Discussion Paper No.462
An evolutionary explanation for why the prior beliefs
can be assumed to coincide with the correct onesin the simple Bayesian hypothesis testings
by Mitsunobu Miyake

February 18, 2022

TOHOKU ECONOMICS RESEARCH GROUP
Discussion Paper

GRADUATE SCHOOL OF ECONOMICS AND
MANAGEMENT TOHOKU UNIVERSITY

27-1 KAWAUCHI, AOBA-KU, SENDAI,
980-8576 JAPANAn evolutionary explanation for why the prior beliefs
can be assumed to coincide with the correct onesin the simple Bayesian hypothesis testings
by Mitsunobu MIYAKE

Graduate School of Economics and Management
Tohoku University, Sendai 980-8576, Japanmitsunobu.miyake.b4@tohoku.ac.jp

February, 2022

Abstract: This paper considers an evolutional process, where a simple Bayesian
hypothesis testing is conducted repeatedly in parallel by many testing agents withdiversified prior beliefs.

Assuming that each of the agents selects a (one-shot) testing

strategy to maximize the fitness value believing that the (own) prior belief is true, it isshown that only the testing agents endowed with the correct prior beliefs are survived
eventually. This result provides an explanation for why the prior belief of the agent canbe assumed to coincide with the correct one in the Bayesian hypothesis testing, as if theagent knows the true probability assigned by the nature, without introducing the longterm learning processes.

Key words: Bayesian hypothesis testing, diagnostic testing, prior belief, NeymanPearson lemma, natural selection, Malthusian competition.

1

1. Introduction
In Bayesian hypothesis testing for simple hypothesis,1 the testing agent is not informed of
the true probability on the events "the null hypothesis is true" or "the alternative
hypothesis is true" assigned by the nature, but the agent has the personal prior beliefs(subjective probabilities) on the events. Although the agent can implement the Bayesian
optimal testing procedure for a given prior belief, which maximizes the (posterior) expectedvalue of the testing outcome conditioned on the signals, there is a problem of at what levelshould such a prior belief be determined.
One possible answer is that the prior belief coincides with the correct one, because
the correct prior belief can be derived from past information in general, applying thestatistical inference theory or Bayesian learning theory.2 For example, before a test, if
sufficiently many times of repetitions of the same test have been conducted, and if theprior belief has revised based on the signal at each repetition, it is possible that thesequence of revised prior beliefs converges to the correct one.
Without introducing such a long-term learning process, this paper atempts toprovide an alternative explanation for why the prior belief of the agent can be assumed tocoincide with the correct one, by showing that the testing agents with correct prior beliefs
are naturally selected in an evolutional process, where a Bayesian testing is conducted_
1

For the hypothesis testings in the philosophical literature, see Sober (2008, Ch.1). Okasha(2013) and Lo and Zhang (2021) derive the Bayesian behavioral hypotheses (up-dating of priorand utility maximization based on posterior) as the properties of long-run equilibria ofevolutionary processes in which non-Bayesian acts are permitted.

Zhang (2013) shows that

non-Bayesian behavioral hypotheses is derived when individual's risk aversion is incorporated
into the processes. In this paper, the Bayesian behavioral hypotheses are major premises, andall agents are assumed to be Bayesian agents throughout this paper.
2

For the statistical inference theory, see DeGroot and Schervish (2012, Ch 7). For the theories

of determiniation of the prior beliefs, see Berger, Bernardo and Sun (2015) and the references.

2

repeatedly in parallel by many testing agents.3
First, we consider the one-shot Bayesian testing model with just one testing agent.
Because the simple Bayesian hypothesis testing model above covers a class of thediagnostic testings, as in a standard textbook (Lesaffre and Lawson, 2012), we formulates
the one-shot Bayesian testing as a diagnostic testing, where testing agent (medical doctor)
conducts a testing with medical treatments for the patient randomly selected in a large
population. For an informational assumptuion, we assume that the agent knows neither
the true prevalence rate nor the true initial disease states of the patients even after the
medical treatments. But we assume that a testing agent is informed of the final healthstate of the patient after the medical treatment.4
Concretely, for a given prior belief q, we introduce the concept of the Bayesian
optimal test procedure with respect to q, which is defined by the test procedure maximi_
3

As a explanation of such an assertion, our approach is closely related to the pragmatistic

explanation why an expert acts as if the expert knows the seemingly unknowable information
which is a crucial factor determining the outcome of the act.

See Milton Friedman (1953,

Section III) for example. Moreover, the result also provides a reason why the common priorassumption in the game theory holds for the testing agents.
4

Consequently, in the evolutional model, this assumption implies that the Bayesian up-datingof belief is conducted within the one-shot testing and the agents cannot use the geneticallyretained up-dating process. If we assume the genetically retained up-dating process or the
existence of the central authority which infer the (true) posterior probabilities, it is not difficult
to derive the (correct) Bayesian testing procedure as shown in Footnote 8 in Section 3 of thispaper. Moreover, in a one-shot testing, if the agent can derive a large sample, the agent inferes
the true hypothesis, making use of the Bayesian updating or the limiting method of the relative
frequency based on Reichenbach's principle. Then, we assume that each agent derives a small
sample in the one-shot testing in this paper.

3

zing the subjective (posterior) probability of the event that the patient is healthy.5 For
any prior belief q, it is well-known that the subjectively optimal test procedure exists andthat the Bayesian optimal test procedure is a likelihood ratio test procedure (Theorem
1(i,ii)).
Moreover, for the given true probability p, we introduce the concept of theobjectively optimal test procedure with respect to p, which is defined by the test procedure
maximizing the objective probability of the event that the patient is healthy, based on thetrue probability p. Assuming that, for each probability q ∊ (0, 1), there exists a testing
agent whose subjective prior belief coincides with q, we prove that the Bayesian optimal
testing procedure with respect to q is objectively optimal if and only if the prior belief q
coincides with the true probability p (Theorem 2).

This implies that the hypothesis

testing scheme has a nice property that a Bayesian optimal testing procedure is objectively
optimal only if the agent's prior belief coincides with the correct one.
Second, we introduce the evolutional testing model, where the Bayesian testing is
conducted repeatedly in parallel by many testing agents, and we assume that the agents
have the diversified initial prior beliefs, which are individually preserved through thegenerations. Moreover, we assume that the fitness value of a medical doctor is determinedby the individual achievement of the agent. Specifically, we assume that the fitness value
coincides with the objective probability of the event that the patient is healthy. Then, it
is shown that only the testing agents with correct beliefs about the probability (the prevalence rate) are survived eventually, if the population ratio of such rational agents is not
neglibible at the initial state of the model (Theorem 3).

The natural selection result is_
5

In a standard textbook as in DeGroot and Schervish (2012, Ch 9, Section 9.8), the cost of atreatment (or decision) is introduced in order to evaluate the testing procedure numerically.
In this paper, we introduce the benefit of a treatment.

There is no essential defference,

because minimizing the cost is equivalent to maximizing the benefit as shown in Footnote 9 in
Section 3 of this paper.

4

derived as a consequence of the Malthusian competition among the testing agents.
Namely, even if the difference of the fitness values between the testing agents with correct
beliefs and the other agents is very small, the selection will appear as a consequence of amathematical property of the exponential function, which specifies the growth rate. 6 Notethat Theorem 3 holds, independent of the natural rate of population growth d > 0, whichis assumed to be common for all types of agents.
Because a Bayesian hypothesis testing can be regarded as a one-shot Bayesian
hypothesis testing which appears after many generations have occured, the latter result(Theorem 3) provides an explanation for why the prior belief of the agent can be assumed
to coincide with the correct one in a Bayesian hypothesis testing, as if the agent knows thetrue probability assigned by the nature.
The next section introduces the Bayesian hypothesis testing, and Section 3
introduces the Bayesian optimal test procedure and re-states its characterization theorem(Theorem 1).

Section 4 evaluates the Bayesian optimal test procedures objectively andshows the main theorem of this paper (Theorem 2).

The dynamic process is introduced

and Theorem 3 is shown in Section 5.

2. The (one-shot) Bayesian hypothesis testingThis section introduces the (one-shot) Bayesian hypothesis testing model with a testing
agent in a human health care setting.

The model can be specified by the following

conditions:
R1: The patient contracts tuberculosis (TB), or the patient suffers from a normal cold (NC).
The nature determines one of the two events, TB or NC exclusively for all patients in acity. The true probability of TB (true prevalence rate) is denoted by p*. We assume thatp* ∊ (0, 1),(1)___________ ____________

6

See Nowak (2006, Ch.2, Section 2.2.1).

5

R2: For a medical doctor, a patient is selected randomly from a large population of patientsin the city. The medical doctor does not know the true probability p*. But each medical
doctor has a subjective probability of the patient is TB (before the test), which we call abelief. The set of all possible beliefs are denoted by P ≡ (0, 1). The two simple hypotheses
are given by:
the null hypothesis (H0): The paitient suffers from a normal cold, NC.
the alternative hypothesis (H1): The patient contracts tuberculosis, TB.

● The patient suffers from a normal cold (NC) : θ = θ0p*●

The stateof patient

1-p*● The patient contracts tuberculosis (TB) : θ = θ

1Figure 1
R3: The medical doctor examins the patient by a tuberculosis check-up kit. The check-up
kit indicates a real number x ∊ X ≡ ℝ, which is assumed to be the sample mean for size mfrom the random device of which probability destribution is f(x|q0) in case that thepaitient suffers from a normal cold (q = q0). Otherwise (q = q1), we assume that the checkup kit indicates a real number x ∊ X, which is assumed to be the sample mean for size mfrom the random device of which probability destribution is f(x|q1). We assume that:f(x|q0) and f(x|q1) are continuous functions of x ∊ X,

(2)

and thatf(x|q0) > 0 and f(x|q1) > 0 for all x ∊ X,
R4: The medical doctor selects one of the two acts,(3)"Not reject H0 and prescribe

streptomycin" or "Reject H0 and prescribe antipyretics". The former act is denoted by d0
and the latter act is denoted by d1. The selection of act can be determined conditioned on

6the number in X = ℝ given by R3, and the selection is represented by a function d : X →{ d0, d1 } such that d−1(d0) ∊ %X, where %X is the s-field of Borel subsets in X.

In thefollowing, a selection of act d is called a testing procedure. Let D be the set of all testingprocedures.
R5: If the event selected by the nature at R1 is consistent to the act d(x) selected at R4,
then there is no error, otherwise an error is happended. Concretely, there are two typesof errors:
An error is happended if and only if d(x) = d1, when q = q0;
An error is happended if and only if d(x) = d0, when q = q1.
The former errors are called Type I errors, and the latter errors are called Type II errors.
Although the doctor remenbers the act d(x) at R4, the doctor is not informed of the true
state, (q = q0) or (q = q1), and the doctor can not guess an error is happended or not.
R6: In case of the type I error is happended, the nature randomly determine the final
outcome, "The paitient is healthy" or "The paitient is not healthy". The probability of"The paitient is healthy" is denoted by pa. In case of the type II error is happended, theprobability of "The paitient is healthy" is denoted by pb. In case of no error is happended,
the nature determine the final outcome "The paitient is healthy" certainly.

We asumethat
1 > pa > pb > 0 .

(4)●

The patient is not healthy.

●

The patient is healthy.

1−paType I error ●(q = q0 and d(x) = d1)
pa

Figure 27●

The patient is not healthy.

●

The patient is healthy.

1−pbType II error ●(q = q1 and d(x) = d0)
pb

Figure 3

3. The Bayesian optimal testing procedures
In the testing model constructed in the previous section, this section introduces theBayesian behavioral assumption for the agent to define the Bayesian optimal testingprocedure.

Namely, for a given prior belief, using the realization of the sample and its

up-dating rule, the testing agent computes the posterior probabilities, and the agent
selects the testing procudure to maximize the probability that the patient is healthy after
the medical treatment.
We assume that the agent's prior belief for the occurrence of the event that a patient
is TB (q = q1) is specified by a subjective probability q in P. For a given a prior beliefq ∊ P, the posterior probability for (q = qi) after recieving the signal x ∊ X is denoted by
P(qi|x : q).7 Using the Bayes theorem, we have the following well-known lemma:_
7

Setting Xn = (x − (1/n), x + (1/n)), we define P(q1|x: q) by P(q1|x : q) = limn q∙P(Xn|q1)/P(Xn),where P(Xn|q1) = ∫Xn f(x|q1)dµ and P(Xn) = ∫Xn [q∙f(x|q1) + (1 – q)∙f(x|q0)]dµ.
P(q0|x: q) by almost the same manner.

8

We can defineLemma 1: P(q1|x : q) = q∙f(x|q1)/[q∙f(x|q1) + (1 – q)∙f(x|q0)] and P(q0|x : q) = (1 – q)∙f(x|q0)/[q∙f(x|q1) + (1 – q)∙f(x|q0)].

q0
q1

d0

d1

1
pb

pa
1

Figure 4

For a given prior belief q ∊ P, the posterior probability of "the paitient is healthy" in caseof the medical treatment di (i=1, 2) after recieving the signal x ∊ X, denoted by h(di|x:q), are defined byh(d1|x: q) = P(q0|x: q)⋅pa +P(q1|x: q)⋅1 and h(d0|x: q) = P(q0|x: q)⋅1+P(q1|x: q)⋅pb. (5)For a given prior belief q ∊ P, the probability of "the paitient is healthy" of a testingprocedure d ∊ D after recieving the signal x can be denoted by h(d(x)|x : q). As a directconsequence of Lemma 1 and (5), we have a lemma:

Lemma 2: (i) h(d1|x: q) = [(1 – q)∙f(x|q0)⋅pa + q∙f(x|q1)]/[q∙f(x|q1) + (1 – q)∙f(x|q0)] andh(d0|x: q) = [(1 – q)∙f(x|q0) + q∙f(x|q1) ⋅pb]/[q∙f(x|q1) + (1 – q)∙f(x|q0)].
(ii) For a given q ∊ P, h(d1|x: q) and h(d0|x: q) are continuous for x ∊ X.

The testing agent atempts to maximize the probabilities h(d(x)|x : q) (x ∊ X) by selecting
the suitable testing procedure d in D.

Namely we can define the optimal testingprocedure d as follows: For a given prior belief q ∊ P, a test procedure d ∊ D is defined to
be a Bayesian optimal test procedure with respect to q if and only if the probability of "Thepaitient is healthy", h(d(x)|x : q) is maximal in the set { h(d1|x : q), h(d0|x : q) } for almost

9all possible x ∊ X. 8 Namely, it holds thath(d(x)|x : q) = max[ h(d1|x : q), h(d0|x : q)] for almost all x ∊ X. 8
The optimality condition above specifies a property of a test procedure only, and there isno condition for relative advantage over the other test procedures.
For the next theorem, we need some definitions and lemmas.

For a given priorbelief q ∊ P, if a testing agent selects a testing procedure d ∊ D, the induced (subjective)
probability of "The paitient is healthy", denoted by G(d: q), is defined byG(d: q) = (1 – q)⋅(1 – a(d))⋅1 + (1 – q)⋅a(d)⋅pa + q⋅(1– b(d))⋅1 + q⋅b(d)⋅pb ,(6)

where a(d) is the probability of Type I errors of d defined by α(δ) = ∫#!" (% ) f(x|θ! )dµ and"b(d) is the probability of Type II errors of d defined by β(δ) = ∫#!" (% ) f(x|θ' )dµ .(#_8When A ∊ $X and µ(A) > 0, a statement of x holds for almost all x ∊ A if and only if there

exists B ⊂ A such that: (i) B ∊ $X, (ii) µ(B) = µ(A), (iii) The statement of x holds for all x ∊ B.
The function w(x) = max[h(d1|x : q), h(d0|x : q)] is Borel measurable, because w(x) is
continuous on X by Lemma 2(ii) and Royden (1988, Problems 4 and 5 in Page 34, and Problem
44 in Page 49). Suppose that there exists a statistician who can collect the information fromthe many agents such as (x, d, h), where x is a realization of the sample, d is the decision,and h is the final health condition of the patient.

If the statistician gets sufficiently manysuch information, then the statistician can infer the probabilities h(d1|x : p*) and h(d0|x : p*),where p* is the true probability, and then derives the Bayesian optimal test procedure withrespect to p* without knowing the true probabilities [f(x|q0), f(x|q1), pa, pb, p*].
9

The numerical value of G(d: q) can be interpreted as the values of the expected utilities

defined by G(d: q)∙(utility of healthy state) + (1 – G(d: q))∙(utility of not healthy state), because
the values coincide with G(d: q), if we set (utility of healthy state) = 1 and (utility of nothealthy state) = 0.

Moreover, because G(d: q) = (1 – q)⋅(1 – a(d))⋅1 + (1 – q)⋅a(d)⋅pa + q⋅(1–b(d))⋅1 + q⋅b(d)⋅pb = 1 – [(1 – q)⋅(1 – pa)⋅a(d) + q⋅(1– pb)⋅b(d)], the maximization of G(d: q) is
equivalent to the minimization of the cost C(d: q) ≡ [(1 – q)⋅(1 – pa)⋅a(d) + q⋅(1– pb)⋅b(d)].

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Lemma 3: (i) For all q ∊ P and all d1, d2 ∊ Dq, it holds that G(d1: q) = G(d2: q).
(ii) G(d: q) = ∫ X h(d(x)|x : q)dlq, where lq is the measure on (X, %X) defined by lq(B) =∫B [q∙f(x|q1) + (1 – q)∙f(x|q0)]dµ for all B ∊ %X.

For a given prior belief q ∊ P, a test procedure d* ∊ D is called a subjectively optimal test
procedure with respect to q if and only if G(d*: q) ≥ G(d: q) for all d ∊ D. For a given priorbelief q ∊ P, a test procedure d* ∊ D is defined to be a likelihood ratio test procedure withrespect to q if and only if
d*(x) = d1 for almost all x ∊ C1(q); d*(x) = d0 for almost all x ∊ C0(q),where C1(q) = { x ∊ X : f(x|q1)/f(x|q0) > (1 – q)⋅(1 – pa)/q⋅(1 – pb) } and C0(q) = { x ∊ X :f(x|q1)/f(x|q0) < (1 – q)⋅(1 – pa)/q⋅(1 – pb) }.

Let Dq be the set of all likelihood ratio test

procedures with respect to q ∊ P.

Lemma 4: For any prior belief q ∊ P, the following assertions hold:(i) Dq ≠ ∅.
(ii) If d1 ∊ Dq and d2 ∊ D, then h(d1(x)|x : q) ≥ h(d2(x)|x : q) for almost all x ∊ X.
(iii) If d1 ∊ Dq and d2 ∊ Dq, then h(d1(x)|x : q) = h(d2(x)|x : q) for almost all x ∊ X.
(iv) Suppose that d1 ∊ Dq. If d2 ∊ D satisfies d2 ∉ Dq, then there exists a compact subset
A in X such that µ(A) > 0 and h(d1(x)|x : q) > h(d2(x)|x : q) for all x ∊ A.
(v) If d1 ∊ Dq and d2 ∉ Dq, then G(d1: q) > G(d2: q).

We have the following well-known theorem10:_10Theorem 1(ii)[(b)⇔(c)] is a re-statement of Neyman-Pearson lemma as in DeGroot andSchervish (2012, Ch.9, Theorem 9.2.1) and Lehmann and Romano (2005, Ch.3). For Theorem
1(ii)[(a)⇔(c)], see DeGroot and Schervish (2012, Ch.9, Section 9.8).

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Theorem 1(Existence and characterization of the Bayesian optimal test procedures):
Select any q ∊ P and let q be a subjective belief of a testing agent.
(i) There exists a Bayesian optimal test procedure with respect to q.
(ii) The following three statements are mutually equivalent:(a) A test procedure d ∊ D is Bayesian optimal with respect to q.
(b) A test procedure d ∊ D is subjectively optimal with respect to q.
(c) A test procedure d ∊ D is a likelihood ratio test procedure with respect to q.
(iii) Let d* be a Bayesian optimal test procedure with respect to q. If a test procedured is not a Bayesian optimal test procedure with respect to q, then G(d*: q) > G(d: q).

Although Theorem 1 is well-known, we provide a proof of the theorem for the completenessof the arguments of this paper:

Proof of Theorem 1: (ii) First, we prove the assertion (ii) of Theorem 1. [(a) ⇒ (c)]:Because Dq ≠ ∅ by Lemma 4(i), we can assume that there exists d* ∊ Dq. If d0 ∊ D is nota likelihood ratio test procedure with respect to q, then it holds by Lemma 4(iv) that there
exists a compact subset A in X such that µ(A) > 0 and h(d*(x)|x : q) > h(d0(x)|x : q) for
all x ∊ A, which implies that d0 ∊ D is not a Bayesian optimal test procedure with respect
to q. Taking the contraposition of this, we have that a Bayesian optimal test procedure
with respect to q is a likelihood ratio test procedure with respect to q. [(c) ⇒ (b)]: Let d*
be a likelihood ratio test procedure with respect to q. Suppose that d* is not subjectively
optimal. Then there exists d0 ∊ D and there exists a compact subset A in X such thatµ(A) > 0 and h(d0(x)|x : q) > h(d*(x)|x : q) for all x ∊ A. However, it holds by Lemma 4(ii)that h(d*(x)|x : q) ≥ h(d(x)|x : q) for almost all x ∊ A.

This is a contradiction, and wehave that d* is subjectively optimal. [(b) ⇒ (c)]: Because Dq ≠ ∅ by Lemma 4(i), we canassume that there exists d* ∊ Dq. If d ∊ D is not a likelihood ratio test procedure with12

respect to q, then it holds by Lemma 4(v) that G(d*: q) > G(d: q), which implies that d is
not subjectively optimal with respect to q.

Taking the contraposition of this, we havethat a subjectively optimal test procedure with respect to q is a likelihood ratio test
procedure with respect to q. [(c) ⇒ (a)]: Let d* be a likelihood ratio test procedure withrespect to q. For any d ∊ D, it holds by Lemma 4(ii) that h(d*(x)|x : q) ≥ h(d(x)|x : q) for
almost all x ∊ X, which implies that d* is a Bayesian optimal test procedure with respect
to q. (iii) Theorem 1(iii) is a direct consequence of Lemma 4(v) and Theorem 1(ii)[(a) ⇔(c)]. (i) Theorem 1(i) is a direct consequence of Theorem 1(ii) and Lemma 4(i).

For any prior belief q ∊ P, the existence of a Bayesian optimal test procedure with respect
to q is ensured by Theorem 1(i).

The optimality concept of (a) is defined by the best

responces against the possible signals, whereas the optimality concept of (b) is defined by
the comparisons of the values directly on the set of all test procedures. In terms of thegame theory, the former corresponds to the optimality in the behavioral strategies in theextenstive form games, the latter corresponds to the optimality in the pure strategies inthe strategic form games.

Theorem 1(ii)[(a) ⇔ (b)] shows that such two optimality

concepts are equivalent in the Bayesian testing scheme. Theorem 1(iii) strengthens the
optimality condition by showing that the value ofGat the Bayesian optimal test

procedure is strictly greater than the values of G at the non-optimal test procedures.

Example 1: Set p* = 0.4. We assume that the real number x ∊ X is the sample mean forsize m from N(0, 1) in case that the paitient suffers from a normal cold (q = q0). Otherwise(q = q1), we assume that x ∊ ℝ is the sample mean for size m from N(1, 1). For simplicity,
if q = q0, we assume that the emergence of x is subject to the nomal distribution N(0, 1/m)
(Note that 1/m is the variance), otherwise (q = q1), we assume that the emergence of x is
subject to the nomal distribution N(1, 1/m). Cosequently, the two conditional probability
destributions, f(x|q0) and f(x|q1) can be defined by13

f(x|q0) = A⋅exp[− (½)⋅m⋅x2] and f(x|q1) = A⋅exp[− (½)⋅m⋅(x−1)2] for all x ∊ X,
where A = m½⋅(2π)−½ and exp(y) = ey for all y ∊ X. When m = 2, setting A* = π−½, thegraph of f(x|q0) = A*⋅exp[−x2] and f(x|q1) = A*⋅exp[− (x−1)2] can be drawn as follows:f(x|𝛉₀)f(x|𝛉₁)x
Figure 5

Lemma 5: (i) L(x) ≡ f(x|q1)/f(x|q0) = exp[m⋅(x−(½))] is an increasing function of x.
(ii) c(q) ≡ L−1((1 – q)⋅(1 – pa)/q⋅(1 – pb)) is a decreasing function of q.
(iii) lim q→0 c(q) = + ∞ and limq→1 c(q) = – ∞ .
(iv) The likelihood ratio test procedure with respect to q can be re-defined byd*(x) = d1 for almost all x ∊ { x ∊ ℝ : x > c(p) };d*(x) = d0 for almost all x ∊ { x ∊ ℝ : x < c(p) }.

For all q ∊ P, c(q) is called the critical value of the likelihood ratio test procedure withrespect to q. When m = 2, pa = 0.3, pb = 0.1, q = q(q1) = 0.2, q(q0) = 0.8, it holds that:c(q) = c(0.2) = (1/2)∙log(4) + (1/2)∙log(7/9) + (1/2) = 1.0675,
L(x) ≡ f(x|q1)/f(x|q0) = A*⋅exp[− (x−1)2]/A*⋅exp[−x2] = exp(2x−1)

14

P(q0|x : q) = (1 – q)∙f(x|q0)/[ q∙f(x|q1) + (1 – q)∙f(x|q0)] = 4/ [L(x) +4] ,P(q1|x : q) = q∙f(x|q1)/[ q∙f(x|q1) + (1 – q)∙f(x|q0)] = 1/[ 1+ 4(1/L(x)] = L(x)/[L(x)+4],h(d1|x: q) = P(q0|x: q)⋅pa + P(q1|x: q)⋅1 = [L(x)+(1.2)] /[L(x) +4],h(d0|x: q) = P(q0|x: q)⋅1 + P(q1|x: q)⋅pb = [(0.1)L(x)+4] /[L(x) +4] .

h(d0|x: q)c(q) = 1.0675

h(d1|x: q)xFigure 6

4. The objective evaluation of the Bayesian optimal testing procedures
It holds by Theorem 1 that the testing agent selects a likelihood ratio test procedure in
Dq, where q is the prior belief of the agent.

This section evaluates the selected test

procedures objectively and proves that the selected test procedure is objectively optimal if
and only if the testing agent's prior belief coincides with the true probability.
Suppose that p* ∊ (0, 1) is the true probability (true prevalence rate). We assume
that:
A1: For any y ∊ ℝ++, there exists some x ∊ ℝ such that y = f(x|q1)/f(x|q0).
The condition A1 means the range of L(x) ≡ f(x|q1)/f(x|q0) coincides with the full set ℝ++.

15

If a testing agent selects a testing procedure d ∊ D, the induced probability of "The paitientis healthy", denoted by G(d: p*), can be defined byG(d: p*) = (1 – p*)⋅(1 – a(d))⋅1 + (1 – p*)⋅a(d)⋅pa + p*⋅(1 – b(d))⋅1 + p*⋅b(d)⋅pb .

(7)

Then we have a lemma:

Lemma 6: (i) For all q ∊ P and all d1, d2 ∊ Dq, it holds that G(d1: p*) = G(d2: p*).
(ii) G(d: p*) = ∫ X h(d(x)|x : p*)dl, where l is the measure on (X, %X) defined by l(B) =∫ B [p*∙f(x|q1) + (1 – p*)∙f(x|q0)]dµ for all B ∊ %X.
(iii) Dq∩Dr = ∅ for all q, r ∊ P such that q ≠ r.

We assume that:
A2: For each q ∊ P, there exists a continuum of agents Nq (µ(Nq) > 0) such that
if i ∊ Nq then i's prior belief coincides with q.
Because the value G(d: p*) is independent of the selection of d in Dq as shown by Lemma
6(i), we can define a function F(⋅, p*) on P by
F(q: p*) = G(d: p*) for some d ∊ Dq.

(8)

The function F(⋅, p*) need not to be an injection on P. See Example 1 (Figure 7) at theend of this section. The value of F(q: p*) can be interpreted as the the population ratio
of "The paitient is healthy", when all agents in Nq take the test procedures in D(q). Hencethe resulting measure µ(Mq) is given byµ(Mq) = µ(Nq) ⋅F(q: p*).

(9)

In the following of this paper, Dq is called a testing strategy corresponding to q.

Notethat Dq ∩ Dr = ∅ for all q, r ∊ P such that q ≠ r. (Lemma 6(iii)). A testing strategy Dq is
objectively optimal with respect to p* if and only if F(q: p*) ≥ F(r: p*) for all r ∊ P. As amain result of this paper, we have the following theorem:

16

Theorem 2(Existence and characterization of the objectively optimal testing strategy):(i) There exists uniquely an objectively optimal testing strategy with respect to p*.
(ii) The following three statements for a prior belief q ∊ P are mutually equivalent:(a) A testing strategy Dq corresponding to q is objectively optimal with respect to p*.
(b) If d* ∊ Dq and d ∉ Dq, then G(d*: p*) > G(d: p*).
(c) The prior belief q coincides with p*.

Proof of Theorem 2: (ii) First, we prove the assertion (ii) of Theorem 2. [(a) ⇒ (c)]:Suppose that (a) and q ≠ p* hold. If d ∊ Dq, it holds by Lemma 6(iii) that d ∉ Dp*. Fix any
d* ∊ Dp*. It holds by Theorem 1(iii) that G(d*: p*) > G(d: p*) , which implies that F(p*:p*) > F(q: p*). This contradicts with (a). Hence we have that [(a) ⇒ (c)]. [(c) ⇒ (b)]:Suppose that d* ∊ Dp* and d ∉ Dp*. It holds by Theorem 1(iii) and Lemma 6(ii) that G(d*:p*) > G(d: p*). [(b) ⇒ (a)]: Let r ∊ P be a prior belief such that r ≠ q. If d ∊ Dr, then it
holds by Lemma 6(iii) that d ∉ Dq. Fix any d* ∊ Dq. We have by (b) that G(d*: p*) > G(d:p*), which implies that F(q: p*) > F(r: p*). (i) Theorem 2(i) holds by Theorem 2(ii)[(b) ⇒(a)] and Lemma 4(i) .

For a given true probability p*∊ P, Theorem 2(i) ensures that there exists uniquely anobjectively optimal testing strategy with respect to p*. Theorem 2(ii)[(a) ⇔ (c)] implies
that a subjectively optimal testing procedure with respect to a prior belief q is objectively
optimal only if the prior belief q coincides with the true probability p*. Namely it holds
that F(p*: p*) > F(q: p*) for all q ≠ p*. The objective optimality condition selects a specificstrategy Dq among the class of all testing strategies { Dq : q ∊ P }, whereas the condition(b) in Theorem 2(ii) selects a specific strategy Dq such that each test procedure in Dq
dominates all of test procedures in (Dq)c = D/Dq, which is an optimality concept in D.
Theorem 2(ii) [(a) ⇔ (b)] shows that such two optimality concepts are equivalent in thetesting scheme.

17Example 1(Continue): Using Lemma 5, we can draw the graph of the function F(q: p*),when m = 2, p* = 0.4, pa = 0.3, pb = 0.1.

F(q; p*)p*=0.4q

Figure 7

5. An dynamic process for selecting the testing agents with correct priorbeliefs
This section introduces an evolutional testing model, where the Bayesian testing is
conducted repeatedly in parallel by many testing agents, and it is shown that only thetesting agents with correct prior belief are survived eventually.
(Time structure): The time is discrete and infinite, and it is denoted by t = 1, 2, ⋯ .
(Initial condition): Let %P be the s-field of Borel subsets in P = (0, 1).

Without

specifying the set of testing agents, we assume that the population distribution on the setof testing strategies at t = 1 is represented by a discrete measure b on (P, %P), where a
discrete measure on (P, %P) is a measure p on (P, %P) such that there exists a finite
subset Q of P such that p(Q) = p(P). In the following, a discrete measure p on (P, %P)
is called a population distribution on P.
(Transition rule): At the biginning of period t, for each medical doctor who are active att, a patient is selected randomly from a large population of patients. The patient is tested,18

and the medical doctor provides the medical treatment depending on the test result(signal) as specified by R3. If a patient's final health condition is good, the medical doctor
can leave an offspring, otherwise the doctor can not leave an offspring. This assumptionmeans that the fitness value of the strategy q ∊ P is assumed to coincide with the valueF(q; p*). The offspring's prior belief coincides with the pearent's prior belief. At the endof period t, all the medical doctors enter retirement, and the offsprings appear.

We

assume that the measure of each type of offsprings is increased by the natural rate of
population growth gt > 0. All the offsprings will become active medical doctors in the next
period t+1. A sequence of strategy distributions { pt } is called a dynamic process. Then
we have the following theorem:

Theorem 3: Let p* be the true probability (true prevalence rate), and let b be the initial
population distribution on P. Suppose that all of the testing agents satisfies the Bayesianbehavioral assumption. The above rules determine uniquely a dynamic process and theprocess is given by a history { pt }t=1∞ such that:(i) p1 = b, and (ii) pt+1({q}) = (1+gt)⋅F(q; p*)⋅pt({q}) for all q ∊ supp(b) and all t = 1, 2, 3, ⋯ .
We suppose additionally that p* ∊ supp(b). Then limt→ +∞ pt({ q })/pt(P) = 1 holds if andonly if q = p*.

Note that the convergence is independent of the value of d, because it is neutral. Hencegt can be variable over time.

For example, gt = sin(t).

Proof of Theorem 3: It holds by Theorem 1 that the initial distribution p coincides withthe distribution b. For all q ∊ supp(b), it holds by (ii) that pt({q}) = [∏n=1t(1+gn)]⋅[F(q; p*)]t⋅p1({q}), which implies
pt({q})/pt(P) = [F(q; p*)]t⋅p1({q})/ [∑q ∊ supp(b) [F(q; p*)]t⋅p1({q})= [F(q; p*)/F(p*; p*)]t/[∑q ∊ supp(b) [F(q; p*)/F(p*; p*)]t .

19

Case 1(q = p*): It holds by Theorem 2 that
limt→ +∞ [F(p*; p*)/F(p*; p*)]t = limt→ +∞1t = 1 andlimt→ +∞ [F(r; p*)/F(p*; p*)]t = 0 for all r ≠ p*.
Hence we have that
limt→ +∞[ ∑r ∊ supp(b) [F(r; p*)/F(p*; p*)]t = limt→ +∞ [F(p*; p*)/F(p*; p*)]t = 1 andlimt→ +∞ pt({q})/pt(P) = limt→ +∞ pt({q*})/pt(P) = 1.
Case 2(q ≠ p*): Because limt→+∞ [ F(q; p*)/F(p*; p*) ]t = 0 and lim t→ +∞ [ ∑r∊ supp(b) [F(r;p*)/F(p*; p*) ]t = 1, we have that limt→+∞pt({q})/pt(P) = 0/1 = 0.

Example 1 (Continue) : When qk = 0.1⋅k for k= 1, 2, ⋅⋅⋅, 8, and m = 2, p* = 0.4, pa = 0.3,
pb = 0.1, d = 0.3, we assume that µ(Nk(1)) = 1/8 for all k= 1, 2, ⋅⋅⋅, 8.

Popuration ratio (%)

q4 = 0.4

q5 = 0.5
q6 =0.6
q3 =0.3

Time

Figure 820

6. Proof of lemmas
Proof of Lemma 1: Setting Xn = (x − (1/n), x + (1/n)), it holds by the definition of P(q1|x :p) that P(q1|x : q) = limn q∙P(Xn|q1)/P(Xn), where P(Xn|q1) = ∫Xn f(x|q1)dµ and P(Xn) =∫Xn [q∙f(x|q1) + (1 – q)∙f(x|q0)]dµ = q∙P(Xn|q1) + (1 – q)∙P(Xn|q0). It follows from the meanvalue theorem that there exists a sequence of { xn } such that: (i) lim xn = x , (ii) xn ∊ Xn
for all n, (iii) p(Xn|qi) = f(xn|qi)(2/n) for i = 0, 1 and all n. Because f is continuous, wehave thatP(q1|x : q) = limn q∙P(Xn|q1)/P(Xn)= limn q∙f(xn|q1)(2/n)/[ q∙f(xn|q1)(2/n)+ (1 – q)∙f(xn|q0)(2/n)]= limn q∙f(xn|q1)/[q∙f(xn|q1)+ (1 – q)∙f(xn|q0)] = q∙f(x|q1)/[q∙f(x|q1) + (1 – q)∙f(x|q0)].
Using almost the same manner, we can prove that P(q0|x : q) = (1 – q)∙f(x|q0)/[q∙f(x|q1) +(1 – q)∙f(x|q0)].

Proof of Lemma 3: (i) We can prove Lemma 3(i) as a direct consequence of (6) and Lemma
A below:

Lemma A: Let g be a bounded continuous function on S in %X such that g(x) > 0 for
all x ∊ S. For a given null set A in %X, the value of the Lebesgue integral of g on S
coincides with the value of the Lebesgue integral of g on S/A.

Lemma A is a direct consequence of Royden (1988, Ch.4, Proposition 5(iii), Page 82).
(ii) Setting k(x) = q∙f(x|q1)+(1 – q)∙f(x|q0), X1 = d–1(d1) and X0 = d–1(d0), we have thatG(d: q) = (1 – q)⋅pa ∙a(d) + q∙(1– b(d)) + (1 – q)∙(1 – a(d)) + q⋅pb∙b(d)= (1 – q)⋅pa ∙ ∫ X1 f(x|q0) dµ + q ∙ ∫ X1 f(x|q1)dµ+ (1 – q)∙∫ X0 f(x|q0) dµ + q⋅pb∙∫ X0 f(x|q1)dµ= ∫ X1 [(1 – q)∙f(x|q0)⋅pa + q∙f(x|q1)]dµ + ∫ X0 [(1 – q)∙f(x|q0) +q∙f(x|q1) ⋅pb]dµ= ∫ X1 h(d1|x : q)⋅k(x)dµ + ∫ X0 h(d0|x : q)⋅k(x)dµ= ∫ X1 h(d1|x : q)dlq + ∫ X0 h(d0|x : q)dlq = ∫ X h(d(x)|x : q)dlq.

21

Proof of Lemma 4: Select any q ∊ P and let q be a subjective belief of a testing agent.
(i) Define a function d : X → { d0, d1 } byd(x) = d1 for all x ∊ { x ∊ X : f(x|q1)/f(x|q0) ≥ (1 – q)⋅(1 – pa)/q⋅(1 – pb) };d(x) = d0 for all x ∊ { x ∊ X : f(x|q1)/f(x|q0) < (1 – q)⋅(1 – pa)/q⋅(1 – pb) }.
It suffices to prove that { x ∊ X : f(x|q1)/f(x|q0) < (1 – q)⋅(1 – pa)/q⋅(1 – pb) } ∊ %X, which is adirect consequence of the fact that L(x) ≡ f(x|q1)/f(x|q0) is continuous on X.
(ii) Fix any d1 ∊ Dq and d2 ∊ D. It follows from Lemma 2(i) that d1 has the property:
d1(x) = d0 for almost all x inh(d1|x: q) = (1 – q)∙f(x|q0)⋅pa + q∙f(x|q1) < (1 – q)∙f(x|q0) + q∙f(x|q1)⋅pb = h(d0|x: q);d1(x) = d1 for almost all x in h(d1|x: q) > h(d0|x: q);
Do either if h(d1|x: q) = h(d0|x: q).
This means that h(d1(x)|x : q) = max[ h(d1|x : q), h(d0|x : q)] for almost all x ∊ X. Becausemax[ h(d1|x : q), h(d0|x : q)] ≥ h(d2(x)|x : q) for all x ∊ X, we have thath(d1(x)|x : q) ≥ h(d2(x)|x : q) for almost all x ∊ X.
(iii) Lemm 4(iii) is a direct consequence of Lemma 4(ii).
(iv) Fix any d1 ∊ Dq, and suppose that d2 ∊ D satisfies d2 ∉ Dq.
Case I( µ(C1(q)) > 0 and µ(C0(q)) > 0): We need a claim:

Claim 1: At least one of the following two statements holds:(a) There exists A*∊ %X such that A* ⊂ C1(q), µ(A*) > 0 and d2(x) = d0 for all x ∊ A*.
(b) There exists B*∊ %X such that B* ⊂ C0(q), µ(B*) > 0 and d2(x) = d1 for all x ∊ B*.
Proof of Claim 1: Set A* = C1(q) ⋂ { x ∊ X : d2(x) = d0 } and B* = C0(q) ⋂ { x ∊ X : d2(x) =d1 }. If µ(A*) = 0 and µ(B*) = 0, then µ(C1(q) ⋂ { x ∊ X : d2(x) = d1 }) = µ(C1(q)) and µ(C0(q)⋂ { x ∊ X : d2(x) = d0 }) = µ(C0(q)). It holds by µ(C1(q)) > 0 and µ(C0(q)) > 0 thatd2(x) = d1 for almost all x ∊ C1(q) and d2(x) = d0 for almost all x ∊ C0(q).
Hence it holds that d2 ∊ Dq, which is a contradiction. Thus we have that µ(A*) > 0 orµ(B*) > 0.

22

First we prove Lemma 4(iv) in case (a). It holds by Claim 1 and Royden (1988, Proposition
15, Ch.3, Page 63) that there exists a compact subset A in C1(q) such that µ(A) > 0 and
d2(x) = d0 for all x ∊ A. Because f(x|q1)/f(x|q0) > (1 – q)⋅(1 – pa)/q⋅(1 – pb) implies (1 –q)∙f(x|q0)⋅pa + q∙f(x|q1) > (1 – q)∙f(x|q0) + q∙f(x|q1)⋅pb for all x ∊ A, it holds by Lemma 2(i)thath(d1|x: q) > h(d0|x: q) for all x ∊ A.

(10)

It holds by the definition of d1 thatd2(x) = d0 and d1(x) = d1 for almost all x ∊ A.

(11)

We have by (10) and (11) that h(d1(x)|x : q) > h(d2(x)|x : q) for almost all x ∊ A. It holds
by Royden (1988, Proposition 15, Ch.3, Page 63) that there exists a compact subset A+ in
A such that µ(A+) > 0 and h(d1(x)|x : q) > h(d2(x)|x : q) for all x ∊ A+. Second, we canprove Lemma 4(iv) in case (b) by almost the same manner.
Case II (µ(C1(q)) > 0 and µ(C0(q)) = 0): We need a claim:

Claim 2: There exists A* ∊ %X such that A* ⊂ C1(q), µ(A*) > 0 and d2(x) = d0 for all x ∊A*.
Proof of Claim 2: Set A* = C1(q) ⋂ { x ∊ X : d2(x) = d0 }. If µ(A*) = 0, then µ(C1(q) ⋂ { x∊ X : d2(x) = d1 }) = µ(C1(q)). It holds by µ(C1(q)) > 0 that d2(x) = d1 for almost all x ∊ C1(q).
Hence it holds that d2 ∊ Dq, which is a contradiction. Thus we have that µ(A*) > 0.

It holds by Claim 2 and Royden (1988, Proposition 15, Ch.3, Page 63) that there exists a
compact subset A such that A* ⊂ C1(q), µ(A) > 0 and d2(x) = d0 for all x ∊ A. Becausef(x|q1)/f(x|q0) > (1 – q)⋅(1 – pa)/q⋅(1–pb) implies (1 – q)∙f(x|q0)⋅pa + q∙f(x|q1) > (1 – q)∙f(x|q0) + q∙f(x|q1)⋅pb for al l x ∊ A, it holds by Lemma 2(i) thath(d1|x: q) > h(d0|x: q) for all x ∊ A.

(12)

It holds by the definition of d1 thatd2(x) = d0 and d1(x) = d1 for almost all x ∊ A.

23

(13)

We have by (12) and (13) that h(d1(x)|x : q) > h(d2(x)|x : q) for almost all x ∊ A. It holds by
Royden (1988, Proposition 15, Ch.3, Page 63) that there exists a compact subset A+ in A
such that µ(A+) > 0 and h(d1(x)|x : q) > h(d2(x)|x : q) for all x ∊ A+.
Case III ( µ(C1(q)) = 0 and µ(C0(q)) > 0): We can prove Lemma 4(iv) in this case by almost
the same manner used in the proof of the case II above.
Case IV( µ(C1(q)) = 0 and µ(C0(q)) = 0): In this case, f(x|q1)/f(x|q0) = (1 – q)⋅(1 – pa)/q⋅(1– pb) for all x. Hence we have that Dq = D, which implies that there is no d2 ∊ D which
satisfies d2 ∉ Dq. Thus Lemma 4(iv) holds in this case.
(v) Lemm 4(v) is a direct consequence of Lemma 3(ii) and Lemma 4(ii, iv).

Proof of Lemma 5: (i) It holds that L'(x) = exp[m⋅(x−(½))] > 0, which implies that L(x)is an increasing function of x.
(ii) Because c(q) = (1/2)+(1/m)∙log [(1 – q)⋅(1 – pa)/q⋅(1 – pb)] = (1/m)∙log(1/q – 1)+(1/m)∙log(1– pa)/(1 – pb) + (1/2), it holds that c'(q) = (1/m)∙1/[1/q – 1]∙(–1/q2) = – (1/m)∙(1/q)∙[1/(1–q)] <0, which implies that c(q) is a decreasing function of q.
(iii) Because lim q→0 log(1/q – 1) = + ∞, we have thatlim q→0 c(q) = lim q→0 [(1/m)∙log(1/q – 1)+(1/m)∙log(1 – pa)/(1 – pb) + (1/2)] = + ∞.
Because lim p→1 log(1/q – 1) = – ∞, we have that
lim p→1 c(q) = lim p→1 [(1/m)∙log(1/q – 1) + (1/m)∙log(1 – pa)/(1 – pb) + (1/2)] = – ∞.
(iv) It holds that { x ∊ X : f(x|q1)/f(x|q0) > (1 – q)⋅(1 – pa)/q⋅(1 – pb) } = { x ∊ X : x > (L−1((1 –q)⋅(1 – pa)/p⋅(1 – pb)) } = { x ∊ X : x > c(q) }, and that { x ∊ X : f(x|q1)/f(x|q0) < (1 – q)⋅(1 –pa)/q⋅(1 – pb) } = { x ∊ X : x < (L−1((1 – q)⋅(1 – pa)/q⋅(1 – pb)) } = { x ∊ X : x < c(q) }.

Proof of Lemma 6: (i) We can prove Lemma 6(i) by almost the same manner used in the
proof of Lemma 3(i).
(ii) We can prove Lemma 6(ii) by almost the same manner used in the proof of Lemma
3(ii).

24

(iii) Because p ≠ q implies (1 – p)⋅(1 – pa)/p⋅(1 – pb) ≠ (1 – q)⋅(1 – pa)/q⋅(1 – pb), Lemma
6(iii) holds by the full range assumption of L(x) and the continuity of L(x).

7. Appendix
The full range assumption: { y ∊ ℝ++ : y = f(x|q1)/f(x|q0) for some x ∊ X. } = ℝ++ isindispensable for Theorems 2 and 3. We construct a counter example, on which the fullrange assumption, Theorems 2 and 3 do not hold :Example 2: Set p* = 0.2, and set m = 2, pa = 0.3, pb = 0.1. The two conditional probability
destributions, f(x|q0) and f(x|q1) are defined byf(x|q0) = A⋅exp[−x2] and f(x|q1) = A⋅exp[− x2]⋅g(x) for all x ∊ X,
where A = 2½⋅(2π)−½ , exp(y) = ey for all y ∊ X and g(x) = 2⋅ex/( ex+1) for all x ∊ X.

f(x|q1)f(x|q0)x

Figure 925

L(x) ≡ f(x|q1)/f(x|q0) = g(x) = 2⋅ex/(ex+1), which implies that the monotonicity condition
holds, but lim x→+∞ L(x) = 2 and lim x→−∞ L(x) = 0. Because (1 – p)⋅(1 – pa)/p⋅(1 – pb) = (1– p)⋅7/p⋅9 = (7/9)⋅(1/p – 1) > 2 iff p < (7/25) = 0.28, we have that
Dq = Dr for all q , r ∊ P such that q, r < 0.28,
and thatF(q: 0.2) = 0.82

for all q < 0.28

= 1 – [0.56⋅ ∫ X1 f(x|q0)dx +0.18⋅∫ X0 f(x|q1)dx]

for all q ≥ 0.28,

where X1 = d–1(d1) and X0 = d–1(d0).

F(q : 0.2)

q

Figure 10

8. References
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Analysis, 10 (1), 189-221.

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DeGroot, M. H., and Schervish, M. J. (2012). "Probability and Statistics, 4rth Ed." Pearson
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Friedman, M. (1953). "Essays in positive economics." University of Chicago Press.

Lehmann, E.L. and J. P. Romano, (2005), "Testing statistical hypotheses, 3rd Ed."
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Lesaffre, E., and Lawson, A. B. (2012). "Bayesian biostatistics." John Wiley & Sons.

Lo, A. W., and R. Zhang, (2021). The evolutionary origin of Bayesian heuristics and finite
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Nowak, M. A. (2006). "Evolutionary dynamics: exploring the equations of life". Harvard
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Okasha, S. (2013). The evolution of Bayesian updating. Philosophy of Science 80 (5), 745757.

Royden, H.L. (1988). "Real analysis 3rd Ed". Macmillan, London.

Sober, E. (2008). "Evidence and evolution: The logic behind the science". Cambridge
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Zhang, H. (2013). Evolutionary justifications for non-Bayesian beliefs. Economics Letters,
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