitat
requirements of the mallard are satisfied on each
quarter-section. But, for the moment, assume prior
knowledge is lacking; we then have to select a prior
distribution for 9 which expresses our ignorance.

The Beta family of prior distributions
p(e) = constant 9°" (1-6)1^
with a >-\, (i >-lis appealing because it is of the
same form as the likelihood function, so that its use
results in a posterior distribution belonging to the
same family. In addition, the Beta family is very
flexible; by varying a and fi the curve can assume a
variety of shapes reflecting various degrees of prior
belief. The choice of exponents a = (3 = -V2 has been
suggested (Jeffreys 1967; Good 1965) on the basis of
invariance under transformation of the parameter
9 . That is, if we have no prior knowledge of 9 , then
we are equally ignorant of 9~\ log 9 , etc., and the
prior should reflect this invariance. The choice of
exponents a , /8 will have little effect on our con-
clusions as long as they are small in absolute value
relative to z and n - z, (defined below). The prior
distribution with a = 0 - -V2 is graphed in Fig. 1.
Note how little it varies over most of the range of 9 .

We next take our random sample of censuses on n
quarter-sections, yielding z = 2.x { quarter-sections
with mallards present. The average frequency of
occurrence is z/n and the likelihood function of 9 ,
given the data, is

L{d\xu X2, ..., x„) - e^'^'n-ef'^'''

For example, Stewart and Kantrud (1972) found
mallards breeding in 79 of their 130 sample quarter-
sections. An estimate of the frequency of occurrence is
then given by 79/130 = 0.608, and the likelihood
function is

L{e\ xuX2, ..., -vn) = 0" {i-ef.

We next combine our prior information with the
information from the sample by multiplying the
corresponding probability functions, to obtain the
posterior distribution of 9 , given the observations,
with probability density denoted by

p{e\ xu X2, . . . , Xn) = constant

XL(e\ .Vl, X2, ■ . . , Xr,) p{6),

where the constant is such that

S'P{9\ Al, X2, . . . ,.Xn) de =1-

o

The term "posterior" is in reference to the fact that it
represents information available "after" the expert-

THETA

Fig. 1. Prior, likelihood, and posterior functions for frequency of occurrence of mallards. "Probability" means probability
density.

ment (or counts). In the present case of a binomial
variate,

L{d\ x\, xi, . . . , Xn) = constant 0'(l-0)" ^ ,

and

pid) = constant 0""'(l-0)"'^'
so the posterior distribution becomes

p(d\ .VI, X2, . . . , Xn) = constant e''"-(l-0)"~'""^

or, in the mallard example,

p(d\ .vi, X2, . . . , Xn) = constant e''\l-ef°\

The appendix describes how the constant is
evaluated. Because of the large sample size and
moderately large mean of the mallard data, the
likelihood function dominates the posterior distribu-
tion to such an extent that graphs of the func-
tions cannot readily be distinguished (Fig. 1). We
can now find points 6] and 62 so that
Pr{0,<0<0,| = 1 - a. The interval (duO:) is the
Bayesian counterpart of a classical confidence

interval. A two-sided confidence interval is ordinarily
chosen so that a/ 2 of the likelihood is eliminated
from each tail of the distribution. On the other hand, a
Bayesian seeks the shortest interval (61, 6:) con-
taining 1 —a of the probability by selecting 0i and
02 such that;

(1) no point outside the interval has its probability
function greater than any point within, and

(2) J Pie I .X-,, X2, . . . , Xn) dd = \ - a .

01

Such an interval is called a Bayesian confidence
region or a Highest Posterior Density (HPD) region
(Box and Tiao 1965). The 95% HPD region for
frequency of occurrence of mallards was
(0.523-0.690). The calculation of HPD regions is
demonstrated in the appendix.

The usual method for obtaining a confidence
interval for a binomial parameter employs the
normal approximation, which is fairly good for large
samples as long as z (the number of samples in which

the species was present) and n - z (the number of
samples in which the species was absent) are not too
small. The normal approximation gives limits on the
mean as plus or minus two standard errors. Exact
limits can be obtained by using the relationship
between a binomial sum and the incomplete beta
function (Johnson and Kotz 1969), which allows
equal-tailed intervals to be obtained from tables of the
F distribution (Bliss 1967). For mallards, with a
moderate value of z (= S.v = 79) as well as a large
sample size (n = 130), all three procedures yielded
very similar confidence limits (Table 1). Consider,
however, the burrowing owl (Speotyto cunicularia),
which was present on only 2 of the 130 sample
quarter-sections. The normal approximation fails
here because the number present was small, and the
Bayesian confidence interval is shorter than the
exact interval (Table 1). Although there are some non-
Bayesian methods for reducing the size of confidence
regions, e.g., Hudson (1971), they do not follow as
naturally as in the Bayesian framework and there is
no unified approach to using available prior
knowledge.

Application to Estimating Population Density

In certain circumstances the probability limits
derived for the frequency of occurrence can also be
used to estimate the density (Dice 1952). Suppose
the sample areas censused are small relative to
the territory size of a species, so that the probability
of more than one pair on an area is zero. Then the
frequency of occurrence is identical to the average
density per unit, and the total number of pairs is equal
to the number of sample units containing a pair. Thus
the method described above will also give probability
limits for the population total.

To equate density to frequency, we have to assume
that Prj more than one pair per sample unit f is
negligible. For a territorial species, the validity of this
assumption can be assured by reducing the size of the
sample unit, although the resultant unit may be
impracticably small. The assumption can be exam-

Table 1. — Confidence limits for frequency of oc-
currence of breeding pairs of mallards and
burrowing owls on 130 sample quarter-sections in
North Dakota.

Method

Mallard Burrowing owl

Normal approximation

to binomial
Exact binomial
Bayesian

(.522, .693)
(.519, .690)
(.523, .690)

(-.006, .037)
( .002, .054)
( .001, .042)

ined if a large number of units are censused by noting
whether or not any unit contains more than one pair.
If not, then Pr| more than ofee pair per sample
unit I is not likely to be appreciable. Even if more
than one pair can occur in a unit, the estimator will be
only slightly affected as long as Pr | more than one
pair per sample unit > is small and the probability of
a large number of pairs on a sample unit is zero. For a
colonial species this latter probability is nonzero, so
the relationship between frequency of occurrence and
density does not hold, and this method should not be
attempted.

Modified Normal Limits for
Population Density

If the counts are reasonably large and sufficiently
numerous, we have some confidence that the central
limit theorem applies. That is, we expect the average
of the counts to be nearly normally distributed. We
will indicate how inferences can be sharpened by
using the slight prior knowledge available. Even
before sampling takes place, we know that the count
on any sample area will be non-negative. Since the
normal distribution extends into negative values, it is
more general then we need, although this becomes a
problem only when the average is small relative to its
standard error.

Let 6 be the mean and a' the variance of the
counts. The usual (Jeffreys 1967) noninformative
prior distribution for ( 0 , a' ) is

with

and

p{e,o') = p{e)p(o') ,

p(6) = constant for all 6 ,

P(a'

constant a for a>0

But for counts, we know 6^0, so we can take
constant Os^O

P{6)

0

0<O ,

to place all of the probability in the range
where 05=0. The likelihood function obtained from
the counts is L(6,a' \ X\, xi, . . . , .Vn)

=, constant a~" exp | — 2(^i ~ ^)^/2a [ .

Combining our prior distribution with the likelihood •
function yields the posterior distribution, given by

/7(0,a' I X\, X2, . . . , Xn)

-' exp { -S(x. - e)V2a' [ 0^0

constant a

0

0<O

Then the distribution of the mean 9 is found by
integrating out a ^ ,' so

p(d I Xl, X2, . . . , Xn)

= r p{d,a^ \ X[, Xl, . . . , Xn) da^

constant [1 + n(e - xf/(n-l)s^] "'"""'' 6^0

0

d<0

On the positive axis, the distribution function
of 0 is proportional to Student's t with mean x and
variance s'^/n, which is nearly normal for moderately
large sample size. Hence, the posterior distribution
of 6 approaches the normal, except that the
negative tail is eliminated and the portion of the
probability mass from the negative range is
redistributed.

For large samples with a moderate coefficient of
variation, the HPD regions obtained from the above
distribution are similar in numerical value, but not in
interpretation, to classical normal confidence limits.
If the mean is small relative to the standard error,
however, the Bayesian method may give con-
siderably shorter intervals because knowledge
that 6 is non-negative contributes more informa-
tion when X is small than when x is large. For
example, the density of the mallard, whose habitat
requirements are so general that they are easily met
in much of North Dakota, was 1.68 pairs per quarter-
section. A standard error of 0.152 resulted in a 95%
confidence interval of (1.38, 1.98), which is no
different from the HPD interval. The long-billed
marsh wren (Telmatodytes palustris), on the other
hand, is mostly restricted to semipermanent
marshes, so that its distribution is much more
localized than that of the mallard. But where it
occurs, it is often abundant, so that the variation in
the counts is large. The 95% confidence interval for
this species, with an average of 0.39 pair per quarter-
section and a standard error of 0.223, was (-0.05, 0.84).
The HPD region (0.00, 0.76) is appreciably shorter
because of the large coefficient of variation. Fig. 2
illustrates for the long-billed marsh wren the
likelihood function on which ordinary confidence
limits are based and the posterior distribution from
which HPD regions are determined.

Blumenthal (1970) developed interval estimators of
a normal mean restricted to be non-negative, in the
case of known variance.

The Use of Prior Information

Thus far we have restricted our attention to the case
in which we lack prior knowledge, except for the fact

that counts are non-negative. One of the features of
Bayesian statistics is the ease with which prior
knowledge can be incorporated intc^he analysis. In
this section we will illustrate this by an example.

The mallard density in the primary waterfowl area
of North Dakota for 1957-66 varied from 0.92 to 2.97
pairs per quarter-section (Table 2). How can these
values be used in combination with the 1967 counts?
One possibility is to use the early data to develop a
prior distribution for estimation of the 1967 count.
Take the prior distribution for the 1967 density to be
normal with mean = 2.11 and variance = 0.365.
Combining this prior with the likelihood function for
the 1967 data, a normal with mean = 2.45 and
variance = 0.044, results in a normal posterior
distribution, with mean = (2.45/0.044 + 2.11/0.365)
-^ (1/0.044 + 1/0.365) = 2.41. The variance is
(1/0.044 + 1/0.365)-! = 0.039. The details of the
calculation are shown in Box and Tiao (1973). Note
that the posterior mean is a weighted average of the
prior mean and the sample mean, with weights
inversely proportional to the corresponding
variances. Thus, the posterior mean will shift from
the observed mean toward the prior mean; the
amount of shift depends upon the relative precision of
the two means. In the example, the shift was minor
because the variance of the 1967 mean was small
relative to the variation among years in the prior
data. Nevertheless, some improvement in precision is
indicated by the 11% reduction in the posterior
variance as compared with the sample variance
(0.039 vs. 0.044).

Although it may be contrary to the intuition of
biologists to use data from other years to estimate the
1967 count, the theoretical justification is con-
siderable. For example, Lindley (1971) and Box and
Tiao (1973) developed Bayesian estimators for a
vector of normal means such that information from
all means is used in the estimation of each mean. A
sampling theory justification was given by James
and Stein (1961), who demonstrated the surprising
result that, in estimating a vector of normal means,
the maximum likelihood estimator (the sample mean)
has a larger mean square error than does a weighted
average of the sample mean and the overall mean.

Judgment is necessary, of course, in determining
whether combining should be done or not. In the
mallard example, if water conditions were uniformly
excellent in 1957-66, but known to be poor in 1967, we
should be reluctant to use the early data to form an
opinion on the 1967 count. Efron and Morris (1973)
discussed strategies for combining data and proposed
a method that allows the data to determine the extent
of combination: larger weights are given to means
that are "close" (in the likelihood sense) to the
observed mean being estimated. So, for example, in
estimating the 1967 mallard density, higher weights

-0.4 -0.2

0.2 0.4

THETA

0.6

0.8

1.0 1.2

Fig. 2a. (TOP) Normal likelihood function for long-billed marsh wren, showing 95% confidence
limits (-0.05, 0.84). Shading indicates the region to which the prior distribution function assigns
zero probability.

Fig. 2b. (BOTTOM) Posterior probability function for long-billed marsh wren, showing 95% HPD
region (0.00, 0.76).

Table 2.— Mallard densities (pairs per quarter-
section) in primary waterfowl area of North
Dakota. 1957-66.

Year

Density

1957
1958
1959
1960
1961
1962
1963
1964
1965
1966

Average
Variance

2.50
2.45
0.92
1.84
1.58
1.68
2.97
2.36
2.23
2.61

2.11
0.365

would be given to years that had densities closer to
the 1967 value. Their procedure is one of a class called
"Empirical Bayes" methods, which are intermediate
between Bayesian and sampling theory methods.
They are similar to Bayesian methods in that a prior
distribution is combined with a likelihood to obtain a
posterior distribution, but, unlike Bayesian practice,
the prior distribution is estimated from the data,
rather than assigned subjectively. See Krutchkoff
(1972) for a brief introduction to Empirical Bayes
methods.

Most of the literature in this area is new and
detailed practical procedures have not yet been fully
worked out. Applications of Empirical Bayes
procedures will no doubt increase in the near future,
but it is beyond the scope of this article to provide a
comprehensive review of the subject.

Acknowledgments

I am grateful to J. S. Ramberg, J. R. Taylor, and P.
F. Springer for comments on an earlier draft of this
report, to L. M. Cowardin for profitable discussions
and review, to C. L. Nustad for the careful typing, and
to R. E. Stewart and H. A. Kantrud, who allowed their
data to be used throughout the development.

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Appendix.

Calculation of Highest

Probability Density Regions

For a posterior distribution p(0j.vi, xi, . . . , Xn),
a (1 - a) Highest Posterior Density (HPD) region
(61,62) must satisfy the following two conditions:

If e e (01,02) and d* i (6^,62), then

p(e I Xl, X2, . . . , Xn) > p{e* I Xl, X2, . . . , Afn) , (1)

and

X^2
pid \ Xl, X2, . . . , X:,) de = I - a . (2)

01

We will develop HPD regions for a binomial
parameter and for a normal mean restricted to non-
negative values.

HPD Regions for
The Binomial Parameter

If I -^i ( are distributed binomially with
parameter 6, then the posterior distribution
of 6 corresponding to the noninformative prior
specified earlier is given by

p{d\ xu X2, . . . , x„) = constant e''^'\l-6y''~"\ (3)\

where z = 2^ x,. The constant term is such that

1 = J P{d I .Vl, A:2, . . . , Xn) dd

O

= constant J" 0'"'^' (1 - d)"'"'^'^ dd

= constant B{z+%, n-z-\-%) ,

where B(a, b) is the beta function (e.g., Feller 1966).
Thus, the required constant is the reciprocal of
B{z+V2, n-2+1/2).

A (1-a) HPD interval is given by (6\, 62) satisfy-
ing equations (1) and (2) above. For a unimodal
distribution the first requirement is equivalent to

P(dl I Xl, X2, . . . , Xn) = p(62 I Xl, X2, . . . , Xn) ,

which, using (3) and noting that the constant occurs
on both sides and thus cancels, becomes

di'-'" (1 - di) "''-'" = 62''"' (1 - 62)""''"',

which has the same solution as does
/i(0i,02) = 0 ,

where

Mdi,e2) = (z-%) (log di - log 02)

+ (n-z-%) (log(l-0,) - log (1-02) ).
It follows from (2) that

1

.0

B(z+%, n-zVh) -^Q

/ V'^\l-0)"-^"'/'d0 = 1 - a ,

or

1

f ' e'-"\\-ef''-"^de

5(2+72, /j-z+'/j) •'o

1

5(z+'/:

~ r- f d'-"\l-dy-'-"'dd - 1 -a

or
where

/2(01,02) = 0

(5)

(4)

/2(01,02) = Ig^ {z+%, «-Z+'/2) - Iq^ iz+%, n-Z+%)

-{I - a) ,

and I u)(a, b) is the incomplete beta integral with
parameters a and b evaluated at the point w (Beyer
1968). This function has been tabled (Pearson 1968),
and computer routines are available for its calcula-
tion (International Business Machines Corporation
1970).

We thus have two equations, (4) and (5), that 0i
and 02 must simultaneously satisfy. Because both
equations are nonlinear, iterative techniques are
necessary. Two such approaches have been
successfully used, although either can fail under
certain extreme conditions.

The first and more direct approach is to use a
computer algorithm designed to solve a system of
nonlinear equations. In the present instance the
system consists of two equations, (4) and (5). Not all
computer facilities have such routines, but Brown
(1973) presented an algorithm and Fortran program
which can be used for this purpose.

The second approach requires only computer
routines that are commonly available. The equations
(4) and (5) can be solved by minimizing the quantity

|/l(01,02)| + 1/2(01,02)1

with respect to 0i and 02. The minimum is zero, and
is attained for values of 0i and 02 such that
equations (4) and (5) are satisfied. Numerous
algorithms have been developed to find the minimum
of a function. Most computing centers have one or
more available.

HPD Regions for
the Normal Mean

The approximate posterior distribution of the mean
of non-negative random variables | Xj > is asymp-
totically

I constant 5~' X
exp \rn{e-xfl2s^] if 0^0
0 if 0<O.

The value of the constant will depend upon the
excluded portion of the normal curve (the shaded area
in Fig. 2a). Let A be this value. Then

A = ^° (In s^lny'"- exp[-«(0-x)V25'] dd

—Jnx Is

— oo

= Pr \w < —\Jnxls\ where W xs a unit normal
deviate

= * (-\/«.v/.s) ,

<!>( • ) being the cumulative normal distribu-
tion function. Because

00

J p(d \ X\, X2, . . . , Xn) dd = \ ,

the distribution for positive values of 0 is \/{\.-A)
times that of a normal variate with mean .v and
variance s^/n, i.e.,

P (6 I Xl, Xl, . . . , Xn)

= (1-/1)"' ilns^ln)'"^ exp [-n(0-x)V2s'] for 0^0.

As with a binomial variable, we want values d\
and 02 which satisfy equations (1) and (2). The form
that the HPD interval takes will depend on whether
or not A < all holds.

Casel:/! < a/2. Here the intervalwillbesymmetric
about X. Equation (2) implies

1 - a = f ' (1-/1)"' {iTTs'ln)'"^

exp [-/I {e-xflls"] dd ,

or

X02
(27rs'lny"'x

exp [-« i6-xfl2s'] dd

^/n(di-x)ls

r)'"' exp [- i W^] dW

sjn(di~x)ls
^ yn{.Bi-x)ls\ -* yn{e^-x)|s] .

it\\/Z. -^ ft ^

Equation (1) requires

p{Q\ I Xl, X2, . . . , Xn)

P(02 I Xl, X2, . . . , Xn)

which implies symmetry about x, so (0i, Qj) simply
corresponds to an ordinary confidence interval with a
confidence coefficient of (1 - a) (1 - >!) rather than
(1 - a)-

For the mallard, having a mean of 1.68 and a
standard error of 0.152, the calculated value of
A-^ (-11.05) is so near zero that it can be ignored;
thus, the HPD region coincides with the ordinary
confidence interval.

Case H: .4 > a/2. Here Equation (1) insures that the
lower limit will be zero, so we need to exclude o of
the probability content from the upper range.
Equation (2) becomes simply

10

r^2 , , ^°

1 - a - (1 - Ay' i27rs'/ny"'x , _ ,

exp [-n{d-xfl2s^] dd.

or

or

6i ^ X + {sl-Jn) <!>''[ 1 - a(l - A)] ,

-1/2 X

a = J (1 - ^)"' (27ri'/«)"

di and

exp [-n{e-xfl2s'-\ dd, 0, = 0 .

or

Case II is exemplified by the long-billed marsh wren.

X2 _^n -2, 1-. L.ase 11 IS exempiinea Dy tne long-Diiiea marsn wren.

(2775 In) exp {-n{e-x) lis ] dd ^ ^^^^ ^f o.39 and standard error of 0.223 resulted in
^^ A =* (-0.39/0.223) = 0.040 > a /2. Hence, e^ = 0.0

and 6*2 = 0.39 + 0.223 $' (0.952)= 0.76.

Calculating HPD regions for the normal mean

requires simply a routine for the normal probability

\Jn{Q2~x)ls function (to determine A) and one for the inverse

normal probability function (to determine the con-
1 — ^ [_\Jn{62-x)ls]. fidence limits).

oo

= J (27r)""' exp [-'/2 ff '] dW

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