1. Introduction

The property of stochastic (or statistical or probabilistic) independence occupies a rather special place in the mathematical theory of probability. It does not belong to the properties that this theory singles out to define a probability measure axiomatically. Indeed, its familiar definitions by the multiplication rule, or the equality of conditional with unconditional probability, do not enter the Kolmogorov axiomatization of a probability measure. Rather, they capture properties of given events (and more generally, of given partitions or random variables) for a given probability measure and can thus be adopted only to model particular situations. At the same time, probability theory obviously makes extensive use of independence assumptions, as evidenced by the laws of large numbers, many theorems on stochastic processes, and some core results of statistical theory. For Kolmogorov himself, this property occupies a “central position in the theory of probability” (Reference Kolmogorov1933, 8), making it significantly different from the theory of positive measures. One would thus expect all theories of the foundations of probability to pay careful attention to stochastic independence (SI), but this is not the case. In this article, I investigate Bayesian decision theory, one of the most influential among these theories and at the same time a curious example of neglect of this major property.

As is well known, Bayesian decision theorists have developed a brand of subjective interpretation for the probability calculus. They claim that an agent’s uncertain beliefs should be represented by a probability measure and ground their claim on a pragmatic argument. They show formally that if the agents’ preferences over uncertain prospects—typically but not exclusively over monetary bets—obey certain requirements of practical rationality, these agents’ beliefs should conform to the axiomatic definition of a probability measure. Bayesian decision theorists typically stop their work when they have completed this demonstration. By not explaining how it can be extended to recover the definition of SI, they are open to the objection that they have not yet fully connected the probability calculus with rational belief. That is, they can be criticized for having handled only the basics of this calculus instead of its actual working.

More technically, Bayesian decision theory offers a representation theorem for preferences over uncertain prospects that involves two sets of quantities, utilities (over the consequences of prospects) and probabilities (over the uncertain events), these two items being combined by the familiar rule of expected utility. After Ramsey’s and de Finetti’s sketches, this strategy was implemented in full axiomatic detail by Savage (Reference Savage1972). At one point, Savage (Reference Savage1972, 44) extends the representation theorem to obtain a posterior probability measure, that is, one that represents beliefs after a partial resolution of uncertainty, and proves that this posterior obeys Bayes’s rule of revision; properly speaking, the “Bayesian” label of the school becomes fully justified only at this stage. This is also where Savage stops. As I report below, however, he acknowledges that a treatment of SI should have come next, but his admission of unfinished business was generally lost to later Bayesian decision theorists. The rare exceptions will be discussed below.

What is missing from Savage’s axiom system and its variants is a condition put in the preference language of these systems that would account for SI specifically. The current article formulates one such condition. I do not use Savage’s system but a more accessible one, and it is another contribution of the article to introduce this alternative system. The desired condition turns out to be highly simple and intuitive. I set up a framework in which there are two distinct sources of uncertainty S and T, and states of nature thus have the form of two-component vectors $(s,t)\in S\times T$ . In this framework, the condition relates to the agent’s conditional preferences and stipulates that those defined conditionally on s should be the same for all s in S, and those defined conditionally on t should be the same for all t in T. What this effectively means is that if the agent knows t but not s, or knows s but not t, then this partial knowledge cannot improve the decisions made under the residual uncertainty. This recovers in axiomatic preference terms one of the standard informal justifications of SI: if the occurrence of an event A carries no information on the occurrence of B, and vice versa, then the two events should be declared to be independent.

An alternative informal justification, which is also common, goes as follows: if the occurrence of an event A does not influence the occurrence of B, and vice versa, then the two events should be declared to be independent. The two lines are semantically distinct but easily get mixed up in probability texts and even some works in the philosophy of probability. When Bayesian decision theory is extended so as to encompass SI, there is no danger of confusion since the information carried by events matters only if it enters the agent’s decision process and, hence, is subjective in character. Objective connections holding between events, for instance, causal connections, play a role only if the agent considers them relevant. Thus, one merit of the extended Bayesian theory is to give the informational reading of SI a foundation that clearly sets it apart from other possible readings. Apart from this semantic contribution, the theory casts some light on the reflective discussion of that property in the philosophy of probability. In particular, it has something to say on the symmetry of the multiplicative definition—A’s independence of B implying B’s independence of A—a property that this literature has sometimes called into question.

I offer two representation theorems in succession. Both are adapted to the multiple uncertainty framework sketched above, and both deliver an expected utility formula in which the subjective probability is multiplicative in the values of two sources and, hence, satisfies SI. The system of theorem 1 has very few axioms, which deliver the two desired outputs jointly. This condensed treatment is both an advantage and a disadvantage. To derive a subjective expected utility formula and the multiplicative property in two successive logical steps helps one better understand how each of the preference axioms contributes to the conclusions. The more advanced theorem 2 is devised for this purpose.

The rest of this article is organized as follows. Section 2 introduces the twofold uncertainty framework and the preference axioms for theorem 1 via a motivating example. Section 3 states this result formally, and section 4 does the same for theorem 2. Section 5 is devoted to comparisons within Bayesian decision theory, including the earlier sketches of a preference analysis of SI. Section 6 draws connections with the philosophy of probability. Three appendixes collect the more specialized material. Appendix A gives proof details on the two theorems. Appendix B generalizes theorem 1 by allowing for zero-probability events. Appendix C pursues some of the comparisons of section 5 in more detail.

2. A Motivating Example

Given any probability space (, P), two events $A,B\in A$ are said to be stochastically independent if

$P(A\cap B)=P\left(A\right)\cdot P\left(B\right)\text{.}$

Building on this elementary definition, probability theory also defines what it means for sets of events, partitions, or random variables to be stochastically independent. Here I will approach SI by specifying that the state set Ω is a product set, a standard move in the theory when it comes to working with this property (see, e.g., Halmos Reference Halmos1974, 191–92). The simple example of this section illustrates this framework and the main decision-theoretic concepts of the article.

Suppose that a corn producer must decide how much land to farm while not knowing what the climatic conditions and the demand for corn will be at the time of the harvest; suppose also this producer evaluates each farming policy in terms of monetary proceeds and no other criterion. Following the basics of decision theory, we can reexpress this example symbolically as follows. There is a set of states of the world, which takes the form of a product set $\Omega =S\times T$ , where S represents the set of unknown climatic conditions and T the set of unknown values for demand. There is a set of consequences, which we take to be the set of real numbers ℝ to represent monetary proceeds. There is a set of uncertain prospects, that is, mappings from the states of the world to consequences, each of which represents a farming policy, which we take to be the set ${ℝ}^{S\times T}$ of all real functions on Ω. Finally, there is a binary relation ≿ on the last set of prospects to capture the producer’s preferences among cultivation policies.

Now suppose that this preference relation obeys the expected utility (EU) rule; that is, there exists a probability function π on $\Omega =S\times T$ and a utility function u on ℝ such that for all uncertain prospects X, Y,

$X\succsim Y\text{ if and only if (iff) }\sum _{s,t}\pi (s,t)u\left(X\right(s,t\left)\right)\ge \sum _{s,t}\pi (s,t)u\left(Y\right(s,t\left)\right)\text{;}$

and suppose moreover that π satisfies the SI property with respect to S and T; that is,

$\pi (s,t)=p\left(s\right)q\left(t\right)\text{, }$

where p and q are probability functions on S and T respectively. This equation, also written as $\pi =p\otimes q$ , determines p and q uniquely; these are the marginals of π on S and T, respectively. The EU rule with an axiomatically derived subjective probability (what will be referred to as subjective expected utility, SEU) and the SI property of this subjective probability are the intended conclusions of the theorems of this article. In the current section, I reason heuristically, working backward from them to a set of preference conditions that could be proposed as axioms.

Assume for simplicity that $S=\left\{s_1,s_2\right\}$ and $T=\left\{t_1,t_2\right\}$ . Then, the probabilities are given as follows:

And an uncertain prospect X is represented by the following table, which gives the consequences of this prospect in each state:

The EU formula for ≿, that is,

$V\left(X\right)=p_{s_1}{q}_{t_1}u\left(x_{11}\right)+p_{s_1}{q}_{t_2}u\left(x_{12}\right)+p_{s_2}{q}_{t_1}u\left(x_{21}\right)+p_{s_2}{q}_{t_2}u\left(x_{22}\right),$

can be restated either as

(1)$\begin{array}{}& V\left(X\right)=p_{s_1}\left[q_{t_1}u\left(x_{11}\right)+q_{t_2}u\left(x_{12}\right)\right]+p_{s_2}\left[q_{t_1}u\left(x_{21}\right)+q_{t_2}u\left(x_{22}\right)\right]\end{array}$

or as

(2)$\begin{array}{}& V\left(X\right)=q_{t_1}\left[p_{s_1}u\left(x_{11}\right)+p_{s_2}u\left(x_{21}\right)\right]+q_{t_2}\left[p_{s_1}u\left(x_{12}\right)+p_{s_2}u\left(x_{22}\right)\right]\text{.}\end{array}$

Observe that the bracketed sums in (1) are numerical representations for conditional preferences on the possible values of s, and those in (2) are numerical representations for conditional preferences on the possible values of t. Thus, the equations entail that (i) these conditional preferences are orderings. Since the same functional form $q_{t_1}u(\cdot )+q_{t_2}u(\cdot )$ appears in the two bracketed sums of (1), and similarly the same functional form $p_{s_1}u(\cdot )+p_{s_2}u(\cdot )$ appears in the two bracketed sums of (2), the equations also entail that (ii) these conditional preferences are the same for different s and the same for different t. Finally, from the same equations, if the conditional orderings for both s ₁ and s ₂, or the conditional orderings for both t ₁ and t ₂, agree to rank prospect X above prospect Y, then the overall preference ≿ ranks X above Y. Thus, it also holds that (iii) preferences over prospects are increasing with respect to either family of conditional preferences.

I have stated i, ii, and iii in pure preference language, thus abstracting entirely from the specifics of a numerical representation. In theorem 1 below, I assume no more than i, ii, and iii, plus some background conditions, and derive a SEU rule with the SI property. Given the definition of a conditional, which is restated below, it is actually possible to reduce iii to i and obtain an even more condensed system. One may wonder how such apparently weak conditions can do the mathematical work. One key point of the formal argument is that the weak ordering property, when applied to both s and t as in i, makes it possible to decompose the preference ≿ not only in terms of s- and t-conditionals but also in terms of conditionals based on any other partitions of Ω and, from there, to get an additive representation for ≿. The other key point is that the invariance property, when applied to both s and t as in ii, permits not only turning this additive representation into an EU formula but also giving the probabilities in this formula the multiplicative form.Footnote ¹

3. A First Representation Theorem Involving Subjective Expected Utility and Stochastic Independence

Formally, there are two variables of interest, $s\in S$ and $t\in T$ , and a state of the world is any pair $(s,t)\in \Omega =S\times T$ . By assumption, S and T are finite with cardinalities |S|, $\left|T\right|\ge 2$ . We keep the same number of sources of uncertainty as in the motivating example, but this is only for mathematical simplicity. The next section will suggest how a larger number can be handled. We take the set of consequences to be ℝ and the set of prospects to be all mappings $S\times T\to ℝ$ , which is identified with ${ℝ}^{S\times T}$ .Footnote ² The sets of all probability functions on S, T and $S\times T$ are denoted by Δ_S, Δ_T and ${\Delta }_{S\times T}$ , respectively.

It is convenient to represent prospects X as $S\times T$ matrices, with each s standing for a row and each t standing for a column. We will thus write $X={\left[x_s^t\right]}_{s\in S}^{t\in T}$ , but sometimes also $X=(x_1,\dots ,x_{\left|S\right|})$ , where each component is a row vector $x_s\in {ℝ}^T$ , or $X=(x^1,\dots ,x^{\left|T\right|})$ , where each component is a column vector $x^t\in {ℝ}^S$ .

We endow the agent with an ex ante preference relation ≿ to compare prospects and specifically assume that

Hence, ≿ is representable by a continuous utility function U(X).Footnote ³ The agent’s other preference relations are obtained from ≿ as conditionals. There are three families of conditionals of interest, that is, ${\left\{{\succsim }_s\right\}}_{s\in S}$ , ${\left\{{\succsim }_t\right\}}_{t\in T}$ , and ${\left\{{\succsim }_{\mathrm{st}}\right\}}_{s\in S,t\in T}$ . The last family represents ex post preferences, and the first two represent interim preferences, since each relation in these families depends on fixing one variable and letting the other vary, which amounts to resolving only part of the uncertainty.

We now formally define how conditionals are defined from the master relation ≿.Footnote ⁴ Take any subset of states $I\subseteq S\times T$ . The conditional of ≿ on I is the relation ≿_I on ℝ^I defined by the property that for all $x_I,y_I\in {ℝ}^I$ , $x_I{\succsim }_I{y}_I$ iff $X\succsim Y$ for some $X,Y\in {ℝ}^{S\times T}$ such that X is x_I on I, Y is y_I on I, and X and Y are equal outside I. Taking $I=\left\{s\right\}\times T$ , $I=S\times \left\{t\right\}$ , and $I=\left\{(s,t)\right\}$ in this definition, we obtain the three families of conditionals mentioned above.Footnote ⁵

In terms of the matrix representation of prospects, the definitions of ≿_s and ≿_t read as follows:

• • for all $x_s,y_s\in {ℝ}^T$ , $x_s{\succsim }_s{y}_s$ iff $X\succsim Y$ for some $X,Y\in {ℝ}^{S\times T}$ such that x_s is the s-row of X, y_s is the s-row of Y, and X and Y are equal outside their s-row,

• • for all $x^t,y^t\in {ℝ}^S$ , $x^t{\succsim }_t{y}^t$ iff $X\succsim Y$ for some $X,Y\in {ℝ}^{S\times T}$ such that x^t is the t-column of X, y^t is the t-column of Y, and X and Y are equal outside their t-column.

Importantly, the definition of conditionals does not by itself make them weak orderings. By a well-known fact of decision theory, ≿_I is a weak ordering iff the choice of X, Y in the definition of ≿_I is immaterial, or more precisely, iff

$X\succsim Y\iff X^{\prime }\succsim Y^{\prime }\text{,}$

for all X′, Y′ that also satisfy the condition stated for X, Y in this definition. In this case, the quantifier “for some” in the definition can be replaced by “for all.” When this holds, ≿ is said to be separable in I. By another well-known fact, separability in I is equivalent to the property that ≿ is monotonically increasing with the family of conditionals {≿_i, $i\in I$ }. We formally illustrate this property on the example of {≿_s, $s\in S$ }. Then, if ≿ is separable in each s ∈ S (or equivalently, each ≿_s is a weak ordering), the following statement holds: for all $X,Y\in {ℝ}^{S\times T}$ , if $x_s{\succsim }_s{y}_s$ for all s, then $X\succsim Y$ , and if moreover $x_s{\succ }_s{y}_s$ for some s, then $X\succ Y$ .Footnote ⁶

Combining the two basic facts just said, we see that the monotonicity condition iii of the last section actually follows from the weak ordering condition i. This will permit reducing the assumptions of the theorem to a bare minimum.

Since conditionals ≿_st compare real numbers, it makes sense to identify them with the natural order of these numbers. This amounts to assuming that, given any realized state (s,t), they are desirable quantities, be they monetary values, as in the producer example, or something else. Thus, we also assume that

Since this equivalence turns the ≿_st into weak orderings, ≿ is increasing with each of them and, hence, also with each entry $x_s^t$ of X.

Let us now say that the conditionals ≿_s (respectively ≿_t) are an invariant family if ${\succsim }_s={\succsim }_{s^{\prime }}$ for all $s,s^{\prime }\in S$ (respectively ${\succsim }_t={\succsim }_{t^{\prime }}$ for all $t,t^{\prime }\in T$ ). These two requirements capture condition ii of last section. None is needed for the conditionals ≿_st, since they are identical relations by construction.

We are now ready for the first representation theorem.

The conclusions state both that the ex ante preference relation can be represented by a SEU formula and that the two sources of uncertainty satisfy SI. Thus, this theorem extends Bayesian decision theory up to the point that, from the argument made in the introduction, it ought to have reached. The conclusion that p and q have only positive values is restrictive, but it can be relaxed by complexifying the assumptions. I pursue this technical line in appendix B.

4. A Second Representation Theorem Involving Subjective Expected Utility and Stochastic Independence

In theorem 1, strong results follow from a short list of assumptions, undoubtedly a feature of mathematical elegance but also a cause for conceptual dissatisfaction. Would it not be better to expand on the assumptions and separate those that are responsible for the existence of the SEU representation from those that account for the SI property occurring in this representation? Such a disentangling would be the more justified since SI is an optional property of a probability measure and, hence, in need of a preference condition that should be detachable from those underlying the existence of this measure. However, the assumptions of theorem 1 cannot be so divided, as the following argument shows. By taking the ≿_s and ≿_t to be merely orderings, not invariant orderings, one would get an additively separable representation that does not distinguish between the utility and probability components of the added terms, unlike EU representations. By taking only one of the two families to satisfy the ordering and invariance assumptions, one would get a representation that is separable in that family alone and would be even more remote from the SEU representation.Footnote ⁷

As it turns out, however, we can obtain a relevant partitioning of assumptions if we enrich the decision-theoretic framework beyond the current two-dimensional stage. Let us suppose that the agent pays attention not only to the uncertainty dimensions s and t of the final consequences but also to a third dimension i, so that these consequences are now represented by real numbers $x_{\mathrm{st}}^i$ . The added dimension can be thought of in several ways, like time, space, or an omitted dimension of uncertainty. Each of these concrete suggestions can fit the motivating example: the added dimension would indicate when, where, or under what further unknown circumstance the monetary proceeds of a farming policy accrue to the producer. I return to the interpretation of the added dimension after stating theorem 2.

By assumption, i takes its values in a finite set I with cardinality $\left|I\right|\ge 2$ . The set of prospects is now the set of all mappings $S\times T\times I\to ℝ$ , that is, ${ℝ}^{S\times T\times I}$ . Prospects can be represented as three-dimensional arrays

$𝕏={\left[x_{\mathrm{st}}^i\right]}_{s\in S,t\in T}^{i\in I}\in {ℝ}^{S\times T\times I}$

or as vectors

$𝕏=(X_1,\dots ,X_{\left|S\right|}),𝕏=(X_1,\dots ,X_{\left|T\right|}),\text{ or }𝕏=(X^1,\dots ,X^{\left|I\right|}),$

the components of which are matrix valued; that is, $X_s={\left[x_{\mathrm{st}}^i\right]}_{t\in T}^{i\in I}\in {ℝ}^{T\times I}$ , $X_t={\left[x_{\mathrm{st}}^i\right]}_{s\in S}^{i\in I}\in {ℝ}^{S\times I}$ , and $X^i={\left[x_{\mathrm{st}}^i\right]}_{s\in S,t\in T}\in {ℝ}^{S\times T}$ , respectively.

Adapting the formalism of last section, we assume that

and from this relation, we define four families of conditionals, that is, {≿_s, $s\in S$ }, {≿_t, $t\in T$ }, {≿ⁱ, $i\in T$ }, and { ${\succsim }_{\mathrm{st}}^i$ , $s\in S$ , $t\in T$ , $i\in I$ }. The ≿_s, ≿_t, and ≿ⁱ respectively compare matrices X_s, X_t, and Xⁱ, and the ≿_st compare vectors $x_{\mathrm{st}}\in {ℝ}^I$ , and the ${\succsim }_{\mathrm{st}}^i$ compare real numbers. Similarly as before, we assume that

which makes it an ordering and furthermore makes the ${\succsim }_{\mathrm{st}}^i$ an invariant family. The other conditional relations may or may not be weak orderings and may or may not form invariant families, depending on which assumptions are put on them.

Unlike theorem 1, theorem 2 is in two parts, corresponding to the SEU formula and the SI property respectively. What appears to be essential to the latter is that one of the two sources (here conventionally taken to be S) gives rise to an invariant family of conditionals. We now reinforce the suggestion that invariance is the crucial condition by a heuristic argument.

Considering for simplicity only four states, suppose that the agent takes (s ₁, t ₁) to be more likely than (s ₁, t ₂) and (s ₂, t ₁) to be less likely than (s ₂, t ₂). That is, from knowing how the uncertainty on s is resolved, the agent is able to draw an inference on how the uncertainty on t would be resolved. If the agent reasoned probabilistically, the joint probabilities would of course not decompose multiplicatively. It is easy to conclude that the conditionals on s cannot be invariant. Take ξ, ξ′ as representing desirable quantities, with $\xi >{\xi }^{\prime }$ , and the following prospects in matrix form:

$\begin{array}{ccc}X& t_1& t_2\\ s_1& \xi & {\xi }^{\prime }\\ s_2& {\xi }^{\prime }& \xi \end{array}\mathrm{and}\begin{array}{ccc}Y& t_1& t_2\\ s_1& {\xi }^{\prime }& \xi \\ s_2& \xi & {\xi }^{\prime }\end{array}.$

The first line of X, which puts the best consequence on the more likely state, should be preferred to the first line of Y, which puts it on the less likely state; that is, $(\xi ,{\xi }^{\prime }){\succ }_{s_1}({\xi }^{\prime },\xi )$ . By a similar comparison, the second line of X should be preferred to the second line of Y; that is, $({\xi }^{\prime },\xi ){\succ }_{s_2}(\xi ,{\xi }^{\prime })$ . Thus, the two conditional preferences differ. Contraposing the argument, we see that if an agent entertains identical ≿_s and learns which s is realized without yet learning t, this agent would not use the knowledge so obtained to draw information on t.

The converse claim is problematic. Preference reversals may occur between ≿_s and ≿_t simply because the agent’s preferences are sensitive to which of s or t is realized, and this evaluative disposition is logically unrelated to the epistemic disposition of not using s-information to infer t-information. That is, the latter disposition can prevail in preference contexts in which the property that the ≿_s are an invariant family should not be assumed. The only way to secure a converse is to exclude the troublesome evaluative disposition by fiat, which all ordinary axiomatizations of SEU theory actually do, a strategy that has the definite shortcoming of narrowing the application range of the theory. For lack of a better solution, I reproduce this standard move here.

To derive (4), it was unnecessary to assume that both the ≿_s and the ≿_t are invariant. The invariance of the latter family follows from the representation (4) itself.Footnote ⁸ If it is sufficient to require one form of invariance, this is because an EU representation (3) holds from the previous stage. Under this assumption, it is impossible to distinguish between s bringing no relevant information on t and t bringing no relevant information on s. This is formally shown in appendix A and further discussed in appendix C.

I now return to the interpretation of the third dimension i introduced in this section. A very natural decision-theoretic account becomes available when i represents time. Then, the alternatives X mean contingent plans, that is, plans for the future whose consequences in a given period depend on the way the uncertainty—still represented by (s,t)—is resolved in that period. The matrix-valued objects X_s, X_t, and Xⁱ mean partly contingent plans (for the first two, when one dimension of uncertainty is fixed) or dated prospects (for the last, when the time dimension is fixed). As to the vector-valued objects x_st, they mean noncontingent plans, since they take the uncertainty to be resolved at each time period.Footnote ⁹

However, time considerations are extraneous to uncertainty, which is the concern here, and it may be more appropriate to find an interpretation for i that relates to these concerns. Suppose we declare i to be a third dimension of uncertainty. We can then add a third part to theorem 2, which puts on the ≿ⁱ the same invariance assumption as was imposed on the ≿_s and the ≿_t. With this addition, it can be proved that (4) gives way to the following more specific representation: for all $𝕏={\left[x_{\mathrm{st}}^i\right]}_{s\in S,t\in T}^{i\in I}$ ,

(5)$\begin{array}{}& W\left(𝕏\right)≔\sum _{s\in S}\sum _{t\in T}\sum _{i\in I}{p}_s{q}_t{r}_{i}u\left(x_{\mathrm{st}}^i\right)\text{,}\end{array}$

where $r={\left(r_i\right)}_{i\in I}$ is a strictly positive probability function on I, and the utility function u in the SEU formula is now independent of the i index. Apart from having a semantic advantage, this extension of theorem 2 carries with it a sense of mathematical generalization. To obtain the SI property for a product space Ω with any finite number of sources of uncertainty is no more difficult than to obtain it for $\Omega =S\times T\times I$ , but this would impose a heavy notational burden.

5. Comparisons with Decision Theory

I start this decision-theoretic section with two comments that Savage (Reference Savage1972) makes on SI. Having axiomatized a qualitative probability relation, he complains that “the notions of independence and irrelevance have … no analogues in qualitative probability; this is surprising and unfortunate, for these notions seem to evoke a strong intuitive response” (44). Later, at the end of a well-known passage on “small worlds,” Savage restates his complaint as follows: “it would be desirable, if possible, to find a simple qualitative personal description of independence between events” (91).Footnote ¹⁰ The two comments clearly express the need for Bayesian decision theory to complement its derivation of subjective probability with an account of SI, but they point in different directions. The first relates to qualitative probability, which is an auxiliary concept in Savage’s construction; he uses it as an intermediary between his preference postulates and his final conclusions, in which subjective probability acquires a numerical form. Today, Savage’s complaint in respect of this concept is no longer justified. There now exist richer systems of qualitative probability than his, which contain a special relation to express the SI of two events or two random variables.Footnote ¹¹ The second comment does not mention qualitative probability, and I read in it a suggestion to base SI directly on the preference relation. In this respect, Savage’s complaint is still topical. The current axiomatic work, which it motivated, appears to have been foreshadowed in only three papers, to be discussed now.

The main connection is with Blume, Brandenburger, and Dekel (Reference Blume, Brandenburger and Dekel1991, 74). These writers use Anscombe and Aumann’s (Reference Anscombe and Aumann1963) axiom system to explore the preference foundations of lexicographic probability, a topic apparently remote from the current one. However, their construction includes ordinary Kolmogorov probabilities as a particular case, and we may just focus on this application. To account for SI, they impose invariance of relevant conditionals of their ex ante weak ordering (which they have redefined so that it bears on a Cartesian product). Leaving aside the Anscombe-Aumann features (see the critical comments below), their condition is the same as mine. Their paper does not contain a proof that this condition delivers SI, but I provide one in appendix C.

In an important follow up, which still uses Anscombe and Aumann’s axiom system, Battigalli and Veronesi (Reference Battigalli and Veronesi1996) push the analysis of Blume et al. in the direction of conditional probability systems (CPS). These amount to taking conditional probabilities, rather than absolute probabilities, as being the primitive terms of the probability calculus, a move that philosophers usually associate with the works of Rényi (Reference Rényi1955) and Popper (Reference Popper1972). For a suitable definition of what it means for a CPS to satisfy SI, Battigalli and Veronesi (Reference Battigalli and Veronesi1996, 243) connect this property with a form of conditional invariance. Given the dissimilarity of frameworks, the step from this condition to mine is far from trivial.

A comparison is also in order with a little-known axiomatization of Bayesian decision theory by Bernardo, Fernandez, and Smith (Reference Bernardo, Fernandez and Smith1985). These writers’ derivation of SEU hinges on the strong assumption that a preexisting probability measure is available on a subalgebra of events; this will serve to calibrate subjective probability values. They define a preference condition relative to two events E and F that entails the equation $P(E\cap F)=P\left(E\right)\cdot P\left(F\right)$ at the stage of proving the representation theorem. Although evocative of the informational reading of SI, this condition differs from ours, and this difference seems connected with the technical choice of approaching SI in general probability spaces rather than product spaces.Footnote ¹²

Theorems 1 and 2 should also be compared with the recent axiomatic work on second-order expected utility.Footnote ¹³ This work involves considering two sources of uncertainty; however, unlike the current one, it aims at establishing a logical hierarchy between these sources. By assumption, uncertainty takes both a primary form and a secondary form, which concerns what the primary form may be. This hierarchy can be interpreted epistemically, with the primary form bearing on the realization of natural events and the secondary form bearing on the realization of the primary form considered as an epistemic event, but it also admits of a temporal interpretation, with the primary and secondary sources corresponding to uncertainties of the second and first periods, respectively. In the current notation, if T represents the uncertain states of the primary form, and S those of the secondary form, a second-order EU representation of ex ante preferences is

$\sum _{s\in S}{p}_{s}v\left(\sum _{t\in T}{r}_{\mathrm{st}}u\right(x_s^t\left)\right)\text{, }$

where $r_s={\left(r_{\mathrm{st}}\right)}_{t\in T}$ is a conditional probability function for t given s, v is a continuous and increasing function on ℝ, and u is a function on a consequence set that may be, but is not necessarily, ℝ. This representation is axiomatized by Nau (Reference Nau2006, 143). Ergin and Gul’s (Reference Ergin and Gul2009, 912) version uses an unconditional probability for t:Footnote ¹⁴

$\sum _{s\in S}{p}_{s}v\left(\sum _{t\in T}{q}_{t}u\right(x_s^t\left)\right)\text{.}$

If one takes v to be a positive affine transformation, the two forms of uncertainty play symmetric roles, and both formulas collapse onto a SEU formula, with a subjective probability that satisfies SI in the latter; thus, one gets the representation of theorem 1 as a particular case. How the axioms for second-order EU should be strengthened so as to get this particular case is a relevant question, which I take up in appendix C.Footnote ¹⁵

I now compare the axiomatization of the SEU formula in this article with those in current use. Being entirely preference based, the former is like Savage’s (Reference Savage1972), but there are important dissimilarities. An obvious one concerns the cardinality of the state set Ω, which is infinite in Savage and finite here. Also, the highly condensed axiom systems for theorems 1 and 2 do not relate to Savage’s seven-postulate system in a transparent way. However, it is well known that given postulate P1 (which requires that the ex ante preference relation ≿ be a weak ordering), postulate P2 (the “sure-thing principle”) can be restated as the requirement that the conditional ≿_E on any event E of the state set be a weak ordering. This restatement facilitates comparison with the current system. I only assume that the conditionals on some distinguished events E of the state set are weak orderings. This weakening is in the spirit of axiomatizations of some non-EU rules that do not entirely relinquish the “sure-thing principle.”

By contrast, the invariance condition used here is stronger than P3—the “event independence” postulate—because it bears on all possible prospects and not only on constant prospects, as is the case in that postulate. Savage has another important postulate, P4, which is a clear step toward the existence of subjective probability and has no analogue here; it can only be verified from the utility representation itself. My best guess is that P4 is made dispensable by the assumptions that consequences are real numbers and conditional preferences respect the order of these numbers. By contrast, Savage puts no restriction at all on his consequence set.

Another comparison to the point is with Anscombe and Aumann’s (Reference Anscombe and Aumann1963) popular variation on Savage’s system. I share with these authors the assumptions of a finite state set and a highly structured consequence set, but they assume the latter to be a set of probabilistically defined lotteries, an assumption I am glad to eschew here. From a Bayesian decision theory perspective, the Anscombe-Aumann system is open to the objection that it is question begging to derive a subjective probability by supposing that other probabilities already exist. A common rejoinder is that the preexisting probabilities are objective and, hence, of a different nature from the subjective probability to be derived, but this is a free commentary without any basis in Anscombe and Aumann’s formal system. I do not deny the practical convenience of this system, but mine is no more complicated, while being perhaps easier to defend theoretically.Footnote ¹⁶

6. Connections with Foundational Discussions on Probability

Underlying the axiomatic work of Savage and Bayesian decision theorists generally are two major claims on the foundations of probability: probability measures represent uncertain beliefs in the normatively appropriate way, and what makes the measures in question normatively appropriate is that practical rationality considerations recommend using them in decision making. Both claims have been disputed, with some objections surfacing already before Bayesian decision theory fully took shape. The first claim can be attacked along at least two different lines. One may question the appropriateness of probabilities on the grounds that they are absolute measures and as an alternative develop a calculus for conditional probabilities taken as primitive terms. This line has recently been defended by Fitelson and Hájek (Reference Fitelson and Hájek2017) in connection with Popper’s (Reference Popper1972, apps. 4 and 5) pioneering work in this area. All existing CPS preserve the additivity of probability measures, and an alternative critical line is indeed to question that property. Decision theory has made thorough contributions on this score; see Gilboa (Reference Gilboa2009) and Wakker (Reference Wakker2010) for overviews. As to the second foundational claim, it can again be attacked from different sides, one representative example being Joyce’s (Reference Joyce1998) “nonpragmatic” argument that probabilities are appropriate representations of uncertain beliefs for directly epistemic reasons.Footnote ¹⁷

These deep foundational questions arise in connection with the current work but exceed its limited purpose. I meant to fill a gap in Bayesian decision theory by following its own principles, rather than defend these principles against outside criticism. However, since SI is the current focus, one should ask whether this theory, as extended here, may contribute to a better understanding of this property.

There is much conceptual tension in the way probability theorists introduce the definition of SI. For one thing, they usually discuss its informal meaning in terms of a provisional definition of SI by the equality of conditional and unconditional probability: for any two events A, $B\in 𝒜$ with $P\left(B\right)>0$ ,

$P(A\mid B)=P\left(A\right)\text{.}$

Once they have made intuitive sense of this equation, they proceed to the multiplicative equation of section 2 as constituting the proper definition of SI, arguing that the latter avoids the sign restriction $P\left(B\right)>0$ and makes SI symmetric. This argument is unconnected with the intuitions supporting the provisional definition, which makes the whole sequence semantically awkward.Footnote ¹⁸ For another thing, probability theorists informally defend their definitions by resorting to more than one concept of unrelatedness. Prominent examples are logical independence, causal independence (or alternative forms of objective independence), and informational independence. While some accounts are relatively clear on which concept they privilege, many others are equivocal, and some even fall into amazing confusions between them.Footnote ¹⁹

The Bayesian decision theory developed here contributes nothing to the first problem. Only a move from absolute to conditional probability could avoid the discrepancy between the provisional and final definitions of SI, and I have not performed this move here. However, on the second problem, the theory has something to say. At the very least, it avoids the equivocations between different informal accounts by firmly opting for informational independence. The invariance property of conditional preferences is the pragmatic criterion by which one can judge that the agent regards s as carrying no information on t.

It remains to be said whether the current theory contents itself with endorsing one of the available accounts of SI or adds something significant to that account. We may credit the theory with two contributions. One is to connect the SI property with the foundations of subjective probability more tightly than is usually done. There has been some vacillation among subjectivists concerning the role of SI assumptions in probability theory. Whereas Savage did not underplay this role, de Finetti considered it with strong reluctance. As Gillies (Reference Gillies2000, 75–76) explains, citing from Probabilismo (de Finetti Reference de Finetti1931), de Finetti argued against the application of SI to repeated trials of the same experiment on the grounds that this assumption blocked the possibility of learning from the successive results of the trials through Bayes’s rule of revision. This argument opened the way to de Finetti’s alternative to SI, which is exchangeability. Without entering the rich debate—well covered by Gillies—on the respective merits of the two concepts, we can make the broader point that learning by Bayes’s rule in repeated trials is just one particular case to be considered by the subjective theory of probability. It is possible to make perfect subjective sense of the opposite particular case in which no learning occurs from one trial to the other; it is actually incumbent on the subject to decide which case is relevant. In other words, there is no logical necessity to associate the subjective theory with a large scope of nontrivial applications of Bayes’s rule. Although this point may be clear by itself, it comes out perhaps more clearly after Bayesian decision theory, which is a brand of subjectivism, has offered an account of SI.

Another possible contribution is to put the symmetry of the definition of SI in perspective. Writers on the foundations of probability have sometimes expressed dissatisfaction toward the fact that asymmetric dependence or independence cannot be formulated within the Kolmogorov axiomatic framework; see Fitelson and Hájek (Reference Fitelson and Hájek2017) for a recent example. This is a fair complaint to make in connection with the logical and causal (or more generally objective) readings of SI, but its force as to the informational reading is not so clear, as Fitelson and Hájek concede. In the Bayesian decision theory of this article, one can assume the s-component of uncertainty to be informationally irrelevant to the t-component without assuming the converse irrelevance, for this amounts to requiring invariance from the s-conditionals and not from the t-conditionals. However, we have seen that this logical independence of the two assumptions vanishes once the preference axioms for SEU are all in place. This result can be understood in two opposite ways—those who take the preference conditions for SEU to be normatively compelling will view it as a justification of the postulated symmetry of SI, whereas others, for whom this symmetry is an arbitrary diktat, will turn the result against the allegedly compelling preference conditions.

7. Conclusion

I have responded to Savage’s request to extend the preference apparatus of Bayesian decision theory to the point where it includes an account of SI. To do so, I have reconstructed this preference apparatus and proved representation theorems that contain both a novel derivation of SEU and the desired addition that the subjective probability makes the sources of uncertainty stochastically independent. These theorems call for richer variants that need to be pursued elsewhere. One such variant would relax the finiteness assumption put on the set of states of the world. Apart from absolute probability as in Kolmogorov, this line of research could more ambitiously concern conditional probability taken as an axiomatic primitive, in Popper’s sense. Each time, the objective would be to map the features of the probability space onto axiomatic preference counterparts. Another project would be to reconsider SI in relation to the nonadditive measures of uncertainty that decision theorists have introduced since they moved away from a primarily Bayesian conception. This is the more challenging of the two lines of research, because it requires one not only to find preference counterparts to already defined mathematical properties but also to discover those new mathematical definitions that capture SI when probability gives way to weaker notions.