What Does Agreement Add?

Two sources say yes.

Suppose we know that each source is informative when considered alone. We also know how often their reports agree. Surely, then, we know how much more confident to become when both say yes?

Not necessarily. Two sets of probability laws can agree about the reliability of each source and about every frequency of paired reports that an observer could count without knowing the outcomes, yet disagree about the probability of the event when the sources agree.

That is a limitation on what the available information identifies. It is not a failure of Bayes’ rule. And it leads to a second, less obvious possibility: even when the combined probability remains uncertain, all probability laws consistent with the information may favor the same consequential action.

We can see both possibilities without complicated machinery. We need an event, two binary signals, and care about which information has actually been supplied.

1. Two ways to agree

Let an event either occur, denoted by $E$, or not occur. Before either report, suppose its probability is $1/2$.

Each of two sources produces a positive (+) or negative (−) signal. We stipulate that, considered separately, each source produces a positive signal in three quarters of event cases and in one quarter of non-event cases:

$$P(S_i=+\mid E)=\tfrac34,\qquad P(S_i=+\mid \neg E)=\tfrac14\quad(i=1,2).$$

These are assumed characteristics of hypothetical binary signal experiments. They are not observations of real forecasting systems, and they are not, by themselves, statements that two numerical probability forecasts are calibrated.

Bayes’ rule tells us what a single positive signal means here. Of all positive reports from either source, three quarters arise in event cases:

$$P(E\mid S_i=+)=\frac{(1/2)(3/4)}{(1/2)(3/4)+(1/2)(1/4)}=\tfrac34.$$

But what if both sources report positive?

For our first scenario, suppose we also know the unconditional distribution of all four paired reports:

This is more than knowing that the sources agree frequently: it is the entire distribution of paired reports without outcome labels. Even so, the event probability after ++ is not determined.

To see why, imagine 160 hypothetical, equally weighted instances, split into 80 event cases and 80 non-event cases. These whole-number counts are a convenient picture of two particular probability laws. They are not a measured sample, nor do they exhaust the continuous family of possible laws we will derive shortly.

World A — constructed counts

World B — constructed counts

In each world the two rows sum to 80 each. Check the stipulated individual characteristics: each source gives 60 positives among event cases, and 20 among non-event cases, in both worlds. That is exactly the required three-quarters and one-quarter.

Now hide the event labels and add the two rows in either world:

Every entry agrees. Even perfect knowledge of the complete unconditional joint distribution of the two reports, not merely the frequency of ++, would leave World A and World B indistinguishable on this information. What differs is how each report profile is allocated across event and non-event cases.

Yet their answers to the question that matters are different.

In World A, 40 of the 60 ++ instances are event cases:

$$P_A(E\mid++)=\frac{40}{60}=\boxed{\tfrac23}.$$

In World B, all 60 are event cases:

$$P_B(E\mid++)=\frac{60}{60}=\boxed{1}.$$

We knew that either source’s positive report alone meant an event probability of $3/4$. Nonetheless, in World A the second positive moves that probability down to $2/3$, while in World B it moves it up to $1$.

There is no paradox. A report’s evidentiary significance is conditional on what is already known. In World A, among cases with a first-source positive, another positive is relatively more common in the non-event cases than in the event cases. In World B the doubly positive pair never appears in non-event cases. The same individual source characteristics can be combined through different state-dependent patterns of overlap.

The correct lesson is not that agreement decreases confidence. It is that individual reliability and the observable frequency of agreement do not, on their own, tell us which conclusion about agreement is justified.

2. Not just two examples: the exact limit of the information

Perhaps Worlds A and B are merely two unusual constructions. Could other compatible worlds put the posterior outside their range?

No. The interval between them is the complete, sharp identified set.

In a literal 160-instance count table, each source has 60 positives among 80 event cases. The two sets of positives must overlap in at least 40 instances and at most 60. If $x$ is the number of event cases among the 60 doubly positive cases, then $x$ is an integer between 40 and 60. A fixed 160-instance picture therefore produces only the discrete values $x/60$. It does not by itself fill a continuum.

The mathematical question, however, concerns all compatible probability laws, not just integer allocations in one finite illustration. For any real number $t$ between 0 and 1, mix the joint laws of Worlds A and B with weights $1-t$ and $t$. The mixture is a valid probability law and preserves every stipulated individual source characteristic and all four unlabeled report probabilities. Its posterior following ++ is

$$P_t(E\mid++)=\frac{40+20t}{60}=\frac23+\frac t3,\qquad 0\le t\le1.$$

Here $40+20t$ is a count-scaled probability mass, not a claim that a fixed 160-instance table contains fractional people or instances. As $t$ ranges continuously, the posterior realizes every real value in

$$\boxed{P(E\mid++)\in[\tfrac23,1].}$$

No compatible law can fall outside these endpoints, by the overlap bounds; the technical section establishes both nonnegativity and complete attainability. This is why the interval is sharp. It is a set of model-compatible probabilities, not a sample-based confidence interval.

Notice how much we stipulated: the prior event rate, each source’s conditional signal behavior, and even the full unconditional distribution of paired reports. These assumptions leave unresolved the allocation of paired reports between event and non-event cases. Unlabeled observations of the same kinds of report pairs cannot settle that allocation just by accumulating more instances. With an arbitrarily large sample from an appropriate stable population, we might estimate the unlabeled frequencies very accurately and still face the same identification problem. Outcome-labeled evidence or additional defensible structural knowledge could distinguish the worlds.

We therefore have a rigorous answer to a limited question: this supplied information does not warrant a unique combined posterior. Whether anyone possesses justified empirical grounds for supplying the information in the first place is a different question.

There is another distinction worth making before leaving the example. A range of compatible probabilities is not equivalent to knowing nothing. Its endpoints can matter for a decision even when the exact probability remains unknown.

3. A decision without an identified probability

Consider a second, different scenario with the same prior and the same stipulated individual source characteristics. We now change the unconditional frequency of ++ from $3/8$ to $3/10$. The whole unlabeled pair distribution consequently changes:

This is not additional data collected about World A or World B, nor a posterior update within the first scenario. It is a different stipulated family of probability laws. Comparing the families reveals what kind of information can matter for a decision.

To picture the endpoints, consider again a 160-instance count scale. In this new scenario, 48 instances have ++; the stipulated individual characteristics require at least 40 of them to be event cases, and no more than all 48 can be. Those integer counts display permissible endpoints and discrete illustrations. The complete sharp interval, however, again ranges over all compatible probability laws—not just the counts representable in a fixed 160-instance table:

$$\boxed{P(E\mid++)\in[\tfrac56,1].}$$

The technical section proves attainability of every intermediate probability by varying the state-dependent overlap continuously.

The event probability remains unidentified. But suppose we must choose between two actions with these invented losses:

A smaller expected loss is preferred. If the event probability is $p$, acting has expected loss 8 and declining has expected loss $10p$. Acting is better exactly when

$$8<10p\quad\Longleftrightarrow\quad p>\tfrac45.$$

After ++ in the second scenario, every compatible probability satisfies $p\ge5/6>4/5$. So acting after two positive reports has the lower expected loss under every admissible joint law, although no unique event probability has been identified.

The same laws also imply that after +− or −+ the event probability lies in $[3/8,5/8]$, and after −− in $[0,1/6]$. Each range lies below the action threshold. Therefore one fixed rule—act after ++; decline otherwise—is strictly preferred, report by report, in every model in this second family.

This is stronger than merely saying that a particular worst-case optimization selects the rule. The preference comparison does not change as we move among the admissible probability laws. It still depends, of course, on having stipulated the two losses and the admissible family.

As an ex ante check, declining always has expected loss $10\times(1/2)=5$. The rule that acts only after ++ improves this loss by at least $1/10$ across the admissible laws, so its worst-case expected loss is 4.9. Using only either single source cannot improve on declining always under these losses: one positive report has posterior $3/4$, below the threshold $4/5$. The guaranteed improvement comes from the additional joint-report-frequency restriction and the fixed rule responsive to both reports, not from an assumption that the signals are independent.

Thus we have reached an initially surprising conclusion:

An exact probability can remain unknown while a particular decision comparison is completely settled under all admissible laws.

The reverse implication does not follow: unidentified probabilities can also span both sides of a decision threshold, leaving the comparison unsettled. Nor does agreement across models determine which action losses society ought to choose, whether the model family is evidentially credible, or whether the decision-maker must adopt a particular ambiguity-sensitive principle when actions are not uniformly comparable.

4. What the mathematical examples have—and have not—earned

The two examples expose three different questions that are easy to compress into one:

What follows mathematically if specified information is known? In our first scenario the stipulated information implies a sharp posterior set, not a point. In the second it implies a sharp posterior set and a determinate preference between the stated actions after each report profile.

What information could be observed? If two sources issue paired reports across an appropriately defined population of comparable event instances, the relative frequencies of the four report pairs may be estimated without learning the outcomes. Such data could help restrict their unconditional joint distribution. But both of our first-scenario worlds have that exact same distribution. To distinguish their event-conditioned overlap requires suitable outcome information or other substantive identifying assumptions.

A possible information history—but not one actually observed here. Imagine two laboratory alert systems, each evaluated separately on outcome-labeled cases. Suppose those evaluations gave adequate grounds for the prior and for each system’s state-conditional positive-report rates. The systems are then used together on new, comparable cases; their reports are correctly paired, but the new case outcomes are unavailable. Under further defensible assumptions, those later unlabeled observations could estimate the joint report-profile frequencies while leaving their association with outcomes unidentified. But the earlier evaluations must genuinely support transfer of the prior and individual signal characteristics to the new population; mere similarity of equipment is not enough. If the new paired cases also had reliable outcome labels, they could provide information that the hypothetical unlabeled record lacks. This imagined sequence shows how differently justified inputs might arise; ARCHIMEDES collected no such data.

What information is actually justified? Estimation requires a target population, valid pairing, a sampling or dependence model, attention to selection and missingness, and some reason to expect observed instances to inform the intended decision setting. Finite samples introduce uncertainty. Estimating source reliability or a prior event rate ordinarily requires outcome information or an alternative identifying method; using past measurements for future decisions requires defensible stability or transport assumptions. A statistical interval for an estimated joint-report frequency is not the same object as the set of posteriors compatible with an exactly stipulated population law.

In this work, no actual forecasting sample, calibrated population, tested stability assumption or empirically justified dependence bound has been established. The numbers are deliberately hypothetical. The mathematical claims are conditional: if the stipulated information describes the relevant world, then the stated inferences and decision comparisons follow. Correct reasoning cannot supply the missing evidence for its own premises.

This does not make the results empty. It identifies both their power and their limit. They show what a decision-maker could infer from certain justified inputs, and why other seemingly rich observations would not be enough. They do not establish that a real decision-maker has those inputs.

5. Where these ideas stand

Uncertain informational dependence and robust forecast aggregation are substantial existing research subjects. The present examples are small constructions within that terrain, not claims of newly discovered mathematical principles.

Levy and Razin study how uncertainty over bounded correlation structures changes the range of predictions considered by a decision-maker; their bounds are formulated using pointwise mutual information rather than the unconditional joint-report frequency stipulated here. De Oliveira, Ishii and Lin analyze robust decisions when individually known information sources have unknown correlations. Their binary-state, binary-action unrestricted-correlation benchmark supports relying on the best single source. Our second example is not a counterexample to that benchmark: it supplies additional joint-report-frequency information that restricts the correlation family. Guo, Li and Mu study aggregation of calibrated probabilistic forecasts, including the decision-relevant joint distribution of their reports and learning under specified feedback assumptions. Their calibration-based setup is not identical to stipulated binary state-conditional experiments, but it substantially precedes and overlaps the broader conceptual territory of our illustration.

The original research account also examined cases where conditional and ex ante worst-case decision procedures disagree. That phenomenon belongs to an established literature on ambiguity and dynamic consistency, including the rectangularity analysis of Epstein and Schneider. We have not needed that additional theory to prove the uniform action preference in Section 3, and claiming it as a fresh general discovery would be misleading.

What, then, can this explanation offer? Not priority. Its intended contribution is to put full unlabeled observational equivalence, sharp but nonunique probability inference, uniform decision comparison, and the justification of the inputs themselves into one transparent argument. Readers should be able to see exactly why two worlds remain observationally indistinguishable and exactly why a decision can sometimes be settled despite the remaining uncertainty.

If that sequence makes the distinctions clearer, the explanation has an intellectual purpose. If it merely restates familiar results without deepening understanding, its correctness alone would not justify publication.

6. What agreement actually adds

We began with two positive reports and an apparently straightforward question: how much more confident should we become?

There is no answer based only on the number of agreeing sources. Even known individual source characteristics and the complete unlabeled joint-report distribution can be compatible with different event probabilities after agreement.

We then changed the supplied joint-report information and introduced explicit consequences. A nonunique posterior still remained, but all compatible probabilities fell on the same side of a decision threshold. Under those assumptions, the uncertainty that defeated point identification did not defeat the action comparison.

Neither result travels to a real setting on its own. The relevant source characteristics, dependence restrictions and consequences must be justified for that setting. An accurate theorem is not evidence that its premises hold.

Agreement can supply useful evidence. The question is what the agreement adds given what the sources share, what we can actually establish about them, and what decision is at stake.

Mathematical examination: verifying the two examples

Technical section for readers who want to check the argument. The main explanation above stands without working through every formula; the same assumptions govern both parts.

This section supplies the complete admissible probability tables, proves both sharp posterior sets—including their continuous attainability—and checks the decision comparison. The tables use no more than four report-profile columns, so the two-world construction can remain readable as ordinary, separately captioned tables on a narrow screen without requiring an interactive mathematical interface.

A. All state-conditional pair laws with the stated individual signal characteristics

Write $a=P(++\mid E)$ and $b=P(++\mid\neg E)$. Because in event cases each signal has positive probability $3/4$, and in non-event cases each has positive probability $1/4$, every compatible conditional pair law has this form:

The entries are nonnegative and sum to one if and only if

$$\tfrac12\le a\le\tfrac34,\qquad 0\le b\le\tfrac14.$$

These are the sharp overlap (Fréchet) restrictions for each state. They do not assume independence of the signals, conditional or unconditional.

With $P(E)=1/2$, the unconditional double-positive frequency is

$$q=P(++)=\frac{a+b}{2}.$$

The unconditional marginal frequency of each source’s positive report is $1/2$, so the entire unconditional pair law must be

$$\big(P(++),P(+{-}),P({-}+),P({-}{-})\big) =\big(q,\tfrac12-q,\tfrac12-q,q\big).$$

Consequently, knowing all four unlabeled pair frequencies is equivalent, under these already stipulated marginal experiments, to knowing $q$. It does not determine $a$ or $b$ separately.

B. First scenario: complete sharp identified set

If $q=3/8$, then $a+b=3/4$. The constraints above permit every $a\in[1/2,3/4]$, paired with $b=3/4-a$. For ++, Bayes’ rule gives

$$P(E\mid++)=\frac{(1/2)a}{(1/2)(a+b)}=\frac{a}{a+b}=\frac{4a}{3}.$$

It follows that the sharp posterior set is $[2/3,1]$. Every real value within it is attained by a compatible probability law, because we may choose continuously varying $a\in[1/2,3/4]$ and set $b=3/4-a$; the nonnegative rows above then define normalized conditional distributions.

The two displayed worlds correspond to $(a,b)=(1/2,1/4)$ and $(3/4,0)$ respectively. For any real $t\in[0,1]$, their mixture has $a=(1-t)/2+3t/4$ and $b=(1-t)/4$, giving $P_t(E\mid++)=2/3+t/3$. The 160-instance endpoint tables have integer counts; this mixture argument, not enumeration of finitely many integer tables, proves that the continuum of intermediate posteriors is feasible. Every compatible law has the same unlabeled distribution $(3/8,1/8,1/8,3/8)$.

C. Second scenario: bounds for all four reports

If instead $q=3/10$, then $a+b=3/5$. The overlap restrictions give exactly

$$\tfrac12\le a\le\tfrac35,\qquad b=\tfrac35-a.$$

Bayes’ rule yields:

Every pair profile has positive unconditional probability in this scenario, so these conditionals are defined throughout the family. Every real $a\in[1/2,3/5]$, with $b=3/5-a$, defines a valid law; the resulting continuous ranges are therefore sharp. The integer counts that fit into a 160-instance illustration would show only a finite subset of these laws.

For the stipulated losses, action has loss 8 and decline has conditional expected loss $10p$. At ++ every admissible $p$ exceeds $4/5$; at all other profiles every admissible $p$ is below $4/5$. Hence the same fixed report policy uniquely minimizes expected loss for each compatible law.

For an ex ante calculation, declining always has loss 5. Switching to action only at ++ changes the expected utility by

$$10P(E,++)-8P(++)=5a-8(3/10)=5a-\tfrac{12}{5}.$$

The smallest gain over decline-all occurs at $a=1/2$ and equals $1/10$. Its worst-case expected utility is therefore $-5+1/10=-4.9$, or equivalently its worst-case loss is 4.9. For one source alone, positive and negative posteriors are $3/4$ and $1/4$; both favor declining at threshold $4/5$, giving loss 5.

This uniform comparison must not be confused with a general statement that maximin or any other decision rule is obligatory. Nor does an identified posterior set under exact $q$ provide a sampling confidence interval for estimating $q$.

Sources and standing

Selected scholarly sources (consulted for comparison, not asserted to exhaust prior art):

1. Gilat Levy and Ronny Razin, “Combining forecasts in the presence of ambiguity over correlation structures,” Journal of Economic Theory 199 (2022), article 105075. Journal article (DOI)
2. Henrique de Oliveira, Yuhta Ishii and Xiao Lin, “Robust Aggregation of Correlated Information,” The Review of Economic Studies (accepted manuscript published online 21 October 2026). Accepted manuscript (DOI) ; arXiv v3: arXiv v3
3. Xinxiang Guo, Yingkai Li and Yifen Mu, “Robust Aggregation of Calibrated Forecasts,” arXiv:2606.31020 (2026). arXiv record
4. Larry G. Epstein and Martin Schneider, “Recursive multiple-priors,” Journal of Economic Theory 113 (2003), 1–31. Journal article (DOI)

Research and editorial provenance. The mathematical constructions draw on ARCHIMEDES’s privately commissioned October 2026 investigation, carried out through separate AI Investigator A and B sessions under the Institutional Principal’s direction. A originated the symmetric posterior-identification construction; B supplied the decisive interior robust-decision example. Their subsequent critiques, reconstructions and corrections informed the reconciled A-006 v1.1 account, which B-08 explicitly confirmed. These were AI-generated research contributions and AI reciprocal examinations, not independent human mathematical specialist validation. The Institutional Principal selected and directed the undertaking, contributed substantive epistemic and editorial challenges, and approved the manuscript as the substantive source of this explanatory work. The manuscript was substantially developed with OpenAI ChatGPT assistance. Neither those facts nor institutional approval would establish novelty, empirical validation or specialist certification.

Editorial accountability and correction. This work is presented under ARCHIMEDES’s institutional responsibility, with its exact public edition and subsequent corrections subject to authorization by the Institutional Principal or a genuinely empowered actor. Institutional approval is not scientific certification or a personal claim to have derived its mathematics. Material corrections, if warranted, should identify the affected claim and edition and remain visibly connected to the original presentation. No public correspondence service or independently certified mathematical review is represented by this work.