3.1. Imprecise Probability

The use of probability ranges by the IPCC invokes what is sometimes known in the theoretical literature as imprecise probability. Central to much of this literature is the use of sets of probability functions to represent the epistemic state of an agent who cannot determine a unique probability for all events of interest. Informally we can think of this set as containing those probability functions that the decision maker regards as permissible to adopt given the information she holds.

To motivate the idea, recall Popper’s paradox of ideal evidence (Reference Popper1974, 407–8), which contrasts two cases in which we are asked to provide a probability for a coin landing heads. In the first, we know nothing about the coin; in the second, we have already observed 1,000 tosses and seen that it lands heads roughly half the time. Our epistemic state in the second case can reasonably be represented by a precise probability of one-half for the outcome of heads on the next toss. By contrast, the thought goes, the evidence available in the first case can justify only a set of probabilities—perhaps, indeed, the set of all possible probabilities. To adhere to a single number, even the “neutral” probability of one-half, would require a leap of faith from the decision maker, and it is hard to see why she should be forced to make this leap. Pragmatic considerations too suggest allowing imprecise probabilities. Given a choice of betting either in the first case or in the second, it seems natural that one might prefer betting in the second, but a Bayesian decision maker cannot have such preferences.Footnote ³

Bayesian decision insists that no matter the scarcity of the decision maker’s information, she must pick a single probability function to be used in all decisions. This is often called her “subjective” probability, and particularly in cases in which the available information (combined with the decision maker’s expertise and personal judgment) provides little guidance, the “subjective” element may be rather hefty indeed. This probability function determines, together with a utility function on consequences, an expected utility for each action available to the decision maker, and the theory enjoins her to choose the action with greatest expected utility.

Despite the severity of uncertainty faced in the climate domain, Bayesian decision theory has its adherents. John Broome, for instance, argues that in climate policy decision making, “The lack of firm probabilities is not a reason to give up expected value theory. You might despair and adopt some other way of coping with uncertainty. … That would be a mistake. Stick with expected value theory, since it is very well founded, and do your best with probabilities and values” (Broome Reference Broome2012, 129).

Paralleling the points made in the coin example above, critics of the Bayesian view argue that the decision maker may be unable to supply the required precise subjective probabilities and that any “filling in” of the gap between probability ranges and precise probabilities may prove too arbitrary to be a reasonable guide to decision. Policy makers may quite reasonably refuse to base a policy decision on a flimsy information base, especially when there is a lot at stake.

Imprecise probabilists, on the other hand, face the problem of spelling out how a decision maker with a set of probability functions should choose. Her problem can be put in the following way. Each probability function in her set determines, together with a utility function on consequences, an expected utility for each available action; but except on rare occasions when one action dominates all others in the sense that its expected utility is greatest relative to every admissible probability function, this does not provide a sufficient basis for choice. Were the decision maker to simply average the expected utilities associated with each action, her decisions would then be indistinguishable from those of a Bayesian. There are, however, other considerations that she can bring to bear on the problem that will lead her to act in a way that cannot be given a Bayesian rationalization. She may wish, for instance, to act cautiously, by giving more weight to the “down-side” risks (the possible negative consequences of an action) than the “up-side” chances or by preferring actions with a narrower spread of (expected) outcomes.

A much-discussed decision rule encoding such caution is the maximin expected utility rule (MMEU), which recommends picking the action with the greatest minimum expected utility relative to the set of probabilities that the decision maker is working with (Gilboa and Schmeidler Reference Gilboa and Schmeidler1989). To state the rule more formally, let $C=\left\{p_1,\dots ,p_n\right\}$ be a set of probability functions,Footnote ⁴ and for any p ∈ C and action f, let EU_p(f) be the expected utility of f computed from p. The rule then ascribes a value V to each action f in accordance with:

MMEU is simple to use but arguably too cautious, paying no attention at all to the full spread of possible expected utilities. This shortcoming is mitigated in some of the other rules for decision making that draw on imprecise probabilities, such as maximizing a weighted average of the minimum and maximum expected utility (often called the α-MEU rule) or that of the minimum and mean expected utility, where the averaging weights can be thought of as reflecting either the decision makers’ pessimism or their degree of caution (see, e.g., Ghirardato, Maccheroni, and Marinacci Reference Ghirardato, Maccheroni and Marinacci2004; Binmore Reference Binmore2008).

A question that all such rules must address is how to specify the set C of probabilities on which they are based. Where evidence is sparse, the Bayesian insistence on a single probability function seems too extreme; but if C contains all probabilities logically consistent with the evidence, then the decision maker is likely to end up with very wide probability intervals, which can in turn lead to overly cautious decision making. A natural thought is that C should determine probability intervals only so broad as to ensure the decision maker is confident that the “true” probabilities lie within them or that they contain all reasonable values (see, e.g., Gärdenfors and Sahlin Reference Gärdenfors and Sahlin1982). The decision maker may, for instance, wish to discard some implausible probability functions even though they are not, strictly speaking, contradicted by the evidence. Or if the sources of these probabilities are the opinions of others, the decision makers need not consider every opinion consistent with the evidence, but rather only those in which they have some confidence. But how confident need they be? We return to this question below after discussing the notion of confidence in more detail.

3.2. Confidence

The decision rules canvassed above can make use of the probability ranges found in IPCC reports, but not the confidence judgments that qualify them. According to the authors of the IPCC guidance notes, “A level of confidence provides a qualitative synthesis of an author team’s judgment about the validity of a finding; it integrates the evaluation of evidence and agreement in one metric” (Mastrandrea et al. Reference Mastrandrea, Mach, Plattner, Edenhofer, Stocker, Field, Ebi and Matschoss2011, 679). Let us address these two contributors to IPCC confidence in turn.

“Evaluation of evidence” depends on the amount, or weight, of the evidence relevant to the judgment in question. Suppose, for instance, that a decision maker is pressed to report a single number for the chance of heads on the next coin toss in each of the two cases described above, namely, where she knows nothing about the coin and where she has already observed many tosses. She may report one-half in both cases but is likely to have more confidence in that assessment in the case in which the judgment is based on abundant evidence (the previously observed tosses). The reason is that a larger body of evidence is likely to be reflected in a higher level of confidence in the judgments that are based on it.

The second contributor to confidence is “agreement.” To tweak the coin example, compare a situation in which a group of coin experts agrees that the probability of heads on the next toss is one-half with a second case in which the same group is evenly split between those that think the probability is zero and those that think it is one. Here too, a decision maker pressed to give a single number might say one-half in both cases, but it seems reasonable to have more confidence in the first case than in the second. Holding the amount of evidence fixed, greater agreement engenders greater confidence.

The two dimensions of IPCC confidence connect to largely distinct literatures. The evidence dimension connects with that on the weight of evidence behind a probability judgment and how this weight can be included in representations of uncertainty (Keynes Reference Keynes1921/1973). The agreement dimension connects with the literature on expert testimony and aggregation of probability functions (for a survey, see Genest and Zidek [Reference Genest and Zidek1986]). Models employing confidence weights on different possible probabilities are to be found in both literatures. In the first, the probabilities are interpreted as different probabilistic hypotheses and the weights as measures of the agent’s confidence in them. In the second, the probabilities are interpreted as the experts’ judgments and the weights as a measure of an agent’s confidence in the experts. So while weight of evidence and expert agreement are two distinct notions, they can be represented similarly and play analogous roles in determining judgments and guiding action. It is thus not unreasonable to proceed in the manner suggested by the IPCC and place both under a single notion of confidence.

What role should these second-order confidence weights play in decision making? To the extent that different probability judgments support different assessments of the expected benefit or utility of an action, one would expect the relative confidence (or lack of it) that a decision maker might have in the former will transfer to the latter. It is then reasonable, ceteris paribus, to favor actions with high expected benefit based on the probabilities in which one has the most confidence over actions whose case for being beneficial depends on probabilities in which one has less confidence.

One way to do this is to use the confidence weights over probability measures to weight the corresponding first-order expected utilities, determining what might be called the confidence-weighted expected utility (CWEU) of an action. Formally, let $C=\left\{p_1,\dots ,p_n\right\}$ be a set of probability functions and {α_i} the corresponding weights, normalized so that ${\sum }_i{\alpha }_i=1$. Then:

Here the weights effectively induce a second-order probability over C, and maximizing CWEU is equivalent to maximizing expected utility relative to the aggregated probability function obtained by averaging the elements of C using the second-order probabilities. But this seems unsatisfactory from a pragmatic point of view as it would preclude a decision maker displaying the sort of caution, or aversion to uncertainty, that we argued could be motivated in contexts like the coin example. Given a choice of betting on either one coin or the other, an agent following CWEU cannot prefer betting on the coin for which she has more evidence. But some degree of discrimination between high- and low-confidence situations does seem appropriate for important policy decisions.

Other decision models proposed in the economics literature allow for this kind of discrimination. The “smooth ambiguity” model of Klibanoff, Marinacci, and Mukerji (Reference Klibanoff, Marinacci and Mukerji2005) is a close variant of CWEU; it too uses second-order probability, but it allows for an aversion to wide spreads of expected utilities by valuing an action f in terms of the expectation (with respect to the second-order probability) of a transformation of the ${\mathrm{EU}}_{p_i}\left(f\right)$ rather than the expected utilities themselves. Formally (and ignoring technicalities due to integration rather than summation), the rule works as follows:

The term ɸ is a transformation of expected utilities capturing the decision maker’s attitudes to uncertainty (the decision maker displays aversion to uncertainty whenever ɸ is concave).

Other suggestions in this literature use general real-valued functions (rather than probabilities) at the second-order level and can be thought of as refinements of the MMEU model discussed in the previous section. Gärdenfors and Sahlin (Reference Gärdenfors and Sahlin1982), for instance, use the weights to determine the set of probability functions $C=\left\{p_1,\dots ,p_n\right\}$ by admitting only those that exceed some confidence threshold; they then apply MMEU. Maccheroni, Marinacci, and Rustichini (Reference Maccheroni, Marinacci and Rustichini2006) value each action as the minimum, across the set of probability functions C, of the sum of the action’s expected utility given p_i and the second-order weight given to p_i; and Chateauneuf and Faro (Reference Chateauneuf and Faro2009) take the minimum of CWEUs over probability functions with second-order weight beyond a certain absolute threshold.

From the perspective of this article, all these proposals suffer from a fundamental limitation. Application of these rules requires a cardinal measure of confidence to serve as the weights on probability measures. That is, the numbers matter; otherwise it would make no sense to multiply or add them as is done by these rules. Whenever such confidence numbers are available, these models are applicable. The IPCC, however, commits to providing only a qualitative, ordinal measure of confidence: it can say whether there is more confidence or less, in one probability judgment compared to another, but not how much more or less.Footnote ⁵ So the aforementioned models of decision require information of a nature different from what is furnished in IPCC reports.

IPCC practice is not unreasonable in this respect. Indeed if the decision maker has trouble forming precise first-order probabilities, why would he have any less trouble forming precise second-order confidence weights? Such considerations plead in favor of a more parsimonious representation of confidence, in line with the ordinal ranking used by the IPCC. To connect this to decision making, however, a model is required that can work with ordinal confidence assessments without requiring cardinality. We have found only one such model in the literature, namely, that proposed by Hill (Reference Hill2013), which we now present in some detail.

3.3. Hill’s Decision Model

Hill’s central insight is that the probability judgments we adopt can reasonably vary with what is at stake in a decision. Consider, for instance, the schema for decision problems represented by table 1, in which the option Act has a low probability of a very bad outcome (utility x ≪ 0) and a high probability of a good outcome (utility y > 0). The table could represent a high-stakes decision, such as whether to build a nuclear plant near a town when there is a small, imprecise probability of an accident with catastrophic consequences. But it could equally well represent a low-stakes situation in which the agent is deciding whether to get on the bus without a ticket when there is a small imprecise probability of being caught.

Table 1. Small Chance of a Bad Outcome

	Probability < .01	Probability ≥ .99
Act	x ≪ 0	y > 0
Don’t Act	0	0

Standard decision rules, such as expected utility maximization or maximin EU, are invariant with respect to the scaling of the utility function. Consequently, they cannot treat a high-stakes and a low-stakes decision problem differently if outcomes in the former are simply a “magnification” of those in the latter, for instance, if the nuclear accident was 100,000 times worse than being fined and the benefits of nuclear energy 100,000 times better than those of traveling for free. But it does not seem at all unreasonable to act more cautiously in high-stakes situations than in low-stakes ones.Footnote ⁶

To accommodate this intuition, Hill allows for the set of probability measures on which the decision is based to be shaped by how much is at stake in the decision. This stakes sensitivity is mediated by confidence: each decision situation will determine an appropriate confidence level for decision making based on what is at stake in that decision. When the stakes are low, the decision maker may not need to have a great deal of confidence in a probability assessment in order to base her decision on it. When the stakes are high, however, the decision maker will need a correspondingly high degree of confidence.

To formulate such a confidence-based decision rule, Hill’s model draws on a purely ordinal notion of confidence, requiring only that the set of probability measures forms a nested family of sets centered on the measures that represent the decision maker’s best-estimate probabilities. This structure is illustrated in figure 1, where each circle is a set of probability measures. The innermost set is assigned the lowest confidence level and each superset a higher confidence level than the one it encloses. These confidence assignments can be thought of as expressing the decision maker’s confidence that the “true” probability measure is contained in that set. Probability statements that hold for every measure in a superset enjoy greater confidence because the decision maker is more confident that the “true” measure endorses the statement.

Figure 1. How much is at stake in a decision determines the set of probability measures that the decision rule can “see.”

For any given decision, the stakes determine the requisite level of confidence, which in turn determines the set of probability measures taken as the basis for choice: intuitively, the smallest set that enjoys the required level of confidence.Footnote ⁷ Once the set of measures has been picked out in this way, the decision maker can make use of one of the rules for decision making under ambiguity discussed earlier, such as MMEU or α-MEU. In the special case in which the set picked out contains just one measure, ordinary expected utility maximization is applicable.

As should be evident, what Hill provides is a schema for confidence-based decisions rather than a specific model. Different notions of stakes will determine different confidence levels. And there is the question of which decision rule to apply in the final step. But these details are less important than the fact that the schema can incorporate roughly the kind of information that the IPCC provides. Spelling this out is our next task.

4.1. A Model of Confidence

Now we develop more formally the notion of confidence required to link IPCC communications to the model of decision making just introduced. As is standard, actions or policies will be modeled as functions from states of the world to outcomes, where outcomes are understood to pick out features of the world that the decision maker seeks to promote or inhibit. States are features of the world that, jointly with the actions, determine what outcome will eventuate. What counts as an outcome or a state depends on the context: when a decision concerns how to prepare for drought, for instance, mean temperatures may serve as states, while in the context of climate change mitigation, they may serve as outcomes.

Central to our model is a distinction between two types of propositions that are the objects of different kinds of uncertainty: propositions concerning “ordinary” events, such as global mean surface temperature exceeding 21°C in 2050, and probability propositions such as there being a 50% chance that temperature will exceed 21°C in 2050. Intuitively, the probability propositions represent possible judgments yielded by scientific models or by experts and, hence, are propositions in which the decision maker can have more or less confidence.

Let $S=\left\{s_1,s_2,\dots ,s_n\right\}$ be a set of n states of the world and $\Omega =\left\{A,B,C,\dots \right\}$ be a Boolean algebra of sets of such states, called events or factual propositions. Let $\Pi =\left\{p_i\right\}$ be the set of all possible probability functions on Ω and Δ(Π) be the set of all subsets of Π.Footnote ⁸ Members of Δ(Π) play a dual role: both as the possible imprecise belief states of the agent and as probability propositions, that is, propositions about the probability of truth of the factual propositions in Ω. For instance, if X is the proposition that it will rain tomorrow, then the proposition that the probability of X is between one-half and three-quarters is given by the set of probability distributions p such that $.5\ge p\left(X\right)\ge .75$. So the probabilistic statements that are qualified by confidence assessments in the IPCC examples given in section 2 correspond to elements of Δ(Π).

To represent the confidence assessments appearing in IPCC reports, we introduce a weak preorder, ⊵, on Δ(Π), that is, a reflexive and transitive binary relation on sets of probability measures. Intuitively, ⊵ captures the relative confidence that a group of IPCC authors has in the various probability propositions about the state of the world, with $P_1⊵P_2$ meaning that they are at least as confident in the probability proposition expressed by P ₁ as that expressed by P ₂, as would be the case if they gave P ₂ a medium confidence assessment and P ₁ high confidence. In practice, a confidence relation in line with IPCC findings will have up to five levels, corresponding to the five qualifiers in their confidence language (sec. 2). It is reasonable to assume that ⊵ is nontrivial (that $\Pi ⊵\mathrm{\varnothing }$) and monotonic with respect to logical implication between probability propositions (i.e., that $P_1⊵P_2$ whenever $P_1\subseteq P_2$), because one should have more confidence in less precise propositions.

We do not, however, assume that ⊵ is complete. Completeness can fail to hold for two different reasons. First, there may be issues represented in the state space about which the agent makes no confidence judgments. For example, the IPCC does not assess the chance of rain in London next week. Second, the agent may make a confidence judgment about a certain probability proposition but no judgments about other probability propositions concerning the same issue. For example, the IPCC may report medium confidence that a certain occurrence is likely (66%–100% chance) but say nothing about how confident one should be that the same occurrence is more likely than not (50%–100% chance).

Given cardinal confidence assessments of probability propositions, it is always possible to extract an underlying order ⊵ summarizing the ordinal information. Hence this setup can also apply when cardinal information is available. But an order ⊵ does not determine any cardinal confidence assessments. There will be a large family of such assessments, each consistent with ⊵, which will not agree on any question concerning the numbers. The order thus drastically underspecifies the cardinal confidence measures required to apply the models discussed in section 3.2. By contrast, the ordinal information is precisely of the sort needed by Hill’s account.

Hill’s model of confidence effectively consists of a chain of probability propositions, {L ₀, L ₁, … , L_n} with $L_i\subset L_{i+1}$. The probability proposition L ₀ is the most precise probability proposition that the agent accepts; it summarizes her beliefs in the sense that every probability proposition that she accepts (with sufficient confidence) is implied by every probability function in L ₀. The other L_i are progressively less precise probability propositions held with progressively greater confidence. The chain {L ₀, L ₁, … , L_n} is equivalent to what we shall call a confidence partition: an ordered partition of the space Π of probability measures. Any nested family of probabilities {L ₀, … , L_n} induces a confidence partition {M ₀, … , M_n}, where $L_0=M_0$ and $M_i=L_i-L_{i-1}$. The element M_i (for i > 0) contains those probability measures that the agent rules out as contenders for the “true” measures at the confidence level i − 1 but not at the higher confidence level i. Inversely, any confidence partition $\pi =\left\{M_0,\dots ,M_n\right\}$ induces a nested family of sets of probability measures {L ₀, … , L_n} such that $L_0=M_0$ and $L_i=M_0\cup \cdots \cup M_i$. A sample confidence partition and corresponding nested family are given in figure 2 for the issue of the weather tomorrow. The agent’s best-estimate probability range for rain, M ₀, is the proposition that the probability of rain tomorrow is between .4 and .6, M ₁ that it is either between .3 and .4 or between .6 and .7, M ₂ that it is between .1 and .3 or .7 and .9, and M ₃ the remaining probabilities. As is generally the case with the Hill model, the agent is represented by this figure as having made confidence judgments regarding any pair of these probability propositions for rain.

Figure 2. Confidence partition for the proposition that it will rain tomorrow. Bracketed intervals show probabilities given by the probability propositions. Nested sets L ₀, L ₁, and L ₃ can be constructed from the partition M ₀, …, M ₃. The overall ordering is $L_2⊵L_1⊵L_0\equiv M_0⊵M_1⊵M_2⊵M_3$.

Which chain of probability propositions (or, equivalently, which confidence partition) does an IPCC-style assessment recommend for decision purposes? The probability measures in the lowest element of the partition are those that satisfy all of the probability propositions, on a given issue, that are affirmed by the IPCC (with any confidence level). The additional measures to be considered as contenders at the next level up, M ₁, need to satisfy only those probability propositions affirmed by the IPCC with this next level up (or higher) level of confidence. Additional probability measures collected in M ₂ should satisfy the IPCC probability propositions that are on or above the next confidence level up, and so on. Only confidence partitions satisfying these conditions faithfully capture the IPCC confidence and probability assessments.

Note that this protocol picks a unique confidence partition only in the case in which the confidence relation ⊵ is complete. Otherwise, several confidence partitions will be consistent with the confidence relation; as noted above, this will generally be the case for IPCC assessments. Since each confidence partition corresponds to a unique complete confidence relation, the use of a particular partition essentially amounts to “filling in” confidence assessments that were not provided. How best to “fill in” is an important question that is beyond the scope of this investigation.Footnote ⁹ (In the final section below, we approach the issue from another perspective: how IPCC authors might reduce the need to perform this completion by providing more confidence assessments.)

Next we illustrate the confidence partition concept by applying it to a concrete example from the IPCC’s fifth assessment report (AR5).

4.2. An Example

Equilibrium climate sensitivity (ECS) is often used as a single-number proxy for the overall behavior of the climate system in response to increasing greenhouse gas concentrations in the atmosphere. The greater the value, the greater the tendency to warm in response to greenhouse gases. The quantity is defined by a hypothetical global experiment: start with the preindustrial atmosphere and instantaneously double the concentration of carbon dioxide; now sit back and allow the system to reach its new equilibrium (this would take hundreds to thousands of years). ECS is the difference between the annual global mean surface temperature of the preindustrial world and that of the new equilibrium world. In short, it answers the question: How much does the world warm if we double CO₂?

The most recent IPCC findings on ECS draw on several chapters of the Working Group I contribution to the AR5. Estimates of ECS are based on statistical analyses of the warming observed so far, similar analyses using simple to intermediate complexity climate models, reconstructions of climate change in the distant past (paleoclimate), as well as the behavior of the most complex, supercomputer-driven climate models used in the last two phases of the colossal Coupled Model Intercomparison Project (CMIP3 and CMIP5). An expert author team reviewed all of this research, weighing its strengths, weaknesses, and uncertainties, and came to the following collective judgments. With high confidence, ECS is likely in the range 1.5°C–4.5°C and extremely unlikely less than 1°C. With medium confidence, it is very unlikely greater than 6°C (Stocker et al. Reference Stocker, Stocker, Qin, Plattner, Tignor, Allen, Boschung, Nauels, Xia, Bex and Midgley2013, 81).

In light of the confidence model discussed above, reports of this kind can be understood in terms of a confidence partition over probability density functions (pdfs). Beginning from all possible pdfs on the real line—each one expressing a (precise) probability claim about ECS—think of what the author team is doing, as they evaluate and debate the evidence, as sorting those pdfs into a partition $\pi =\left\{M_0,\dots {,M}_n\right\}$. The findings cited above then communicate aspects of this confidence partition. To illustrate, we present a toy partition that exemplifies the IPCC’s findings on ECS.

Suppose the confidence partition has four elements $\pi =\left\{M_0,\dots ,M_3\right\}$. For concreteness, we suppose that each element contains only lognormal distributions.Footnote ¹⁰ Figure 3 displays the pdfs in the first two elements of the partition. The solid lines are the functions from M ₀; collectively, these indicate what the IPCC’s experts regard as a plausible range of probabilities for ECS in light of the available evidence. The dashed lines are the pdfs in M ₁, which collectively represent a second tier of plausibility. The element M ₂ is another step down from there, and M ₃ is the bottom of the barrel: all of the pdfs more or less ruled out by the body of research that the experts evaluated (M ₂ and M ₃ are not represented in fig. 3).

Figure 3. Illustration of a confidence partition consistent with IPCC findings on equilibrium climate sensitivity. The hatched area corresponds to the finding that ECS is very unlikely greater than 6°C (medium confidence).

Recall that the partition π generates a nested family of subsets {L ₁, … , L_n}, where L_i is the union of P ₀ through P_i and each L is associated with a level of confidence. Here we are concerned mainly with $L_0=M_0$ and $L_1=M_0\cup M_1$, and we suppose in this case that L ₀ corresponds to medium confidence and L ₁ to high confidence. To see how an IPCC-style finding follows from the confidence partition, consider what our partition says about ECS values above 6°. If we restrict attention to L ₀, there are only two pdfs to examine; one assigns (nearly) zero probability to values above 6° while the other assigns just under .1 probability (the hatched area in fig. 3). In the IPCC’s calibrated language, the probability range 0–.1 is called very unlikely; thus the finding that ECS is very unlikely greater than 6°C (medium confidence).

The IPCC’s two other findings on ECS are reflected in our partition as follows. Reporting with high confidence means broadening our view from L ₀ to L ₁, taking the M ₁ pdfs into account in addition to those in M ₀. The interval 1.5–4.5 is indicated in figure 3 in gray. The smallest probability given to that interval by any of the functions pictured is a little more than .6, and the highest probability is nearly one, an interval that corresponds roughly with the meaning of likely (.66–1). Regarding ECS values below one, several pdfs give that region zero probability, while the most left leaning of them gives it .05. The range 0–.05 is called extremely unlikely. Thus we have that ECS is likely in the range 1.5°C–4.5°C and extremely unlikely less than 1°C (both with high confidence).

The three findings discussed above are far from the only ones that follow from the example partition. To report again on ECS values above 6°C, only now with high confidence rather than medium, the probability interval should be expanded from 0–.1 to 0–.2 (.2 being the area to the right of 6° under the fattest-tailed of the M ₁ pdfs). Or to report on values below 1°C with medium confidence rather than high, the probability interval should be shrunk from 0–.5 to 0–.01 (exceptionally unlikely). The confidence partition determines an imprecise probability at medium confidence, and another at high confidence, for any interval of values for ECS.

It should be emphasized that these additional findings do not necessarily follow from the three that the IPCC in fact published. They follow from this particular confidence partition, which is constrained—though not fully determined—by the IPCC’s published findings.Footnote ¹¹ Asking what could be reported about ECS at very high confidence further highlights the limits of what the IPCC has conveyed. Suppose the set L ₂ corresponds to very high confidence. As the IPCC has said nothing with very high confidence, we have no information about the pdfs that should go into M ₂, and thus L ₂, so we have no indication of how much the reported probability ranges should be expanded in order to claim very high confidence. The reason may be that in the confidence partition representation of the experts’ group beliefs, M ₂ is a sprawling menagerie of pdfs. In this case, probabilities at the very high confidence level would be so imprecise as to appear uninformative. On the other hand, it may sometimes be of interest to policy makers just how much (or how little) can be said at the very high confidence level.