Discussion Paper No.465

Evolution of the prior beliefs in the simple
Bayesian hypothesis tests: A selection of thetesting agents with the correct beliefs
by Mitsunobu MIYAKE
August 24, 2022

TOHOKU ECONOMICS RESEARCH GROUP
Discussion Paper

GRADUATE SCHOOL OF ECONOMICS AND
MANAGEMENT TOHOKU UNIVERSITY
27-1 KAWAUCHI, AOBA-KU, SENDAI,
980-8576 JAPAN1

Evolution of the prior beliefs
in the simple Bayesian hypothesis tests:
A selection of the testing agents with the correct beliefs

by Mitsunobu MIYAKE

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

August, 2022

Abstract: In general, a successful decision maker need not have been endowed with thecorrect prior belief on the states of nature.

This paper, however, demonstrates that asimple Bayesian hypothesis test scheme has a good property: The probability of a testing
agent to obtain the optimal outcome is maximized only if the prior belief of the agent
coincides with the “correct” one, when the agent selects the Bayesian-optimal strategy
with respect to his or her (own) prior belief.

Consequently, in an evolutional setting,where the Bayesian test is conducted repeatedly in parallel by many testing agents withdiverse prior beliefs, if the fitness value is determined by the outcome, then only the agentsendowed with the correct prior beliefs survive. This result expains why an agent’s priorbelief can be assumed to coincide with the correct one in the Bayesian hypothesis test, as
if the agent knows the true probability that was assigned by nature.

Key words: Bayesian hypothesis test, diagnostic test, prior belief, Neyman-Pearsonlemma, natural selection, Malthusian competition, Laplace’s principle2

1. Introduction
In Bayesian hypothesis tests of a simple hypothesis,1 the testing agent is not informed of
the true probability of the events “the null hypothesis is true” or “the alternative
hypothesis is true” assigned by the nature; however the agent has the personal priorbeliefs (i.e., subjective probabilities) regarding the events.

Although the agent can

implement the Bayesian-optimal test procedure for a given prior belief, which maximizes
the (posterior) expected value of the test outcome conditioned on the sample (signal), there
is one problem: At what level should such a prior belief be determined ?
One possible answer is that such prior belief coincides with the “correct” belief,
because the correct prior belief can be derived from past information in general by applying
the statistical inference theory or Bayesian learning theory.2 Namely, before a test, if a
sufficient number of repetitions of the same test have been conducted, and if the priorbelief is revised based upon the sample (or signal) at each repetition, it is possible that thesequence of revised prior beliefs will converge to the correct one.
Such a long-term learning process can be conducted when the agent has a sufficient
information and memory regarding the samples.

Without assuming such an informa-tional assumption, this paper attempts to provide an alternative explanation for why theprior belief of the agent can be assumed to coincide with the correct one, by showing that_1

For the hypothesis tests in the philosophical literature, see Sober (2008, Ch.1). Okasha(2013) and Lo and Zhang (2021) derive the Bayesian behavioral hypothesis (updating of priorand utility maximization based on posterior) as a property of long-run equilibria ofevolutionary process in which non-Bayesian acts are permitted. Zhang (2013) shows that nonBayesian behavioral hypothesis is derived when individual’s risk aversion is incorporated into
the process.

In this paper, the Bayesian behavioral hypothesis is major premise; we do not

introduce the non-Bayesian hypotheses in this paper.
2

For statistical inference theory, see DeGroot and Schervish (2012, Ch 7). In particular, forthe theories of determiniation of the prior beliefs, see Berger, Bernardo and Sun (2015) andthe references. By way of the statistical inference, it is possible that a testing agent eventuallyknows the true probability with some awareness.

3the testing agents with the correct prior belief are naturally selected in an evolutional test
model.3
We first demonstrate that a simple Bayesian hypothesis test scheme has the good
property whereby the probability of the testing agent to obtain the optimal outcome ismaximized only if the prior belief of the agent coincides with the correct one, when theagent selects the Bayesian-optimal test procedure with respect to his or her (own) priorbelief, although a successful decision maker need not have been endowed with the correctprior belief in general.4

If the fitness value is determined by the test outcome in an

evolutional setting, we can re-state this result as follows: The simple Bayesian hypothesis
test scheme has a good property whereby a testing agent obtains the evolutionally optimal
outcome only if the prior belief of the agent coincides with the correct one (Proposition 2).
More precisely, the test scheme has this property under an additional assumption
for the sample distribution, which is not introduced in the test theory. However, we showthat the additional assumption is necessary for the test scheme to have this property by
constructing an example of a test in which the property does not hold if the assumption isremoved.
Second, we apply the aforementioned result to an evolutional test model in which
the Bayesian test is conducted repeatedly in parallel by many testing agents. We assume
that the agents have the diverse initial prior beliefs, which can be derived from a_3

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

explanation why an expert acts as if the expert knows the underlying information that is a
crucial factor for determining the outcome of the act. See Milton Friedman (1953, Section III)for example. In the literature of evolutionary epistemology, Downes (2000) and Boudry and
Vlerick (2014) consider the possibility that the true beliefs are naturally selected.
4

In the horse races, the single win scheme has the property, but the double win scheme does

not, because not only the full information for the true arrival order but also the full information
for the other betters’ betting behaviors is required to obtain the best result in the double win
scheme.

4hypothetical experiment with the agents who base their decision on Laplace’s principle ofinsufficient reason. Then, only the testing agents with the correct belief survive throughthe Malthusian competition (Proposition 3).
Because a Bayesian hypothesis test can be regarded as a one-shot Bayesian
hypothesis test that appears after many generations have occurred, the latter result(Proposition 3) provides an explanation for why the prior belief of the agent can beassumed to coincide with the correct one in a Bayesian hypothesis test, as if the agent
knows the true probability assigned by the nature.5
The next section introduces the Bayesian hypothesis test, and Section 3 introduces
the Bayesian-optimal test procedure and shows its well-known characterization (Proposition 1). Section 4 evaluates the Bayesian-optimal test procedures objectively and states
the main result of this paper (Proposition 2).

The dynamic process is introduced, andProposition 3 is stated in Section 5.

2. The Bayesian hypothesis test
This section introduces the (one-shot) Bayesian hypothesis test model with a testing agent.
As in a standard textbook (Lesaffre and Lawson, 2012), we formulate the Bayesian test as
a diagnostic test in which the testing agent (a medical doctor) conducts a test for the
medical treatments of a patient randomly selected from a large population. We make thefollowing assumptions:
A1: The patient contracts tuberculosis (TB), or the patient suffers from a normal cold (NC).
Nature determines one of the two events, TB or NC, exclusively for all patients in a city.
Let p* be the true probability of TB (the true prevalence rate), and we assume that 1 >p* > 0.
_5

Even though the agents know the true probability without awareness, this result provides areason why the common prior assumption in the game theory holds for the agents.

5A2: For a medical doctor, the patient is selected randomly from a large population ofpatients in the city. The medical doctor does not know the true probability, p*. But the
medical doctor has a subjective probability that the patient is TB, which we call a priorbelief. Let P ≡ (0, 1) be the set of all possible prior beliefs. The two simple hypotheses
are given as follows:
the null hypothesis (H0): The patient suffers from a normal cold, NC.
the alternative hypothesis (H1): The patient contracts tuberculosis, TB.
A3: A medical doctor examines a patient by a TB check-up kit. The check-up kit indicatesa real number x ∊ X ≡ ℝ, which is assumed to be the sample derived from the random
device of which the probability distribution is f(x|q0) in case the patient suffers from aNC. (This means that the sample size is one.) Otherwise, we assume that the check-up
kit indicates a real number x ∊ X, which is assumed to be the sample derived from therandom device of which probability destribution is f(x|q1). We assume that f(x|q0) andf(x|q1) are differentiable functions of x ∊ X, and that f(x|q0) > 0 and f(x|q1) > 0 for allx ∊ X.

We define the likelihood ratio function L(x) by L(x) ≡ f(x|q1)/f(x|q0), and weassume that L(x) is a monotone function of x ∊ X. Namely, we assume the following:L΄(x) > 0 for all x ∊ X, or L΄(x) < 0 for all x ∊ X.

(1)

The monotonicity assumption is a standard assumption in the test theory. See DeGroot
and Schervish (2012, Ch.9, Definition 9.3.2).
A4: The medical doctor selects one of the two acts, “Do 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 an act can be determined conditionedon the number in X given by A 3 , and the selection is represented by a function
d : X → { d0, d1 } such that d−1(d0) ∊ 𝔅X, where 𝔅X is the s-field of Borel subsets in X. The(completed) Lebesgue measure on 𝔅X is denoted by µ. In the following, the selection of6●q = q0

1−p*

I(xo)

●

Originx ∼ f(x ∣ q0)

I(x1)




x0

x1

x0



x∊ℝ

x1



x∊ℝ

p*

●

x ∼ f(x ∣ q1)

q = q1Figure 1

act d is called a test procedure, and the set of all test procedures is denoted by D.
A5: If the event selected by nature at A1 is consistent with the act d(x) selected at A4, thenthere is no error, otherwise an error occurred. Concretely, there are two types of errors:
An error is happened if and only if d(x) = d1, when q = q0;
An error is happened 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 remembers the act d(x) at A4, the doctor is not informed of the true
state, (q = q0) or (q = q1), and the doctor can not guess if an error occurred or not, as shownin Figure 2 below: 6_
6This assumption implies that the Bayesian updating is conducted within the one-shot test

and that the agents cannot use the genetically retained updating process. If we assume thegenetically retained updating or the existence of a central authority that attempts to infer the(true) posterior probabilities, it is not difficult to derive the (correct) Bayesian test procedure71 − pa

d1

●

d0

I(xo)

1 − pbNC (q = q0)

●

●

●

I(xo, d1)pb●●

I(xo, do)pax0
Bad Health(d = d0)Good Health(d = d0)Good Health(d = d1)

Bad Health(d = d1)x0

1 − pa

pb

●

pa

●

d1

●

d0

●

1 − pb●●●TB (q = q1)

Figure 2

A6: After the medical treatment, nature randomly determines the final outcome, “Thepatient is healthy” or “The patient is not healthy”.7

In case of no error, the probability of“The patient is healthy” is denoted by pa. In case of errors, the probability of “The patientis healthy” is denoted by pb. We assume that 1 > pa > pb > 0.

3. The Bayesian-optimal test procedures
In the test model constructed in the previous section, this section introduces the Bayesian_as shown in Footnote 9 in Section 3 of this paper. Moreover, in a one-shot test, if the agent
can derive a large sample, the agent inferes the true hypothesis, making use of the Bayesian
updating. Because the sample size is always equal to one in this paper, it is very difficult for
the agent to take such an inference strategy.
7

In a standard textbook as DeGroot and Schervish (2012, Ch 9, Section 9.8), only the cost of atreatment (or decision) is introduced in order to evaluate the test procedure.
the probabilities for good health mean the benefit of a treatment.

In this paper,

There is no essential

defference, because minimizing the cost is equivalent to maximizing the benefit as shown in
Section 3 of this paper. We can derive almost the same results, even if we do not assume thatthe probabilities for the two types errors are the same.

8

behavioral assumption for the agent and the Bayesian-optimal test procedure. Namely,
for a given prior belief, using the realization of the sample and the updating rule of thebelief, the testing agent computes the posterior probabilities, and the agent selects the test
procedure to maximize the probability that the patient is healthy after the medical
treatment.
Concretely, the agent’s prior belief for the occurrence of the event that a patient
has TB (q = q1) is specified by a subjective probability q in P. For a prior belief q ∊ P, let
P(qi|x: q) be the posterior probability of (q = qi) after receiving signal x ∊ X for i = 0, 1.8
Using the Bayes theorem, we have the following well-known lemma:
Lemma 1: P(θ! |x:q) =#∙%('|)! )#∙%('|)! )+(!–#)∙%('|)" )

q0
q1

d0

d1

pa
pb

pb
pa

and P(θ- |x:q) =(!–#)∙%('|)" )#∙%('|)! )+(!–#)∙%('|)" ).

Table 1.

For a prior belief q ∊ P, let h(di|x: q) be the posterior probability of “The patient is healthy”
in case of the medical treatment di after receiving signal x ∊ X for i = 0, 1. The following
holds by the definition of P(qi|x:q) and A6:h(d1|x:q)=P(q0|x:q)⋅pb +P(q1|x:q)⋅pa and h(d0|x:q)=P(q0|x:q)⋅pa+P(q1|x:q)⋅pb. (2)As a direct consequence of Lemma 1 and Equation (2), we may posit another lemma:
Lemma 2: (i) h(d1|x: q) =(1.q)⋅pβ ∙f(x|θ0)+q⋅pα ∙f(x|θ1)q∙f(x|θ1)+(1.q)∙f(x|θ0)

and h(d0|x: q) =(1.q)⋅pα ∙f(x|θ0)+q⋅pβ ∙f(x|θ1)q∙f(x|θ1)+(1.q)∙f(x|θ0).

_8

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. We can define P(q0|x:q) byalmost the same manner.

9(ii) For all q ∊ P, h(d1|x: q) and h(d0|x: q) are continuous for x∊ X.
For prior belief q ∊ P, h(d(x)|x : q) can be recognized as the probability of “The patientis healthy” of a test procedure d ∊ D after receiving signal x. The testing agent attempts
to maximize the probabilities h(d(x)|x : q) (x ∊ X) by selecting the suitable test procedured ∊ D. We define the optimal test procedure d as follows: For a given prior belief q ∊ P, atest procedure d ∊ D is defined to be a Bayesian-optimal test procedure with respect to q if
and only if:h(d(x)|x : q) = max[h(d1|x : q), h(d0|x : q)] for almost all x ∊ X, 9(3)which means that the probability of “The patient is healthy”, h(d(x)|x : q) is maximal in
the set { h(d1|x : q), h(d0|x : q) } for almost all x ∊ X.

Equation (3) above specifies theproperty of a test procedure only, and there is no specification for a relative advantage
over the other test procedures.
For the next Proposition, we require some definitions and Lemmas. For prior beliefq ∊ P, let G(d: q) be the induced probability of “The patient is healthy” if a testing agent
selects a test procedure d ∊ D. Then, according to A5 and A6 the following holds:
G(d: q) = (1–q)⋅(1–a(d))⋅pa+(1–q)⋅a(d)⋅pb +q⋅(1–b(d))⋅pa+q⋅b(d)⋅pb ,(4)

where a(d) is the probability of Type I errors of d defined by α(δ) = ∫2%! (3 ) f(x|θ- )dµ and!

b(d) is the probability of Type II errors of d defined by β(δ) = ∫2%! (3 ) f(x|θ! )dµ .!"_9

When A ∊ 𝔅X 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 ∊ 𝔅X, (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 from the 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 sufficient information, then thestatistician can infer the probabilities h(d1|x : p*) and h(d0|x : p*), where p* is the trueprobability, and then derives the Bayesian-optimal test procedure with respect to p* without
knowing the true probabilities [f(x|q0), f(x|q1), pa, pb, p*].
10

The numerical value of G(d: q) can be interpreted as the value of the expected utility definedby G(d: q)∙(utility of a healthy state)+(1–G(d: q))∙(utility of a not healthy state), because the10Lemma 3: (i) For all q ∊ P and all d1, d2 ∊ Dq, it holds that G(d1: q) = G(d2: q). (ii) It holds
that G(d: q) = ∫X h(d(x)|x : q)dlq, where lq is the measure of (X, 𝔅X) defined by lq(B) =∫B [q∙f(x|q1)+(1–q)∙f(x|q0)]dµ for all B ∊ 𝔅X.
For prior belief q ∊ P, a test procedure d* ∊ D is defined to be a subjectively optimal
test procedure with respect to q if and only if:
G(d*: q) ≥ G(d: q) for all d ∊ D.

(5)Statement (5) means that the test procedure d* maximizes the probability of “The patientis healthy”, which is computed from belief q. According to (4), it holds that G(d: q) = pa–[(1–q)⋅(pa–pb)⋅a(d)+q⋅(pa–pb)⋅ b(d)], which implies that the maximization of G(d: q) is
equivalent to the minimization of the cost, C(d: q) ≡ (1–q)⋅(pa–pb)⋅a(d)+q⋅(pa–pb)⋅b(d), bywhich we can define the likelihood ratio test procedure in the standard way as in DeGroot
and Schervish (2012, Ch.9, Section 9.2, Corollary 9.2.1).

Concretely, using the weights(1–q)⋅(pa–pb) and q⋅(pa–pb) for the probabilities of errors a(d) and b(d), we can define thecritical likelihood ratio at q ∊ P by D(q) = (1–q)⋅(pa–pb)/q⋅(pa–pb) = (1–q)/q. Then, the test
procedure d* ∊ D is defined to be a likelihood ratio test procedure with respect to q if andonly if:d*(x) = d1 for almost all x ∊ C1(q); d*(x) = d0 for almost all x ∊ C0(q),(6)

where C1(q) = { x ∊ X: f(x|q1)/f(x|q0) > D(q) } and C0(q) = { x ∊ X: f(x|q1)/f(x|q0) < D(q) }. 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) { x ∊ X:
f(x|q1)/f(x|q0) > (1–q)/q } = { x ∊ X: h(d1|x: q) > h(d0|x: q) }, { x ∊ X: f(x|q1)/f(x|q0) < (1– q)/q }= { x ∊ X: h(d1|x: q) < h(d0|x: q) }, and { x ∊ X: f(x|q1)/f(x|q0) = (1–q)/q } = { x ∊ X: h(d1|x: q)= h(d0|x: q) }. (iii) If d1 ∊ Dq and d2 ∊ D, then h(d1(x)|x: q) ≥ h(d2(x)|x: q) for almost all x ∊ X.
(iv) If d 1 ∊ D q and d 2 ∊ D q , then h(d 1 (x)|x: q) = h(d 2 (x)|x: q) for almost all x ∊ X.
_
utility value coincides with G(d:q), if we set (utility of a healthy state) = 1 and (utility of a nothealthy state) = 0.

11

(v) 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. (vi) If d1 ∊ Dq and d2∉ Dq, then G(d1: q) > G(d2: q).
We have the following well-known proposition11:

Proposition 1(Existence and characterization of the Bayesian-optimal test procedures):
Select any q ∊ P and let q be the 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) The test procedure d ∊ D is Bayesian-optimal with respect to q.
(b) The test procedure d ∊ D is subjectively optimal with respect to q.
(c) The 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).
Proof of Proposition 1: (ii) First, we prove the assertion (ii) of Proposition 1. [(a) ⇒ (c)]:Because Dq ≠ ∅ by Lemma 4(i), we can assume that there exists d* ∊ Dq. If d0 ∊ D is not a
likelihood ratio test procedure with respect to q, then it holds by Lemma 4(v) 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 then find 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. It holds according to (5) that there exists d0 ∊ D and a compact subset A in X suchthat µ(A) > 0 and h(d0(x)|x : q) > h(d*(x)|x : q) for all x ∊ A. However, it holds by Lemma_
11

Specifically, Proposition 1(ii)[(b)⇔(c)] is a re-statement of Neyman-Pearson lemma as in

DeGroot and Schervish (2012, Ch.9, Theorem 9.2.1) and Lehmann and Romano (2005, Ch.3).
For Proposition 1(ii)[(a)⇔(c)], see DeGroot and Schervish (2012, Ch.9, Section 9.8).

12

4(iii) that h(d*(x)|x : q) ≥ h(d0(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 with respect
to q, then it holds by Lemma 4(vi) 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 find that 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 with respect to q.
For any d ∊ D, it holds by Lemma 4(iii) 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)

Proposition 1(iii) is a direct consequence of Lemma 4(vi) and Proposition 1(ii)[(a) ⇔ (c)].
(i) Proposition 1(i) is a direct consequence of Proposition 1(ii) and Lemma 4(i).
For any prior belief q ∊ P, the existence of a Bayesian-optimal test procedure withrespect to q is ensured by Proposition 1(i). The optimality concept of (a) is defined by thebest responses to possible signals, whereas the optimality concept of (b) is defined by thecomparison of the values directly on the set of all test procedures. In terms of the gametheory, the former corresponds to the optimality in the behavioral strategies of extenstive
form games; the latter corresponds to the optimality in the pure strategies of strategic
form games.

Proposition 1(ii)[(a) ⇔ (b)] shows that these optimality concepts areequivalent in the Bayesian test scheme.

Proposition 1(iii) strengthens the optimality

condition by showing that the value of G in the Bayesian-optimal test procedure is strictly
greater than the values of G in the non-optimal test procedures.

Example 1: We assume that the conditional probability distributions, f(x|q0) and f(x|q1)
in A3 are given by f(x|q0) = A⋅exp[–(½)⋅x2] and f(x|q1) = A⋅exp[–(½)⋅(x–1)2] for all x ∊ X,
where A = (2π)−½ and exp(y) = ey for all y ∊ X.

13

f(x|q1)f(x|q0)x
Figure 3

The critical likelihood ratio at q ∊ P is given by D(q) = (1–q)/q. Then it holds by (4) thatthe likelihood ratio function L(x) is given by L(x) = exp(x–(½)). When q = 0.2, D(0.2) = 4,
and L−1(D(0.2)) = log(4)+0.5 ≒ 1.886.

L(x)D(0.2) = 4

x
L−1(D(0.4)) = 1.886

Figure 4

Example 2: We assume that the conditional probability distributions, f(x|q0) and f(x|q1)
in A3 are given by f(x|q0) = A⋅exp[–(½)⋅x2] and f(x|q1) = A⋅exp[–(½)⋅x2]⋅g(x) for all x ∊ X,
where A = (2π)–½ and g(x) = 2⋅ex/(ex+1) for all x ∊ X.

14

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

Figure 5

The likelihood ratio function L(x) is given by L(x) = 2⋅ex/(ex+1).
L(x)D(0.2) = 4

D(0.4) = 1.5

xL−1(D(0.4)) = 1.0986

Figure 6

When q = 0.2, D(0.2) = 4, C1(0.2) = { x ∊ X: L(x) > 4 } = ∅ and C0(0.2) = { x ∊ X: L(x) < 4 } =X. Hence a typical likelihood ratio test procedure d* in D0.2 is given by d*(x) = d0 for all x∊ X.

When q = 0.4, D(0.4) = 1.5 and L−1(D(0.4)) = log(3) ≒ 1.0986.

Hence a typical

likelihood ratio test procedure d* in D0.4 is given by d*(x) = d1 for all x ≥ 1.0986; d*(x) = d0
for all x < 1.0986.

15

4. The test strategy concept and the evolutionally optimal test strategy
It holds by Proposition 1 that if a testing agent attempts to maximize the subjective
probability of the event that the patient is healthy, then the agent selects a likelihood ratio
test procedure in Dq, where q is the prior belief of the agent. We assume the following:
A7: The fitness value coincides with the (objective) probability of the event that the patientis healthy for each testing agent.
Suppose that p* ∊ P is the true probability (the true prevalence rate). If a testing agent
selects a test procedure d ∊ D, the fitness value of the agent denoted by G(d: p*) can be
defined by A7 and (4) as follows:G(d: p*) = (1–p*)⋅(1–a(d))⋅pa+(1–p*)⋅a(d)⋅pb +p*⋅(1–b(d))⋅pa+p*⋅b(d)⋅pb .
Lemma 5: Set 𝛼 ∗ (q) = ∫5 ! (6) f(x|θ- )dµ and 𝛽 ∗ (q) = ∫5 " (6) f(x|θ! )dµ .

(7)

It holds that a*(q)= a(d) and b*(q) = b(d) for all q ∊ P and all d ∊ Dq.

We define a function, F(⋅, p*) on P as follows:
F(q: p*) = (1–p*)⋅(1–a*(q))⋅pa+(1–p*)⋅a*(q)⋅pb+p*⋅(1–b*(q))⋅pa+p*⋅b*(q)⋅pb for all q ∊ P.

(8)

Lemma 6: (i) F(q: p*) = G(d: p*) for all q ∊ P and all d ∊ Dq. (ii) F(q: p*) is continuous withrespect to q ∊ P.
The function F(⋅, p*) need not to be an injection on P. The value of F(q: p*) coincides
with the fitness value, when the testing agent takes the test procedure in Dq.

Later inthis paper, Dq is called a test strategy corresponding to q. Test strategy Dq is defined to bean evolutionally optimal strategy with respect to p* if and only if:
F(q: p*) > F(r: p*) for all r ∊ P such that r ≠ q.

(9)

Let us consider the two examples again:
Example 1.(continued) Set p* = 0.4. We can draw the graph of the function F(q: 0.4) for
this example, using the argument for deriving the procedure d* in D0.2. Therefore, D0.4 is16

an evolutionally optimal test strategy with respect to p* = 0.4.
F(q; 0.4)

q = 0.4q

Figure 7

Example 2.(continued) Set p* = 0.2, pa = 0.3, pb = 0.1.

Following the arguments forderiving the procedure d* in D0.2 and (7), we have that Dq = Dr for all q, r < d–1(2) = 1/3, andthat:
F(q: 0.2) = p*⋅pb+(1–p*)⋅pa = 0.26

for all q < 1/3= 0.3–[0.16a*(q)+0.04b*(q)] for all q ≥ 1/3.
F(q : 0.2)

q
Figure 8

Therefore, there is no evolutionally optimal test strategy with respect to p* in Example 2.

17

To ensure the existence of an evolutionally optimal test strategy with respect to thetrue probability (the true prevalence rate), p*, we additionally assume the following
condition:
A8: For any y ∊ ℝ++, there exists some x ∊ ℝ such that y = f(x|q1)/f(x|q0).
Because A8 means the range of L(x) ≡ f(x|q1)/f(x|q0) coincides with the full set ℝ++, we callA8 the full range condition of the likelihood ratio. Under the full range condition of thelikelihood ratio, we obtain the following lemma:

Lemma 7: Dq ∩ Dr = ∅ for all q, r ∊ P such thatq ≠ r.

As a main result of this paper, we make the following proposition:

Proposition 2(Existence and characterization of the evolutionally optimal test strategy):
Under all the assumptions stated above, the following assertions hold:(i) There exists an evolutionally optimal test strategy with respect to p*.
(ii) Test strategy Dq corresponding to a prior belief q is evolutionally optimal with respect
to p* if and only if the prior belief q coincides with p*.
(iii) Let Dq be an evolutionally optimal test strategy with respect to p*.

Then, it holds

that G(d*: p*) > G(d: p*) for all d* ∊ Dq and all d ∉ Dq.
Proof of Proposition 2: (ii) Suppose that Dq is an evolutionally optimal test strategy withrespect to p* and that q ≠ p* holds. We have by (9) that F(q: p*) > F(p*: p*). If d ∊ Dq, itholds by Lemma 7 that d ∉ Dp*. Fix any d* ∊ Dp*. It holds by Proposition 1(iii) that G(d*:p*) > G(d: p*), which implies that F(p*: p*) > F(q: p*). This is a contradiction. Therefore,
we have that q = p*.

Conversely, suppose that q = p*.

Let r ∊ P be a prior belief suchthat r ≠ q. Suppose that d ∊ Dr holds. Then, it holds by Lemma 7 that d ∉ Dp*. Fix any d*∊ Dp*. We have by Proposition 1(iii) that G(d*: p*) > G(d: p*), which implies that F(q: p*)> F(r: p*).

(i) Proposition 2(i) holds by Proposition 2(ii) and Lemma 4(i). (iii) Supposethat d* ∊ Dq and d ∉ Dq. It holds by Proposition 2(ii) that d* ∊ Dp* and d ∉ Dp*. We have by18

Proposition 1(iii) that G(d*: p*) > G(d: p*).
For a true probability p*∊ P, Proposition 2(i) ensures that there exists an evolutionallyoptimal test strategy with respect to p*.

Proposition 2(ii) implies that the uniqueevolutionally optimal test strategy is Dp*. Proposition 2(iii) implies that any test procedurein the evolutionally optimality strategy Dp* dominates all of the test procedures in (Dp*)c ≡D/Dp*, which implies that Dp* dominates any mutation strategies.

5. A dynamic process for selecting the testing agents with the correctprior beliefs
This section introduces a dynamic test model in which the Bayesian test is conducted
repeatedly in parallel by many medical doctors (testing agents), and it is shown that only
the medical doctors with the correct prior belief survive through the Malthusiancompetition. We additionally assume the following conditions:
A9 (Time structure and population measures): Time is discrete and infinite, and it isdenoted by t = 1, 2, ⋯ .

Without specifying the set of medical doctors, the population

measure of the doctors at t is represented by the measure pt on P for t = 1, 2, ⋯ .
A10 (Initial condition): At the beginning of time 1, we assume that there is a continuumof doctors N1 = (0, 1), and that all the doctors in N1 form the initial priors based on Laplace’s
principle of insufficient reason.

Concretely, we consider a hypothetical experiment inwhich each doctor determines the initial prior by drawing a lottery defined by the uniform
probability distribution on the set of all possible priors.
lottery independently and simultaneously.

Then, all the doctors draw the

As the result of the experiment, there is a

population measure p1 on P,12 which coincides with the Lebesgue µ on P.

The density

function h1 of p1 is defined as follows:
h1(q) = 1 for all q ∊ P.

(10)
_

12

Namely, we apply Laplace’s principle for an experiment with a double infinity of events anddoctors.

19

A11 (Transition rule): At the beginning of period t, for each medical doctor who is activeat t, a patient is selected randomly from a large population of patients.

The patient istested, and the doctor provides the medical treatment depending on the test result (signal)
specified by A3.

If the patient’s final health condition is good, the doctor can leave anoffspring, otherwise the doctor cannot leave an offspring. At the end of period t, all the
doctors enter retirement, and the offsprings will appear and become active doctors in thenext period t+1.

We assume that the offspring’s prior belief coincides with the parent’s

prior belief. Moreover, we assume that the measure of each type of offspring is increasedby the natural rate of population growth gt > 0. Formally, we assume the following:
lim

7→+9

:&'! (;(#,=)):& (;(#,=))= F(q; p*)⋅(1+gt) for all q ∊ P and all t = 1, 2, 3, ⋯ ,(11)where B(q, k) ≡ { r ∊ P: ∣ r–q ∣ < (1/k) } for all q ∊ P and all k = 1, 2, 3, ⋯ . A sequence ofmeasures { π> }+9>?! satisfying (11) is called a dynamic process.
Lemma 8: Let { π> }+9>?! be a dynamic process, and let h1 be the Radon-Nikodý m density
function of p1. Then all population measures pt (t = 2, 3, ⋯ ) are absolutely continuous withrespect to the Lebesgue µ on P, and it holds that@&'! (#)@& (#)= F(q; p*)⋅(1+gt) for all q ∊ P and all t = 1, 2, ⋯ ,where ht be the density function of pt for t = 2, 3, ⋯ .
We then make the following proposition:

Proposition 3: Let p* be the true probability (true prevalence rate), and let p1 be theinitial population measure on P. Let h1 be the density function of p1, and suppose that
all of the assumptions hold.

Then, a dynamic process is determined uniquely by thesequence of the density functions { ℎ> }+9>?! such that
ht+1(q) = F(q; p*)⋅(1+gt)⋅ht(q) > 0 for all q ∊ P and all t = 1, 2, ⋯ .
Moreover, it holds that lim@& (B)

A→+9 @& (C∗)= 0 for all r ≠ p*.

20

(12)

Proof of Proposition 3: Euation (12) holds by Lemma 8, and according to Equation (12)and Proposition 2 the following holds:&

D ∏&()! F(B; C∗)⋅(!+H( )I ∙@! (B)@& (B)= lim&
A→+9 @& (C∗)
A→+9 D ∏& F(C∗; C∗)⋅(!+H( )I ∙@! (C∗)

lim

()!

F(B; C∗) A
@ (B)
] ∙ !
@! (C∗)
A→+9 F(C∗; C∗)= lim [

= 0 for all 𝑟 ≠ 𝑝 ∗.

Even if the difference in the fitness values between the medical doctors with correct beliefs
and the other medical doctors is very small, it follows from Proposition 3 that the selection
will appear as a consequence of a mathematical property of the exponential function.13
Proposition 3 holds, independent of the natural rate of the population growth rate gt > 0,
because the rate gt is assumed to be common for all types of agents. Therefore, even if gt
is variable over time, Proposition 3 still holds. For example, gt = sin(t).
Example 1.(continued): When p* = 0.4, pa = 0.3, pb = 0.1, set h1(q) = 1 for all q ∊ P and gt= 0.23 for all t. Then, we define a function Ht by Ht(q) = pt(q)/pt(p*) for all q ∊ P and all t.

Ht(q)

q
Figure 9_
13

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

21

6. Proof of lemmas
This section proves Lemmas 1, 3, 4, 5 and 6. We need some technical claims:
Claim 1: Let g be a bounded continuous function on S in 𝔅X such that g(x) > 0 for allx ∊ S.

For a given null set A in 𝔅X, the value of the Lebesgue integral of g on S

coincides with the value of the Lebesgue integral of g on S/A.
Proof of Claim 1: Claim 1 is a direct consequence of Royden (1988, Ch.4, Proposition
5(iii), Page 82).
Claim 2: For any Z ∊ 𝔅X such that µ(Z) > 0, there exists a compact set Z* ∊ 𝔅X such thatZ* ⊂ Z and µ(Z*) > 0.
Proof of Claim 2: Fix any Z ∊ 𝔅X such that µ(Z) > 0. It holds by Royden (1988,
Proposition 15, Ch.3, Page 63) or Ito (1984, Theorem 2.4.1, Ch.2, Page 61) that there exists
a closed set W ∊ 𝔅X such that W ⊂ Z and µ(W) > 0. Set Vn = [– n, n] for all n = 1, 2,⋯.

If W⋂Vn = ∅ for all n, then µ(W) = 0, which is a contradiction. Hence it holds thatthat W⋂Vn* ≠ ∅ for some n*. Thus Z* ≡ W⋂Vn* is compact and µ(Z*) > 0.

Proof of Lemma 1: Setting Xn = (x−(1/n), x+(1/n)) for all n ≥ 2, it holds by the definitionof P(q1|x : p) that P(q1|x : q) = limn→+∞q∙P(Xn |θ1 ), where P(Xn|q1) = ∫Xn f(x|q1)dµ and P(Xn)
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 themean-value theorem that there exists a sequence of { xn } such that: (i) lim xJ = x, (ii)n→+∞xn ∊ Xn for all n ≥ 2, (iii) p(Xn|qi) = f(xn|qi)(2/n) for i = 0, 1 and all n ≥ 2. Because f
is continuous, we have thatP(q1|x : q) = lim

n→+∞

=

q∙P(Xn |θ1 )P(Xn )q∙f(xn |θ1 )∙(2/n)
n→+∞ q∙f(xn |θ1 )∙(2/n) + (! – #)∙f(xn |θ0 )∙(2/n)= limq∙f(x|θ1 )q∙f(x|θ1 )# (% – ')∙f(x|θ0 ).

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)].

22

Proof of Lemma 3: (i) Lemma 3(i) is a direct consequence of (4) and Claim 1. (ii) Settingk(x) ≡ q∙f(x|q1)+(1 – q)∙f(x|q0), X1 = d–1(d1) and X0 = d–1(d0), we have by (4) thatG(d: q) = (1 – q)⋅(1 – a(d))⋅pa + (1 – q)⋅a(d)⋅pb + q⋅(1– b(d))⋅pa + q⋅b(d)⋅pb= (1 – q)⋅pa∙∫ X₀f(x|q0)dµ + (1 – q)⋅pb∙∫ X₁f(x|q0)dµ+ q⋅pa∙∫ X₁f(x|q1)dµ + q⋅pb∙∫ X₀ f(x|q1)dµ= ∫ X₁ [(1 – q)⋅pb ∙f(x|q0) + q⋅pa∙f(x|q1)]dµ + ∫ X₀ [(1 – q)⋅pa∙f(x|q0) + q⋅pb∙f(x|q1)]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.

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) /q };d(x) = d0 for all x ∊ { x ∊ X : f(x|q1)/f(x|q0) < (1 – q) /q }.
It suffices to prove that { x ∊ X : f(x|q1)/f(x|q0) < (1 – q)/q} ∊ 𝔅X, which is a direct consequence of the continuity of L(x) ≡ f(x|q1)/f(x|q0) on X. (ii): Setting k(x) ≡ q∙f(x|q1)+(1 –q)∙f(x|q0), it holds that h(d1|x: q) – h(d0|x: q) = { [(1 – q)⋅pb∙f(x|q0)+q⋅pa∙f(x|q1)] – [(1 –q)⋅pa∙f(x|q0) +q⋅pb∙f(x|q1)] }/k(x) = [(pa – pb)/k(x)]∙[ q⋅f(x|q1) – (1 – q) ∙f(x|q0)]. Hence Lemma
4(ii) holds by (pa – pb)/k(x) > 0. (iii) Fix any d1 ∊ Dq and d2 ∊ D. It follows from Lemma
2(i) and Lemma 4(ii) that d1 has the property:
d1(x) = d1 for almost all x in { x ∊ X : h(d1|x: q) > h(d0|x: q) }.
d1(x) = d0 for almost all x in { x ∊ X : h(d0|x: q) > h(d1|x: q) }.
Do either for all x in { x ∊ X : h(d0|x: q) = h(d1|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.
(iv) Lemm 4(iv) is a direct consequence of Lemma 4(iii). (v) Fix any d1 ∊ Dq, and supposethat d2 ∊ D satisfies d2 ∉ Dq.

23

Case I( µ(C1(q)) > 0 and µ(C0(q)) > 0): We need a claim:
Claim 3: At least one of the following two statements holds:(a) There exists A*∊ 𝔅X such that A* ⊂ C1(q), µ(A*) > 0 and d2(x) = d0 for all x ∊ A*.
(b) There exists B*∊ 𝔅X such that B* ⊂ C0(q), µ(B*) > 0 and d2(x) = d1 for all x ∊ B*.
Proof of Claim 3: 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.

First we prove Lemma 4(v) in the case (a) in Claim 3. It holds by Claim 2 that there exists
a compact subset A in C1(q) such that µ(A) > 0 and d2(x) = d0 for all x ∊ A. It holds byLemma 4(ii) and Lemma 2(i) thath(d1|x: q) > h(d0|x: q) for all x ∊ A.

(13)

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

(14)

We have by (13) and (14) that h(d1(x)|x : q) > h(d2(x)|x : q) for almost all x ∊ A. It holds
by Claim 2 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 can prove Lemma 4(v) in the case (b) byalmost the same manner.
Case II (µ(C1(q)) > 0 and µ(C0(q)) = 0): We need a claim:
Claim 4: There exists A* ∊ 𝔅X such that A* ⊂ C1(q), µ(A*) > 0 and d2(x) = d0 for all x ∊A*.
Proof of Claim 4: 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.

24

It holds by Claim 4 and Claim 2 that there exists a compact subset A such that A* ⊂C1(q), µ(A) > 0 and d2(x) = d0 for all x ∊ A. It holds by Lemma 4(ii) and Lemma 2(i) thath(d1|x: q) > h(d0|x: q) for all x ∊ A.

(15)

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

(16)

We have by (15) and (16) that h(d1(x)|x : q) > h(d2(x)|x : q) for almost all x ∊ A. It holds by
Claim 2 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(v) 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)/q for all x.
Hence we have that Dq = D, which implies that there is no d2 ∊ D which satisfies d2 ∉ Dq.
Thus Lemma 4(v) holds in this case. (vi) Lemm 4(vi) is a direct consequence of Lemma
3(ii) and Lemma 4(iii, v).

Proof of Lemma 5: Lemma 5 holds by (1), (4) and (6).

Proof of Lemma 6: (i) Lemma 6(i) holds by (8) and Lemma 5. (ii) First, we prove thata*(q) is continuous on P. Fix any q ∊ P.
Case I (C1(q) ≠ ∅): We have by (1) that C1(q) is an interval (−∞, c(q)] or
where c(q) = L−1(D(q)).

[c(q), +∞),

Becasue D(q) = (1 – q)/q and L−1 are continuous, c(q) = L−1(D(q))

is continuous at q. Hence it holds by the absolute continuity of the definite integral thata*(p) is continuous.
Case II (C1(q) = ∅): If there exists an open interval Q in P such that q ∊ Q and C1(r) =∅ for all r ∊ Q, then a*(r) = 0 for all r ∊ Q, which implies the continuity of a*(q). Hence,
we can assume that there exists a sequence { qn } such that limn qn = q and a*(qn) > 0
for all n. Because a*(qn) > 0 implies C1(qn) ≠ ∅, we have by (1) that C1(qn) is an interval(−∞, c(qn)] or [c(qn), +∞), where c(q) = L−1(D(q)). If C1(qn) = (−∞, c(qn)], we have by C1(q)25

= ∅ that lim c(qn) = −∞ and lim a*(qn) = 0 = a*(q). Otherwise, we have by C1(q) = ∅ that
lim c(qn) = +∞ and lim a*(qn) = 0 = a*(q).

Proof of Lemma 7: Because p ≠ q implies (1 – p)/p ≠ (1 – q)/q, Lemma 7 holds by A8 andthe continuity of L(x).

Proof of Lemma 8: Let { π> }+9>?! be a sequence of measures satisfying (11), and let ht be
the density function satisfying (12) for t = 1, 2, 3, ⋯ . Fix t = 1 and select any q ∊ P. Thenit holds by the definition of the Radon-Nikodym density (Shilov and Gurevich, 2012,
Sec.10.1, (3)) and (11) that
, (/(0,2)))*+ '→)* 4(/(0,2)), (/(0,2)))*+ 5'→)* 4(/(0,2 ))= lim/- (0(',2))= F(q; p*)⋅(1+g1).

,→#. /5 (0(',2))Because it holds by Lemma 6 and (12) that lim/- (0(',2)),→#. 3(0(',2))= F(q; p*)⋅(1+g1)⋅h1(q) for all q∊ P and F(q; p*)⋅(1+g1)⋅h1(q) is continuous and positive on P, p2 is absolutely continuouswith respect to the Lebesgue µ on P.

Using an induction argument, we have that all

population measures pt (t = 2, 3, ⋯ ) are absolutely continuous with respect to theLebesgue µ on P, and it holds that@&'! (#)@& (#)= F(q; p*)⋅(1+gt) for all q ∊ P and all t = 1,

2, ⋯, where ht be the density function of pt for t = 2, 3, ⋯ .

7. References
Berger, J. O., Bernardo, J. M., and Sun, D. (2015). Overall objective priors. Bayesian
Analysis, 10 (1), 189-221.
Boudry, M., and Vlerick, M. (2014). Natural selection does care about truth. International
Studies in the Philosophy of Science, 28(1), 65-77.

26

DeGroot, M. H., and Schervish, M. J. (2012). “Probability and Statistics, 4rth Ed.” Pearson
Education, Harlow, UK.
Downes, S. M. (2000). Truth, selection and scientific inquiry. Biology and Philosophy, 15(3),425-442.
Friedman, M. (1953). “Essays in positive economics ”, University of Chicago Press.
Ito, K. (1984). “Introduction to probability theory”, Cambridge University Press,
Cambridge.
Lehmann, E.L. and J. P. Romano, (2005), “Testing statistical hypotheses, 3rd Ed.”,
Springer, New York.
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
memory. iScience 102853.
Nowak, M. A. (2006). “Evolutionary dynamics: exploring the equations of life”, Harvard
University Press.
Okasha, S. (2013). The evolution of Bayesian updating. Philosophy of Science 80 (5), 745757.
Royden, H.L. (1988). “Real analysis 3rd Ed.”, Macmillan, London.
Shilov, G. E., and Gurevich, B. L. (2012). “Integral, measure and derivative: a unified
approach”, Dover, New York.
Sober, E. (2008). “Evidence and evolution: The logic behind the science”, Cambridge
University Press.
Zhang, H. (2013). Evolutionary justifications for non-Bayesian beliefs. Economics Letters,
121 (2), 198-201.

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