Friday, 17 May 2019

What is conditionalization and why should we do it?

The three central tenets of traditional Bayesian epistemology are these:

Precision Your doxastic state at a given time is represented by a credence function, $c$, which takes each proposition $X$ about which you have an opinion and returns a single numerical value, $c(X)$, that measures the strength of your belief in $X$. By convention, we let $0$ represent your minimal credence and we let $1$ represent your maximal credence.

Probabilism Your credence function should be a probability function. That is, you should assign minimal credence (i.e. 0) to necessarily false propositions, maximal credence (i.e. 1) to necessarily true propositions, and your credence in the disjunction of two propositions whose conjunction is necessarily false should be the sum of your credences in the disjuncts.

Conditionalization You should update your credences by conditionalizing on your total evidence.

Note: Precision sets out the way in which doxastic states will be represented; Probabilism and Conditionalization are norms that are stated using that representation.

Here, we will assume Precision and Probabilism and focus on Conditionalization. In particular, we are interested in what exactly the norm says; and, more specifically, which versions of the norm are supported by the standard arguments in its favour. That is, we are interested in what versions of the norm we can justify using the existing arguments. We will consider three versions of the norm; and we will consider four arguments in its favour. For each combination, we'll ask whether the argument can support the norm. In each case, we'll notice that the standard formulation relies on a particular assumption, which we call Deterministic Updating and which we formulate precisely below. We'll ask whether the argument really does rely on this assumption, or whether it can be amended to support the norm without that assumption. Let's meet the interpretations and the arguments informally now; then we'll be ready to dive into the details.

Here are the three interpretations of Conditionalization. According to the first, Actual Conditionalization, Conditionalization governs your actual updating behaviour.

Actual Conditionalization (AC)

If
  • $c$ is your credence function at $t$ (we'll often refer to this as your prior);
  • the total evidence you receive between $t$ and $t'$ comes in the form of a proposition $E$ learned with certainty;
  • $c(E) > 0$;
  • $c'$ is your credence function at the later time $t'$ (we'll often refer to this as your posterior);
then it should be the case that $c'(-) = c(-|E) = \frac{c(-\ \&\ E)}{c(E)}$.
According to the second, Plan Conditionalization, Conditionalization governs the updating behaviour you would endorse in all possible evidential situations you might face:

Plan Conditionalization (PC)

If
  • $c$ is your credence function at $t$;
  • the total evidence you receive between $t$ and $t'$ will come in the form of a proposition learned with certainty, and that proposition will come from the partition $\mathcal{E} = \{E_1, \ldots, E_n\}$;
  • $R$ is the plan you endorse for how to update in response to each possible piece of total evidence,
then it should be the case that, if you were to receive evidence $E_i$ and if $c(E_i) > 0$, then $R$ would exhort you to adopt credence function $c_i(-) = c(-|E_i) = \frac{c(-\ \&\ E_i)}{c(E_i)}$.

According to the third, Dispositional Conditionalization, Conditionalization governs the updating behaviour you are disposed to exhibit.
 
Dispositional Conditionalization (DC)

If
  • $c$ is your credence function at $t$;
  • the total evidence you receive between $t$ and $t'$ will come in the form of a proposition learned with certainty, and that proposition will come from the partition $\mathcal{E} = \{E_1, \ldots, E_n\}$;
  • $R$ is the plan you are disposed to follow in response to each possible piece of total evidence,
then it should be the case that, if you were to receive evidence $E_i$ and if $c(E_i) > 0$, then $R$ would exhort you to adopt credence function $c_i(-) = c(-|E_i) = \frac{c(-\ \&\ E_i)}{c(E_i)}$.

Next, let's meet the four arguments. Since it will take some work to formulate them precisely, I will give only an informal gloss here. There will be plenty of time to see them in high-definition in what follows.

Diachronic Dutch Book or Dutch Strategy Argument (DSA) This purports to show that, if you violate conditionalization, there is a pair of decisions you might face, one before and one after you receive your evidence, such that your prior and posterior credences lead you to choose options when faced with those decisions that are guaranteed to be worse by your own lights than some alternative options (Lewis 1999).

Expected Pragmatic Utility Argument (EPUA) This purports to show that, if you will face a decision after learning your evidence, then your prior credences will expect your updated posterior credences to do the best job of making that decision if they are obtained by conditionalizing on your priors (Brown 1976).

Expected Epistemic Utility Argument (EEUA) This purports to show that your prior credences will expect your posterior credences to be best epistemically speaking if they are obtained by conditionalizing on your priors (Greaves & Wallace 2006).

Epistemic Utility Dominance Argument (EUDA) This purports to show that, if you violate conditionalization, then there will be alternative priors and posteriors  that are guaranteed to be better epistemically speaking, when considered together, than your priors and posteriors (Briggs & Pettigrew 2018).

The framework


In the following sections, we will consider each of the arguments listed above. As we will see, these arguments are concerned directly with updating plans or dispositions, rather than actual updating behaviour. That is, the items that they consider don't just specify how you in fact update in response to the particular piece of evidence you actually receive. Rather, they assume that your evidence between the earlier and later time will come in the form of a proposition learned with certainty (Certain Evidence); they assume the possible propositions that you might learn with certainty by the later time form a partition (Evidential Partition); and they assume that each of the propositions you might learn with certainty is one about which you had a prior opinion (Evidential Availability); and then they specify, for each of the possible pieces of evidence in your evidential partition, how you might update if you were to receive it.

Some philosophers, like David Lewis (1999), assume that all three assumptions---Certain Evidence, Evidential Partition, Evidential Availability---hold in all learning situations. Others, deny one or more. So Richard Jeffrey (1992) denies Certain Evidence and Evidential Availability; Jason Konek (2019) denies Evidential Availability but not Certain Evidence; Bas van Fraassen (1999), Miriam Schoenfield (2017), and Jonathan Weisberg (2007) deny Evidential Partition. But all agree, I think, that there are certain important situations when all three assumptions are true; there are certain situations where there is a set of propositions that forms a partition and about each member of which you have a prior opinion, and the possible evidence you might receive at the later time comes in the form of one of these propositions learned with certainty. Examples might include: when you are about to discover the outcome of a scientific experiment, perhaps by taking a reading from a measuring device with unambiguous outputs; when you've asked an expert a yes/no question; when you step on the digital scales in your bathroom or check your bank balance or count the number of spots on the back of the ladybird that just landed on your hand. So, if you disagree with Lewis, simply restrict your attention to these cases in what follows.

As we will see, we can piggyback on conclusions about plans and dispositions to produce arguments about actual behaviour in certain situations. But in the first instance, we will take the arguments to address plans and dispositions defined on evidential partitions primarily, and actual behaviour only secondarily. Thus, to state these arguments, we need a clear way to represent updating plans or dispositions. We will talk neutrally here of an updating rule. If you think conditionalization governs your updating dispositions, then you take it to govern the updating rule that matches those dispositions; if you think it governs your updating intentions, then you take it to govern the updating rule you intend to follow.

We'll introduce a slew of terminology here. You needn't take it all in at the moment, but it's worth keeping it all in one place for ease of reference.

Agenda  We will assume that your prior and posterior credence functions are defined on the same set of propositions $\mathcal{F}$, and we'll assume that $\mathcal{F}$ is finite and $\mathcal{F}$ is an algebra. We say that $\mathcal{F}$ is your agenda.

Possible worlds  Given an agenda $\mathcal{F}$, the set of possible worlds relative to $\mathcal{F}$ is the set of classically consistent assignments of truth values to the propositions in $\mathcal{F}$. We'll abuse notation throughout and write $w$ for (i) a truth value assignment to the propositions in $\mathcal{F}$, (ii) the proposition in $\mathcal{F}$ that is true at that truth value assignment and only at that truth value assignment, and (iii) what we might call the omniscient credence function relative to that truth value assignment, which is the credence function that assigns maximal credence (i.e. 1) to all propositions that are true on it and minimal credence (i.e. 0) to all propositions that are false on it.

Updating rules An updating rule has two components:
  • a set of propositions, $\mathcal{E} = \{E_1, \ldots, E_n\}$. This contains the propositions that you might learn with certainty at the later time $t'$; each $E_i$ is in $\mathcal{F}$, so $\mathcal{E} \subseteq \mathcal{F}$; $\mathcal{E}$ forms a partition;
  • a set of sets of credence functions, $\mathcal{C} = \{C_1, \ldots, C_n\}$. For each $E_i$, $C_i$ is the set of possible ways that the rule allows you to respond to evidence $E_i$; that is, it is the set of possible posteriors that the rule permits when you learn $E_i$; each $c'$ in $C_i$ in $\mathcal{C}$ is defined on $\mathcal{F}$.

Deterministic updating rule We say that an updating rule $R = (\mathcal{E}, \mathcal{C})$ is deterministic if each $C_i$ is a singleton set $\{c_i\}$. That is, for each piece of evidence there is exactly one possible response to it that the rule allows.

Stochastic updating rule  A stochastic updating rule is an updating rule $R = (\mathcal{C}, \mathcal{E})$ equipped with a probability function $P$. $P$ records, for each $E_i$ in $\mathcal{E}$ and $c'$ in $C_i$, how likely it is that I will adopt $c'$ in response to learning $E_i$. We write this $P(R^i_{c'} | E_i)$, where $R^i_{c'}$ is the proposition that says that you adopt posterior $c'$ in response to evidence $E_i$.
  • We assume $P(R^i_{c'} | E_i) > 0$ for all $c'$ in $C_i$. If the probability that you will adopt $c'$ in response to $E_i$ is zero, then $c'$ does not count as a response to $E_i$ that the rule allows.
  • Note that every deterministic updating rule is a stochastic updating rule for which $P(R^i_{c'} | E_i) = 1$ for each $c'$ in $C_i$. If $R = (\mathcal{E}, \mathcal{C})$ is deterministic, then, for each $E_i$, $C_i = \{c_i\}$. So let $P(R^i_{c_i} | E_i) = 1$.

Conditionalizing updating rule An updating rule $R = (\mathcal{E}, \mathcal{C})$ is a conditionalizing rule for a prior $c$ if, whenever $c(E_i) > 0$, $C_i = \{c_i\}$ and $c_i(-) = c(-|E_i)$.

Conditionalizing pairs  A pair $\langle c, R \rangle$ of a prior and an updating rule is a conditionalizing pair if $R$ is a conditionalizing rule for $c$.

Pseudo-conditionalizing updating rule Suppose $R = (\mathcal{E}, \mathcal{C})$ is an updating rule. Then let $\mathcal{F}^*$ be the smallest algebra that contains all of $\mathcal{F}$ and also $R^i_{c'}$ for each $E_i$ in $\mathcal{E}$ and $c'$ in $C_i$. (As above $R^i_{c'}$ is the proposition that says that you adopt posterior $c'$ in response to evidence $E_i$.) Then an updating rule $R$ is a pseudo-conditionalizing rule for a prior $c$ if it is possible to extend $c$, a credence function defined on $\mathcal{F}$, to $c^*$, a credence function defined on $\mathcal{F}^*$, such that, for each $E_i$ in $\mathcal{E}$ and $c'$ in $C_i$, $c'(-) = c^*(-|R^i_{c'})$. That is, each posterior is the result of conditionalizing the extended prior $c^*$ on the evidence to which it is a response and the fact that it was your response to this evidence.

Pseudo-conditionalizing pair A pair $\langle c, R \rangle$ of a prior and an updating rule is a pseudo-conditionalizing pair if $R$ is a pseudo-conditionalizing rule for $c$.

Let's illustrate these definitions using an example. Condi is a meteorologist. There is a hurricane in the Gulf of Mexico. She knows that it will make landfall soon in one of the following four towns: Pensecola, FL, Panama City, FL, Mobile, AL, Biloxi, MS. She calls a friend and asks whether it has hit yet. It has. Then she asks whether it has hit in Florida. At this point, the evidence she will receive when her friend answers is either $F$---which says that it made landfall in Florida, that is, in Pensecola or Panama City---or $\overline{F}$---which says it hit elsewhere, that is, in Mobile or Biloxi. Her prior is $c$:

Her evidential partition is $\mathcal{E} = \{F, \overline{F}\}$. And here are some posteriors she might adopt:



And here are four possible rules she might adopt, along with their properties:


As we will see below, for each of our four arguments for conditionalization---DSA, EPUA, EEUA, and EUDA---the standard formulation of the argument assumes a norm that we will call Deterministic Updating:

Deterministic Updating (DU) Your updating rule should be deterministic.

As we will see, this is crucial for the success of these arguments. In what follows, I will present each argument in its standard formulation, which assumes Deterministic Updating. Then I will explore what happens when we remove that assumption.

The Dutch Strategy Argument (DSA)


The DSA and EPUA both evaluate updating rules by their pragmatic consequences. That is, they look to the choices that your priors and/or your possible posteriors lead you to make and they conclude that they are optimal only if your updating rule is a conditionalizing rule for your prior.

DSA with Deterministic Updating


Let's look at the DSA first. In what follows, we'll take a decision problem to be a set of options that are available to an agent: e.g. accept a particular bet or refuse it; buy a particular lottery ticket or don't; take an umbrella when you go outside, take a raincoat, or take neither; and so on. The idea behind the DSA is this. One of the roles of credences is to help us make choices when faced with decision problems. They play that role badly if they lead us to make one series of choices when another series is guaranteed to serve our ends better. The DSA turns on the claim that, unless we update in line with Conditionalization, our credences will lead us to make such a series of choices when faced with a particular series of decision problems.

Here, we restrict attention to a particular class of decision problems you might face. They are the decision problems in which, for each available option, its outcome at a given possible world obtains for you a certain amount of a particular quantity, such as money or chocolate or pure pleasure, and your utility is linear in that quantity---that is, obtaining some amount of that quantity increases your utility by the same amount regardless of how much of the quantity you already have. The quantity is typically taken to be money, and we'll continue to talk like that in what follows. But it's really a placeholder for some quantity with this property. We restrict attention to such decision problems because, in the argument, we need to combine the outcome of one decision, made at the earlier time, with the outcome of another decision, made at the later time. So we need to ensure that the utility of a combination of outcomes is the sum of the utilities of the individual outcomes.

Now, as we do throughout, we assume that the prior $c$ and the possible posteriors $c_1, \ldots, c_n$ permitted by a deterministic updating rule $R$ are all probability functions. And we will assume further that, when your credences are probabilistic, and you face a decision problem, then you should choose from the available options one of those that maximises expected utility relative to your credences.

With this in hand, let's define two closely related features of a pair $\langle c, R \rangle$ that are undesirable from a pragmatic point of view, and might be thought to render that pair irrational. First:

Strong Dutch Strategies  $\langle c, R \rangle$ is vulnerable to a strong Dutch strategy if there are two decision problems, $\mathbf{d}$, $\mathbf{d}'$ such that
  1. $c$ requires you to choose option $A$ from the possible options available in $\mathbf{d}$;
  2. for each $E_i$ and each $c'$ in $C_i$, $c'$ requires you to choose $B$ from $\mathbf{d}'$;
  3. there are alternative options, $X$ in $\mathbf{d}$ and $Y$ in $\mathbf{d}'$, such that, at every possible world, you'll receive more utility from choosing $X$ and $Y$ than you receive from choosing $A$ and $B$. In the language of decision theory, $X + Y$ strongly dominates $A + B$.
Weak Dutch Strategies  $\langle c, R \rangle$ is vulnerable to a weak Dutch strategy if there are decision problems $\mathbf{d}$ and, for each $c'$ in $C_i$ in $\mathcal{C}$, $\mathbf{d}_{c'}$ such that
  1. $c$ requires you to choose $A$ from $\mathbf{d}$;
  2. for each $E_i$ and each $c'$ in $C_i$, $c'$ requires you to choose $B^i_{c'}$ from $\mathbf{d}'_{c'}$;
  3. there are alternative options, $X$ in $\mathbf{d}$ and, for $E_i$ and $c'$ in $C_i$, $Y^i_{c'}$ in $\mathbf{d}'_{c'}$, such that (a) for each $E_i$, each world in $E_i$, and each $c'$ in $C_i$, you'll receive at least as much utility at that world from choosing $X$ and $Y^i_{c'}$ as you'll receive from choosing $A$ and $B^i_{c'}$, and (b) for some $E_i$, some world in $E_i$, and some $c'$ in $C_i$, you'll receive strictly more utility at that world from $X$ and $Y^i_{c'}$ than you'll receive from $A$ and $B^i_{c'}$.
Then the Dutch Strategy Argument is based on the following mathematical fact (de Finetti 1974):

Theorem 1 Suppose $R$ is a deterministic updating rule. Then:
  1. if $R$ is not a conditionalizing pair for $c$, then $\langle c, R \rangle$ is vulnerable to a strong Dutch strategy;
  2. if $R$ is a conditionalizing rule for $c$, then $\langle c, R \rangle$ is not vulnerable even to a weak Dutch strategy.
That is, if your updating rule is not a conditionalizing rule for your prior, then your credences will lead you to choose a strongly dominated pair of options when faced with a particular pair of decision problems; if you satisfy it, that can't happen.

Now that we have seen how the argument works, let's see whether it supports the three versions of conditionalization that we met above: Actual (AC), Plan (PC), and Dispositional (DC) Conditionalization. Since they speak directly of rules, let's begin with PC and DC.

The DSA shows that, if you endorse a deterministic rule that isn't a conditionalizing rule for your prior, then there is pair of decision problems, one that you'll face at the earlier time and the other at the later time, where your credences at the earlier time and your planned credences at the later time will require you to choose a dominated pair of options. And it seems reasonable to say that it is irrational to endorse a plan when you will be rendered vulnerable to a Dutch Strategy  if you follow through on it. So, for those who endorse deterministic rules, DSA plausibly supports Plan Conditionalization.

The same is true of Dispositional Conditionalization. Just as it is irrational to plan to update in a way that would render you vulnerable to a Dutch Strategy if you were to stick to the plan, it is surely irrational to be disposed to update in a way that will renders you vulnerable in this way. So, for those whose updating dispositions are deterministic, DSA plausibly supports Dispositional Conditionalization.

Finally, AC. There various different ways to move from either PC or DC to AC, but each one of them requires some extra assumptions. For instance:

(I) I might assume: (i) between an earlier and a later time, there is always a partition such that you know that the strongest pieces of evidence you might receive between those times is a proposition from that partition learned with certainty; (ii) if you know you'll receive evidence from some partition, you are rationally required to plan how you will update on each possible piece of evidence before you receive it; and (iii) if you plan how to respond to evidence before you receive it, you are rationally required to follow through on that plan once you have received it. Together with PC + DU, these give AC.

(II) I might assume: (i) you have updating dispositions. So, if you actually update other than by conditionalization, then it must be a manifestation of a  disposition other than conditionalizing. Together with DC + DU, this gives  AC.

(III) I might assume: (i) that you are rationally required to update in any way that can be represented as the result of updating on a plan that you were rationally permitted to endorse or as the result of dispositions that you were rationally permitted to have, even if you did not in fact endorse any plan prior to receiving the evidence nor have any updating dispositions. Again, together with PC + DU or DC + DU, this gives AC.

Notice that, in each case, it was essential to invoke Deterministic Updating (DU). As we will see below, this causes problems for AC.

DSA without Deterministic Updating


We have now seen how the DSA proceeds if we assume Deterministic Updating. But what if we don't? Consider, for instance, rule $R_3$ from our list of examples above:
$$R_3 = (\mathcal{E} = \{F, \overline{F}\}, \mathcal{C} = \{\{c^\circ_F, c^+_F\}, \{c^\circ_{\overline{F}}, c^+_{\overline{F}}\}\})$$
That is, if Condi learns $F$, rule $R_3$ allows her to update to $c^\circ_F$ or to $c^+_F$. And if she receives $\overline{F}$, it allows her to update to $c^\circ_{\overline{F}}$ or to $c^+_{\overline{F}}$. Notice that $R_3$ violates conditionalization thoroughly: it is not deterministic; and, moreover, as well as not mandating the posteriors that conditionalization demands, it does not even permit them. Can we adapt the DSA to show that $R_3$ is irrational? No. We cannot use Dutch Strategies to show that $R_3$ is irrational because it isn't vulnerable to them.

To see this, we first note that, while $R_3$ is not deterministic and not a conditionalizing rule, it is a pseudo-conditionalizing rule.  And to see that, it helps to state the following representation theorem for pseudo-conditionalizing rules.

Lemma 1 $R$ is a pseudo-conditionalizing pair for $c$ iff
  1. for all $E_i$ in $\mathcal{E}$ and $c'$ in $C_i$, $c'(E_i) = 1$, and
  2. $c$ is in the convex hull of the possible posteriors that $R$ permits.
But note:$$c(-) = 0.4c^\circ_F(-) + 0.4c^+_F(-) + 0.1c^\circ_{\overline{F}}(-) + 0.1 c^+_{\overline{F}}(-)$$
So $R_3$ is pseudo-conditionalizing. What's more:


Theorem 2
  • If $R$ is not a pseudo-conditionalizing rule for $c$, then $\langle c, R \rangle$ is vulnerable at least to a weak Dutch Strategy, and possibly also a strong Dutch Strategy.
  • If $R$ is a pseudo-conditionalizing rule for $c$, then $\langle c, R \rangle$ is not vulnerable to a weak Dutch Strategy.
Thus, $\langle c, R_3 \rangle$ is not vulnerable even to a weak Dutch Strategy. The DSA, then, cannot say what is irrational about Condi if she begins with prior $c$ and either endorses $R_3$ or is disposed to update in line with it. Thus, the DSA cannot justify Deterministic Updating. And without DU, it cannot support PC or DC either. After all, $R_3$ violates each of those, but it is not vulnerable even to a weak Dutch Strategy. And moreover, each of the three arguments for AC break down because they depend on PC or DC. The problem is that, if Condi updates from $c$ to $c^\circ_F$ upon learning $F$, she violates AC; but there is a non-deterministic updating rule---namely, $R_3$---that allows $c^\circ_F$ as a response to learning $F$, and, for all DSA tells us, she might have rationally endorsed $R_3$ before learning $F$ or she might rationally have been disposed to follow it. Indeed, the only restriction that DSA can place on your actual updating behaviour is that you should become certain of the evidence that you learned. After all:

Theorem 3 Suppose $c$ is your prior and $c'$ is your posterior. Then there is a rule $R$ such that:
  1. $c'$ is in $C_i$, and
  2. $R$ is a pseudo-conditionalizing rule for $c$
iff $c'(E_i) = 1$.

Thus, at the end of this section, we can conclude that, whatever is irrational about planning to update using non-deterministic but pseudo-conditionalizing updating rules, it cannot be that following through on those plans leaves you vulnerable to a Dutch Strategy, for it does not. And similarly, whatever is irrational about being disposed to update in those ways, it cannot be that those dispositions will equip you with credences that lead you to choose dominated options, for they do not. With PC and DC thus blocked, our route to AC is therefore also blocked.

The Expected Pragmatic Utility Argument (EPUA)


Let's look at EPUA next. Again, we will consider how our credences guide our actions when we face decision problems. In this case, there is no need to restrict attention to monetary decision problems. We will only consider a single decision problem, which we face at the later time, after we've received the evidence, so we won't have to combine the outcomes of multiple options as we did in the DSA. The idea is this. Suppose you will make a decision after you receive whatever evidence it is that you receive at the later time. And suppose that you will use your later updated credence function to make that choice---indeed, you'll choose from the available options by maximising expected utility from the point of view of your new updated credences. Which updating rules does your prior expect will lead you to make the choice best?

EPUA with Deterministic Updating


Suppose you'll face decision problem $\mathbf{d}$ after you've updated. And suppose further that you'll use a deterministic updating rule $R$. Then, if $w$ is a possible world and $E_i$ is the element of the evidential partition $\mathcal{E}$ that is true at $w$, the idea is that we take the pragmatic utility of $R$ relative to $\mathbf{d}$ at $w$ to be the utility at $w$ of whatever option from $\mathbf{d}$ we should choose if our posterior credence function were $c_i$, as $R$ requires it to be at $w$. But of course, for many decision problems, this isn't well defined because there is no unique option in $\mathbf{d}$ that maximises expected utility by the lights of $c_i$; rather there are sometimes many such options, and they might have different utilities at $w$. Thus, we need not only $c_i$ but also a selection function, which picks a single option from  any set of options. If $f$ is such a selection function, then let $A^{\mathbf{d}}_{c_i, f}$ be the option that $f$ selects from the set of options in $\mathbf{d}$ that maximise expected utility by the lights of $c_i$. And let
$$u_{\mathbf{d},f}(R, w) = u(A^{\mathbf{d}}_{c_i, f}, w).$$
Then the EPUA argument turns on the following mathematical fact (Brown 1976):

Theorem 4 Suppose $R$ and $R^\star$ are both deterministic updating rules. Then:
  • If $R$ and $R^\star$ are both conditionalizing rules for $c$, and $f$, $g$ are selection functions, then for all decision problems $\mathbf{d}$ $$\sum_{w \in W} c(w) u_{\mathbf{d}, f}(R, w) = \sum_{w \in W} c(w) u_{\mathbf{d}, g}(R^\star, w)$$
  • If $R$ is a conditionalizing rule for $c$, and $R^\star$ is not, and $f$, $g$ are selection functions, then for all decision problems $\mathrm{d}$, $$\sum_{w \in W} c(w) u_{\mathbf{d}, f}(R, w) \geq \sum_{w \in W} c(w) u_{\mathbf{d}, g}(R^\star, w)$$with strict inequality for some decision problems $\mathbf{d}$.
That is, a deterministic updating rule maximises expected pragmatic utility by the lights of your prior just in case it is a conditionalizing rule for your prior.

As in the case of the DSA above, then, if we assume Deterministic Updating (DU), we can establish PC and DC, and on the back of those AC as well. After all, it is surely irrational to plan to update in one way when you expect another way to guide your actions better in the future; and it is surely irrational to be disposed to update in one way when you expect another to guide you better. And as before there are the same three arguments for AC on the back of PC and DC.

EPUA without Deterministic Updating


How does EPUA fare when we widen our view to include non-deterministic updating rules as well? An initial problem is that it is no longer clear how to define the pragmatic utility of such an updating rule relative to a decision problem at a possible world. Above, we said that, relative to a decision problem $\mathbf{d}$ and a selection function $f$, the pragmatic utility of rule $R$ at world $w$ is the utility of the option that you would choose when faced with $\mathbf{d}$ using the credence function that $R$ mandates at $w$ and $f$: that is, if $E_i$ is true at $w$, then
$$u_{\mathbf{d}, f}(R, w) = u(A^{\mathbf{d}}_{c_i, f}, w).$$
But, if $R$ is not deterministic, there might be no single credence function that it mandates at $w$. If $E_i$ is the piece of evidence you'll learn at $w$ and $R$ permits more than one credence function in response to $E_i$, then there might be a range of different options in $\mathbf{d}$, each of which maximises expected utility relative to a different credence function $c'$ in $C_i$. So what are we to do?

Our response to this problem depends on whether we wish to argue for Plan or Dispositional Conditionalization (PC or DC). Suppose, first, that we are interested in DC. That is, we are interested in a norm that governs the updating rule that records how you are disposed to update when you receive certain evidence. Then it seems reasonable to assume that the updating rule that records your dispositions is stochastic. That is, for each possible piece of evidence $E_i$ and each possible response $c'$ in $C_i$ to that evidence that you might adopt in response to receiving that evidence, there is some objective chance that you will respond to $E_i$ by adopting $c'$. As we explained above, we'll write this $P(R^i_{c'} | E_i)$, where $R^i_{c'}$ is the proposition that you receive $E_i$ and respond by adopting $c'$. Then, if $E_i$ is true at $w$,  we might take the pragmatic utility of $R$ relative to $\mathbf{d}$ and $f$ at $w$ to be the expectation of the utility of the options that each permitted response to $E_i$ (and selection function $f$) would lead us to choose:
$$u_{\mathbf{d}, f}(R, w) = \sum_{c' \in C_i} P(R^i_{c'} | E_i) u(A^{\mathbf{d}}_{c', f}, w)$$
With this in hand, we have the following result:

Theorem 5 Suppose $R$ and $R^\star$ are both updating rules. Then:
  • If $R$ and $R^\star$ are both conditionalizing rules for $c$, and $f$, $g$ are selection functions, then for all decision problems $\mathbf{d}$, $$\sum_{w \in W} c(w) u_{\mathbf{d}, f}(R, w) = \sum_{w \in W} c(w) u_{\mathbf{d}, g}(R^\star, w)$$
  • $R$ is a conditionalizing rule for $c$, and $R^\star$ is a stochastic but not conditionalizing rule, and $f$, $g$ are selection functions, then for all decision problems $\mathbf{d}$,$$\sum_{w \in W} c(w) u_{\mathbf{d}, f}(R, w) \geq \sum_{w \in W} c(w) u_{\mathbf{d}, g}(R^\star, w)$$with strictly inequality for some decision problems $\mathbf{d}$.
This shows the first difference between the DSA and EPUA. The latter, but not the former, provides a route to establishing Dispositional Conditionalization (DC). If we assume that your dispositions are governed by a chance function, and we use that chance function to calculate expectations, then we can show that your prior will expect your posteriors to do worse as a guide to action unless you are disposed to update by conditionalizing on the evidence you receive.

Next, suppose we are interested in Plan Conditionalization (PC). In this case, we might try to appeal again to Theorem 5. To do that, we must assume that, while there are non-deterministic updating rules that we might endorse, they are all at least stochastic updating rules; that is, they all come equipped with a probability function that determines how likely it is that I will adopt a particular permitted response to the evidence. That is, we might say that the updating rules that we might endorse are either deterministic or non-deterministic-but-stochastic. In the language of game theory, we might say that the updating strategies between which we choose are either pure or mixed. And then Theorem 5 will show that we should adopt a deterministic-and-conditionalizing rule, rather than any deterministic-but-non-conditionalizing or non-deterministic-but-stochastic rule. The problem with this proposal is that it seems just as arbitrary to restrict to deterministic and non-deterministic-but-stochastic rules as it was to restrict to deterministic rules in the first place. Why should we not be able to endorse a non-deterministic and non-stochastic rule---that is, a rule that says, for at least one possible piece of evidence $E_i$ in $\mathcal{E}$, there are two or more posteriors that the rule permits as responses, but does not endorse any chance mechanism by which we'll choose between them? But if we permit these rules, how are we to define their pragmatic utility relative to a decision problem and at a possible world?

Here's one suggestion. Suppose $E_i$ is the proposition in $\mathcal{E}$ that is true at world $w$. And suppose $\mathbf{d}$ is a decision problem and $f$ is a selection rule. Then we might take the pragmatic utility of $R$ relative to $\mathbf{d}$ and $f$ and at $w$ to be the average utility of the options that each permissible response to $E_i$ and $f$ would choose when faced with $\mathbf{d}$. That is,$$u_{\mathbf{d}, f}(R, w) = \frac{1}{|C_i|} \sum_{c' \in C_i}  u(A^{\mathbf{d}}_{c', f}, w)$$where $|C_i|$ is the size of $C_i$, that is, the number of possible responses to $E_i$ that $R$ permits. If that's the case, then we have the following:

Theorem 6 Suppose $R$ and $R^\star$ are updating rules. Then if $R$ is a conditionalizing rule for $c$, and $R^\star$ is not deterministic, not stochastic, and not a conditionalizing rule for $c$, and $f$, $g$ are selection functions, then for all decision problems $\mathbf{d}$,
$$\sum_{w \in W} c(w) u_{\mathbf{d}, f}(R, w) \geq \sum_{w \in W} c(w) u_{\mathbf{d}, f}(R^\star, w)$$with strictly inequality for some decision problems $\mathbf{d}$.

Put together with Theorems 4 and 5, this shows that our prior expects us to do better by endorsing a conditionalizing rule than by endorsing any other sort of rule, whether that is a deterministic and non-conditionalizing rule, a non-deterministic but stochastic rule, or a non-deterministic and non-stochastic rule.

So, again, we see a difference between DSA and EPUA. Just as the latter, but not the former, provides a route to establishing DC without assuming Deterministic Updating, so the latter but not the former provides a route to establishing PC without DU. And from both of those, we have the usual three routes to AC. This means that EPUA explains what might be irrational about endorsing a non-deterministic updating rule, or having dispositions that match one. If you do, there's some alternative updating rule that your prior expects to do better as a guide to future action.

Expected Epistemic Utility Argument (EEUA)


The previous two arguments criticized non-conditionalizing updating rules from the standpoint of pragmatic utility. The EEUA and EUDA both criticize such rules from the standpoint of epistemic utility. The idea is this: just as credences play a pragmatic role in guiding our actions, so they play other roles as well---they represent the world;  they respond to evidence; they might be more or less coherent. These roles are purely epistemic. And so just as we defined the pragmatic utility of a credence function at world when faced with a decision problem, so we can also define the epistemic utility of a credence function at a world---it is a measure of how valuable it is to have that credence function from a purely epistemic point of view.

EEUA with Deterministic Updating


We will not give an explicit definition of the epistemic utility of a credence function at a world. Rather, we'll simply state two properties that we'll take measures of such epistemic utility to have. These are widely assumed in the literature on epistemic utility theory and accuracy-first epistemology, and I'll defer to the arguments in favour of them that are outlined there (Joyce 2009, Pettigrew 2016, Horowitz 2019).

A local epistemic utility function is a function $s$ that takes a single credence and a truth value---either true (1) or false (0)---and returns the epistemic value of having that credence in a proposition with that truth value. Thus, $s(1, p)$ is the epistemic value of having credence $p$ in a truth, while $s(0, p)$ is the epistemic value of having credence $p$ in a falsehood. A global epistemic utility function is a function $EU$ that takes an entire credence function defined on $\mathcal{F}$ and a possible world and returns the epistemic value of having that credence function when the propositions in $\mathcal{F}$ have the truth values they have in that world.

Strict Propriety  A local epistemic utility function $s$ is strictly proper if each credence expects itself and only itself to have the greatest epistemic utility. That is, for all $0 \leq p \leq 1$,$$
ps(1, x) + (1-p) s(0, x)$$
is maximised, as a function of $x$ at $p = x$.

Additivity  A global epistemic utility function is additive if, for each proposition $X$ in $\mathcal{F}$, there is a local epistemic utility function $s_X$ such that the epistemic utility of a credence function $c$ at a possible world is the sum of the epistemic utilities at that world of the credences it assigns. If $w$ is a possible world and we write $w(X)$ for the truth value (0 or 1) of proposition $X$ at $w$, this says:$$EU(c, w) = \sum_{X \in \mathcal{F}} s_X(w(X), c(X))$$

We then define the epistemic utility of a deterministic updating rule $R$ in the same way we defined its pragmatic utility above: if $E_i$ is true at $w$, and $C_i = \{c_i\}$, then
$$EU(R, w) = EU(c_i, w)$$Then the standard formulation of the EEUA turns on the following theorem (Greaves & Wallace 2006):

Theorem 7 Suppose $R$ and $R^\star$ are deterministic updating rules. Then:
  • If $R$ and $R^\star$ are both conditionalizing rules for $c$, then$$\sum_{w \in W} c(w) EU(R, w) = \sum_{w \in W} c(w) EU(R^\star, w)$$
  • If $R$ is a conditionalizing rule for $c$ and $R^\star$ is not, then$$\sum_{w \in W} c(w) EU(R, w) > \sum_{w \in W} c(w) EU(R^\star, w)$$
That is, a deterministic updating rule maximises expected epistemic utility by the lights of your prior just in case it is a conditionalizing rule for your prior.
So, as for DSA and EPUA, if we assume Deterministic Updating, we obtain an argument for PC and DC, and indirectly one for AC too.

EEUA without Deterministic Updating


If we don't assume Deterministic Updating, the situation here is very similar to the one we encountered above when we considered EPUA. Suppose $R$ is a non-deterministic but stochastic updating rule. Then, as above, we let its epistemic utility at a world be the expectation of the epistemic utility that the various possible posteriors permitted by $R$ take at that world. That is, if $E_i$ is the proposition in $\mathcal{E}$ that is true at $w$, then$$EU(R, w) = \sum_{c' \in C_i} P(R_{c'} | E_i) EU(c', w)$$Then, we have a similar result to Theorem 5:

Theorem 8 Suppose $R$ and $R^\star$ are updating rules. Then if $R$ is a conditionalizing rule for $c$, and $R^\star$ is stochastic but not a conditionalizing rule for $c$, then
$$\sum_{w \in W} c(w) EU(R, w) > \sum_{w \in W} c(w) EU(R^\star, w)$$

Next, suppose $R$ is a non-deterministic but also a non-stochastic rule. Then we let its epistemic utility at a world be the average epistemic utility that the various possible posteriors permitted by $R$ take at that world. That is, if $E_i$ is the proposition in $\mathcal{F}$ that is true at $w$, then
$$EU(R, w) = \frac{1}{|C_i|}\sum_{c' \in C_i} EU(c', w)$$And again we have a similar result to Theorem 6:

Theorem 9 Suppose $R$ and $R^\star$ are updating rules. Then if $R$ is a conditionalizing rule for $c$, and $R^\star$ is not deterministic, not stochastic, and not a conditionalizing rule for $c$. Then:
$$\sum_{w \in W} c(w) EU(R, w) > \sum_{w \in W} c(w) EU(R^\star, w)$$

So the situation is the same as for EPUA. Whether we assess a rule by looking at how well the posteriors it produces guide our future actions, or how good they are from a purely epistemic point of view, our prior will expect a conditionalizing rule for itself to be better than any non-conditionalizing rule. And thus we obtain PC and DC, and indirectly AC as well.

Epistemic Utility Dominance Argument (EUDA)


Finally, we turn to the EUDA. In EPUA and EEUA, we assess the pragmatic or epistemic utility of the updating rule from the viewpoint of the prior. In DSA, we assess the prior and updating rule together, and from no particular point of view; but, unlike the EPUA and EEUA, we do not assign utilities, either pragmatic or epistemic, to the prior and the rule. In EUDA, like in DSA and unlike EPUA and EEUA, we assess the the prior and updating rule together, and again from no particular point of view; but unlike in DSA and like in EPUA and EEUA, we assign utilities to them---in particular, epistemic utilities---and assess them with reference to those.

EUDA with Deterministic Updating


Suppose $R$ is a deterministic updating rule. Then, as before, if $E_i$ is true at $w$, let the epistemic utility of $R$ be the epistemic utility of the credence function $c_i$ that it mandates at $w$: that is,$$EU(R, w) = EU(c_i, w).$$
But this time also let the epistemic utility of the pair $\langle c, R \rangle$ consisting of the prior and the updating rule be the sum of the epistemic utility of the prior and the epistemic utility of the updating rule: that is,$$EU(\langle c, R \rangle, w) = EU(c, w) + EU(R, w) = EU(c, w) + EU(c_i, w)$$
Then the EUDA turns on the following mathematical fact (Briggs & Pettigrew 2018):

Theorem 10  Suppose $EU$ is an additive, strictly proper epistemic utility function. And suppose $R$ and $R^\star$ are deterministic updating rules. Then:
  • if $\langle c, R \rangle$ is non-conditionalizing, there is $\langle c^\star, R^\star \rangle$ such that, for all $w$ $$EU(\langle c, R \rangle, w) < EU(\langle c^\star, R^\star \rangle, w))$$
  • if $\langle c, R \rangle$ is conditionalizing, there is no $\langle c^\star, R^\star \rangle$ such that, for all $w$ $$EU(\langle c, R \rangle, w) < EU(\langle c^\star, R^\star \rangle, w))$$
That is, if $R$ is not a conditionalizing rule for $c$, then together they are $EU$-dominated; if it is a conditionalizing rule, they are not. Thus, like EPUA and EEUA and unlike DSA, if we assume Deterministic Updating, EUDA gives PC, DC, and indirectly AC.

EUDA without Deterministic Updating


Now suppose we permit non-deterministic updating rules as well as deterministic ones. In this case, there are two approaches we might take. On the one hand, we might define the epistemic utility of non-deterministic rules, both stochastic and non-stochastic, just as we did for EEUA. That is, we might take the epistemic utility of a stochastic rule at a world to be the expectation of the epistemic utility of the various posteriors that it permits in response to the evidence that you obtain at that world; and the epistemic utility of a non-stochastic rule at a world is the average of those epistemic utilities. This gives us the following result:

Theorem 11  Suppose $EU$ is an additive, strictly proper epistemic utility function. Then, if $\langle c, R \rangle$ is not a conditionalizing pair, there is an alternative pair $\langle c^\star, R^\star \rangle$ such that, for all $w$, $$EU(\langle c, R \rangle, w) < EU(\langle c^\star, R^\star \rangle, w)$$And this therefore supports an argument for PC and DC and indirectly AC as well.

On the other hand, we might consider more fine-grained possible worlds, which specify not only the truth value of all the propositions in $\mathcal{F}$, but also which posterior I adopt. We can then ask: given a particular pair $\langle c, R \rangle$, is there an alternative pair $\langle c^\star, R^\star \rangle$ that has greater epistemic utility at every fine-grained world by the lights of $EU$? If we judge updating rules by this standard, we get a rather different answer. If $E_i$ is the element of $\mathcal{E}$ that is true at $w$, and $c'$ is in $C_i$ and $c^{\star \prime}$ is in $C^\star_i$, then we write $w\ \&\ R^i_{c'}\ \&\ R^{\star i}_{c^{\star \prime}}$ for the more fine-grained possible world we obtain from $w$ by adding that $R$ updates to $c'$ and $R^\star$ updates to $c^{\star\prime}$ upon receipt of $E_i$. And let
  • $EU(\langle c, R \rangle, w\ \&\ R^i_{c'}\ \&\ R^{\star i}_{c^{\star \prime}} ) = EU(c, w) + EU(c', w)$
  • $EU(\langle c^\star, R^\star \rangle, w\ \&\ R^i_{c'}\ \&\ R^{\star i}_{c^{\star \prime}} ) = EU(c^\star, w) + EU(c^{\star\prime}, w)$
Then:
Theorem 12  Suppose $EU$ is an additive, strictly proper epistemic utility function. Then:
  • If $\langle c, R \rangle$ is a pseudo-conditionalizing pair, there is no alternative pair $\langle c^\star, R^\star\rangle$ such that, for all $E_i$ in $\mathcal{E}$, $w$ in $E_i$, $c'$ in $C_i$ and $c^{\star\prime}$ in $C^\star_i$, $$EU(\langle c, R \rangle, w\ \&\ R^i_{c'}\ \&\ R^{\star i}_{c^{\star \prime}} ) < EU(\langle c^\star, R^\star \rangle, w\ \&\ R^i_{c'}\ \&\ R^{\star i}_{c^{\star \prime}})$$
  • There are pairs $\langle c, R \rangle$ that are non-conditionalizing and non-pseudo-conditionalizing for which there is no alternative pair $\langle c^\star, R^\star\rangle$ such that, for all $E_i$ in $\mathcal{E}$, $w$ in $E_i$, $c'$ in $C_i$ and $c^{\star\prime}$ in $C^\star_i$, $$EU(\langle c, R \rangle, w\ \&\ R^i_{c'}\ \&\ R^{\star i}_{c^{\star \prime}} ) < EU(\langle c^\star, R^\star \rangle, w\ \&\ R^i_{c'}\ \&\ R^{\star i}_{c^{\star \prime}})$$
Interpreted in this way, then, and without the assumption of Deterministic Updating, EUDA is the weakest of all the arguments. Where DSA at least establishes that your updating rule should be pseudo-conditionalizing for your prior, even if it does not establish that it should be conditionalizing, EUDA does not establish even that.

Conclusion


One upshot of this investigation is that, so long as we assume Deterministic Updating (DU), all four arguments support the same conclusions, namely, Plan and Dispositional Conditionalization, and also Actual Conditionalization. But once we drop DU, that agreement vanishes.

Without DU, DSA shows only that, if we plan to update using a particular rule, it should be a pseudo-conditionalizating rule for our prior; and similarly for our dispositions. As a result, it cannot support AC. Indeed, it can support only the weakest restrictions on our actual updating behaviour, since nearly any such behaviour can be seen as an implementation of a pseudo-conditionalizing rule.

EPUA and EEUA are much more hopeful. Let's consider our updating dispositions first. It seems natural to assume that, even if these are not deterministic, they are at least governed by some objective chances. If so, this gives a natural definition of the pragmatic and epistemic utilities of my updating dispositions at a world---they are expectations of pragmatic and epistemic utilities the posteriors, calculated using the objective chances. And, relative to that, we can in fact establish DU---we no longer need to assume it. With that in hand, we regain DC and two of the routes to AC.

Next, let's consider the updating plans we endorse. It also seems natural to assume that those plans, if not deterministic, might not be stochastic either. And, if that's the case, we can take their pragmatic or epistemic utility at a world to be the average pragmatic or epistemic utility of the different possible credence functions they endorse as responses to the evidence you gain at that world. And, relative to that, we can again establish DU. And with it PC and two of the routes to AC.

Finally, EUDA is a mixed bag. Understanding the epistemic and pragmatic utility of an updating rule as we have just described gives us DU and with it PC, DC, and AC. But if we take a fine-grained approach, we cannot even establish that your updating rule should be a pseudo-conditionalizing rule for your prior.

Proofs

For proofs of the theorems in this post, please see the paper version here.