Everettian chance in no uncertain terms

Jer Steeger and James Read

Supporters agree chance should be derived with uncertain belief, but not on how.

Critics argue these approaches are viciously circular.

We suggest a shift in focus to what Everettians take to be certain.

• They tacitly hold a functional link between chance and certainty, and they agree when we're certain about a branch's isolation.
• This gives a partial interpretation defining trivial (0 or 1) chance values, from which we derive the others.

Everettian chance in no uncertain terms

1. 'Chance' tacitly invokes certainty

2. Certainty of isolation helps with reference class

3. But 'isolation' is ambiguous

4. Probing Everettian 'isolation' with theory sectors

5. Sectors unite and strengthen Born rule derivations

Everettian chance in no uncertain terms

1. 'Chance' tacitly invokes certainty

2. Certainty of isolation helps with reference class

3. But 'isolation' is ambiguous

4. Probing Everettian 'isolation' with theory sectors

5. Sectors unite and strengthen Born rule derivations

1. 'Chance' tacitly invokes certainty

1. 'Chance' tacitly invokes certainty

Saunders (2010), following Papineau (1996):

• (C1) The inferential link. The chance of an event is measured
  (roughly) by (actual) relative frequencies of that event.
• (C2) The credential link. All else being equal, one’s subjective
  degree of belief or credence in an event ought to equal the
  chance of that event.

1. 'Chance' tacitly invokes certainty

Saunders (2010) suggests another:

• (C1) The inferential link. The chance of an event is measured
  (roughly) by (actual) relative frequencies of that event.
• (C2) The credential link. All else being equal, one’s subjective
  degree of belief or credence in an event ought to equal the
  chance of that event.
• (C3) The uncertainty link. Chance events, prior to their
  occurrence, are uncertain.
  • Do we need this?
  • If all outcomes are actual, maybe we don't want it.
    (Greaves (2007) develops this line of thinking.)

1. 'Chance' tacitly invokes certainty

• (C3) The uncertainty link. Chance events, prior to their
  occurrence, are uncertain.
  • Still, we can make sense of 'not knowing what branch we're on.'
    (See Sebens & Carroll (2018) and McQueen & Vaidman (2019).)
  • Even so: should this have anything to do with a specific measure of chance?
  • One way it might: the principle of indifference.
  • Alas, it's well-known that this principle struggles to give consistent recommendations.

1. 'Chance' tacitly invokes certainty

E.g., van Fraassen's (1989) box factory.

1. 'Chance' tacitly invokes certainty

smallest side length:
$$4\text{cm}$$

largest side length:
$$5\text{cm}$$

• Indifference over length? Estimate for face area:  \( 4.5 \times 4.5 = 20.25 \text{cm}^2 \)
• Indifference over face area? New estimate:  \( (16 + 25) /2 = 20.5 \text{cm}^2 \)
• Which do we choose?

1. 'Chance' tacitly invokes certainty

smallest side length:
$$4\text{cm}$$

largest side length:
$$5\text{cm}$$

Ideally, we'd like to get more information about how the box factory works to say that any one estimate is best.

1. 'Chance' tacitly invokes certainty

Motivates a fourth link:

• (C1) The inferential link. The chance of an event is measured
  (roughly) by (actual) relative frequencies of that event.
• (C2) The credential link. All else being equal, one’s subjective
  degree of belief or credence in an event ought to equal the
  chance of that event.
• (C3) The uncertainty link. Chance events, prior to their
  occurrence, are uncertain.
• (C4) The certainty link. Chance values supervene on the
  totality of physical information that is relevant to the
  (actual) relative frequencies of the event.

Everettian chance in no uncertain terms

1. 'Chance' tacitly invokes certainty

2. Certainty of isolation helps with reference class

3. But 'isolation' is ambiguous

4. Probing Everettian 'isolation' with theory sectors

5. Sectors unite and strengthen Born rule derivations

2. Certainty of isolation helps with reference class

What goes wrong in the box factory? Maybe it's the reference class problem.

• I.e., we don't know whether the right reference class varies lengths or varies areas.
• Most Everettians follow subjectivists by analyzing chance in terms of credence. Isn't this only a problem frequentists?

2. Certainty of isolation helps with reference class

Doesn't the reference class problem apply only to frequentists?

• Hájek (2007) argues not: subjectivists fill in the credential link, (C2), by asking us to set our credence according to some expert's: $$\textit{cr} (X \mid \textit{cr}_\text{expert}(X) = x) = x,$$ where \(\textit{cr}_\text{expert}(X)\) could be a Lewisian chance function, a function given by a physical theory, a logical probability, etc.
• And these plausibly give different answers: ‘You can’t serve all your masters at once, so you have to play favorites. But who trumps whom, and which trumps which?’ (2007, p. 597).

2. Certainty of isolation helps with reference class

Doesn't the reference class problem apply only to frequentists?

• For us, physical theory trumps all else.
• We view this as a consistency constraint: if we think QM is authoritative enough to describe the totality of relevant empirical info, it had better also give us our expert credence function!
• First, let's see if we can pin down a 'reference class' (or: unique expert credence function) for the box factory.

2. Certainty of isolation helps with reference class

Maybe information about when a system is isolated will help.

2. Certainty of isolation helps with reference class

Maybe isolation and symmetries will help.

\(1\text{cm}/\text{s}\)

2. Certainty of isolation helps with reference class

Maybe isolation and symmetries will help.

\(1\text{cm}/\text{s}\)

Does this pin down a chance measure?

$$ \textit{ch}_t (s) = \textit{ch}_{t+t'} (s + t'\, \text{mod} \,1) $$

Rods' distribution described by (unknown) \(\textit{ch}_t(x) = \textit{ch}_t(s)\)

$$ \textit{ch}_{t+.5} (s + .5\, \text{mod} \,1) $$

2. Certainty of isolation helps with reference class

Isolation and symmetries will help, sometimes.

\(1\text{cm}/\text{s}\)

Rods' distribution described by (unknown) \(\textit{ch}_t(x) = \textit{ch}_t(s)\)

changing \(t\) does not change frequencies of \(s\)

2. Certainty of isolation helps with reference class

Isolation and symmetries will help, sometimes.

\(1\text{cm}/\text{s}\)

changing \(t\) does not change frequencies of \(s\):

Rods' distribution described by (unknown) \(\textit{ch}_t(x) = \textit{ch}_t(s)\)

2. Certainty of isolation helps with reference class

Isolation and symmetries will help, sometimes.

1. Find a subsystem that is isolated. For \(U\) the factory and \(S\) the cutter dropping at \(t=0\),$$\textit{ch}(s\mid U) = \textit{ch}(s\mid S)$$
2. Find a symmetry of that subsystem. For \(S'\) the cutter dropping at \(t\) and \(s' = s + t \, \text{mod}\, 1\),$$\textit{ch}(s\mid S) = \textit{ch}(s' \mid S')$$
3. Find states identified by that symmetry. If \(t\) doesn't change the outcomes, then we should set \(S=S'\), and $$ \textit{ch}(s\mid S) = \textit{ch}(s' \mid S') = \textit{ch}(s'\mid S)$$

2. Certainty of isolation helps with reference class

Isolation and symmetries will help, sometimes.

1. Find a subsystem that is isolated. For \(U\) the factory and \(S\) the cutter dropping at \(t=0\),$$\textit{ch}(s\mid U) = \textit{ch}(s\mid S)$$
2. Find a symmetry of that subsystem. For \(S'\) the cutter dropping at \(t\) and \(s' = s + t \, \text{mod}\, 1\),$$\textit{ch}(s\mid S) = \textit{ch}(s' \mid S')$$
3. Find states identified by that symmetry. If \(t\) doesn't change the outcomes, then we should set \(S=S'\), and $$ \textit{ch}(s\mid S) = \textit{ch}(s' \mid S') = \textit{ch}(s'\mid S)$$

2. Certainty of isolation helps with reference class

Isolation and symmetries will help, sometimes.

• Ontic Separability Principle (OSP). Suppose that a physical theory \(T\) specifies the totality of physical information that is relevant to the (actual) relative frequencies of an event \(X\) in a system \(U\). Moreover, suppose that \(X\) occurs in an isolated subsystem \(S\) of \(U\), and that a map from \(X\) in \(S\) to \(X'\) in \(S'\) preserves its relative frequency. Then \(T\) should assign chance values to \(X\) in \(S\) that are independent of the environment \(U \setminus S\) and preserved by this map: $$\textit{ch}(s\mid U) = \textit{ch}(s\mid S) = \textit{ch}(s' \mid S')$$
• OSP gets steps 1 and 2; step 3 might uncover some \(S'=S\).

2. Certainty of isolation helps with reference class

The isolated system \(S\) in OSP acts as our reference class.

• When OSP gives strong enough constraints to pin down a single measure, it affords a pragmatic resolution of the reference class problem that applies equally well to Lewisians and frequentists (and others besides).
• ...as long as we're clear about that first step of identifying an isolated subsystem...

Everettian chance in no uncertain terms

1. 'Chance' tacitly invokes certainty

2. Certainty of isolation helps with reference class

3. But 'isolation' is ambiguous

4. Probing Everettian 'isolation' with theory sectors

5. Sectors unite and strengthen Born rule derivations

3. But 'isolation' is ambiguous

'Isolation', like 'chance', is tricky.

• Our strategy is to do what we just did for 'chance': give 'isolation' a thin, functional analysis based on long-standing desiderata among physicists.
• We identify two key links:
  • (I1) The recursion link. Roughly, 'isolated subsystems' should instantiate a recursive structure.
  • (I2) The control link. We can study them in a laboratory.

3. But 'isolation' is ambiguous

Wallace's (2022a; 2022b) theory sectors help us get a handle on recursion.

• Roughly, a sector of a theory identifies a system where the following can be described without reference to any other system:
  • kinematics
  • dynamics
  • symmetries
  • relations of restriction and extension between subsystems

3. But 'isolation' is ambiguous

Wallace's (2022a; 2022b) theory sectors help us get a handle on recursion.

• Example: a subset \(M\) of \(N\) charged particles very far from the others identifies a sector of \(N\)-particle Coulombic electrostatics.
  • kinematics: charges, masses, and positions of the \(M\) particles
  • dynamics: Coulomb's law, \(F = k \frac{|q_1q_2|}{  r^2}\)
  • symmetries: translation, rotation, permutation, etc.
  • relations of restriction and extension between subsystems (e.g., projection onto a further subset \(L\subset M\))

3. But 'isolation' is ambiguous

• So, we flesh out the recursion link with Wallace's (2022a; 2022b) theory sectors, and the case of Newtonian gravity motivates the control link.
  • (I1) The recursion link. Roughly, 'isolated subsystems' should instantiate a recursive structure (like Wallace's theory sectors).
  • (I2) The control link. We can study them in a laboratory.
• There are plausibly many valid ways of fleshing out these links—just like Lewisians and frequentists differ on how to fill in (C1)–(C4).
• Using theory sectors for (I1) meshes nicely with OSP.

3. But 'isolation' is ambiguous

• Using theory sectors for (I1) meshes nicely with OSP. Recall from OSP:$$\textit{ch}(s\mid U) = \textit{ch}(s\mid S) = \textit{ch}(s' \mid S')$$
• The second equality gives a function of states \(S\) and outcomes \(s\) that is preserved by applying a symmetry map to both.
• In other words, it says the chances are the same up to a re-labeling of outcomes. 
• This closely matches what Wallace would call an intrinsic property of a sector.
• Thus, we'll refer to chance's invariance under re-labeling as OSP's intrinsicality condition.

3. But 'isolation' is ambiguous

Our strategy from here follows the recipe for the modified box factory:

1. For isolation, pin down appropriate theory sectors for Everettian quantum mechanics
2. For symmetries, apply OSP's intrinsicality condition to the appropriate symmetries of those sectors
3. Look for symmetry-identified states to pin down a specific chance measure

Everettian chance in no uncertain terms

1. 'Chance' tacitly invokes certainty

2. Certainty of isolation helps with reference class

3. But 'isolation' is ambiguous

4. Probing Everettian 'isolation' with theory sectors

5. Sectors unite and strengthen Born rule derivations

4. Probing Everettian 'isolation' with theory sectors

• According to orthodoxy about quantum mechanics, there are two dynamical rules:
  • unitary dynamics describes self-evolution
  • projective dynamics describes measurement
• In contrast, a traditional selling point of Everettian
  approaches is that they treat all dynamics as unitary.
• Thus, one might think that orthodoxy will frustrate attempts
  to define a theory sector, while Everettian approaches will
  accommodate sectors more straightforwardly.
  • Both of these impressions are false!

4. Probing Everettian 'isolation' with theory sectors

• Thus, one might think that orthodoxy will frustrate attempts
  to define a theory sector, while Everettian approaches will
  accommodate sectors more straightforwardly.
  • Both of these impressions are false!
    • On the orthodox approach, one can helpfully use von Neumann’s measurement scheme to relegate projective dynamics to interactions with macroscopic objects.
    • Similarly, Everettians rely on projections to specify the sense in which their macrostates—branches—are dynamically isolated.
• Everettian or not, use two sectors: one for microstates and one for macrostates.

4. Probing Everettian 'isolation' with theory sectors

• In both cases, microstates 1 and 2 are isolated iff
  • their kinematical states are separable: $$\psi_{12} = \psi_1\otimes \psi_2$$
  • and their dynamics are separable: $$U_{12} = U_1\otimes U_2$$
• In both cases, the isolation of macrostates 1 and 2 implies
  • their kinematical states are orthogonal: $$\psi_1\bot\psi_2$$
  • their dynamics are completely described by their own unitaries
    • Orthodoxy adds: either 1 or 2 occurs after a projection
    • Everettians add: both 1 and 2 occur but are dynamically isolated

4. Probing Everettian 'isolation' with theory sectors

How defensible is the orthodox approach?

• Nice notions of extension and restriction in both 'space' (smaller and larger systems) and 'time' (before and after measurement)
• Two problems:
  • Using projections means measurements must be repeatable; no room for noise.
    • Too strict: like how Earth-bound labs couldn't test our first pass at Newtonian gravity. (Measurement theory addresses this!)
  • It's not clear why some projections are allowed and not others.
    • Too liberal: like saying we don't need the electric field to vanish in electrostatics.

4. Probing Everettian 'isolation' with theory sectors

• The branching criterion gives a natural partial chance assignment.
  • First, partially interpret the Hilbert space norm to mean negligible or nearly impossible.
  • Then, since by (C1) chance ought to be measured by relative frequencies, $$|\psi_{ij}^{t_1 t_2}|= 0 \Longleftrightarrow \textit{ch}(\psi_{ij}^{t_1 t_2})=0$$
• Realistic systems only satisfy the branching criterion approximately; we need to justify applying the partial interpretation to \(|\psi_{ij}^{t_1 t_2}|\approx 0\).
• Following Dawid and Thébault (2025), we can plausibly do so without chance (by, e.g., appealing to the \(\hbar\to 0\) limit).

4. Probing Everettian 'isolation' with theory sectors

What about symmetries?

• Unitary maps are symmetries of both microstates and macrostates; they preserve the form of both Schrödinger dynamics and projective dynamics/restriction.
• We also want repartitionings to be symmetries of macrostates; roughly, different ways of coarse-graining microstates shouldn't make a difference to relative frequencies.
• Unitary maps and repartitionings play the role of time translation in the box factory!

Everettian chance in no uncertain terms

1. 'Chance' tacitly invokes certainty

2. Certainty of isolation helps with reference class

3. But 'isolation' is ambiguous

4. Probing Everettian 'isolation' with theory sectors

5. Sectors unite and strengthen Born rule derivations

5. Sectors unite and strengthen Born rule derivations

The core approach:

1. Find an isolated system, e.g., a two-branch case:$$ (a|\uparrow\rangle + b |\downarrow\rangle)|A\rangle |E\rangle \to a|\uparrow\rangle |A_\uparrow \rangle |E_\uparrow\rangle + b |\downarrow \rangle|A_\downarrow \rangle |E_\downarrow \rangle $$ where the RHS specifies a microstate \(S\) with macrostates \(X_1\) and \(X_2\) corresponding to its summands.  
2. Using OSP, impose the intrinsicality constraint: $$\textit{ch}(X_i\mid S) = \textit{ch}(X_{i'}\mid S) = \textit{ch}({X_i}' \mid S')$$ where \(X_{i'}\) and \({X_i}'\) denote the actions of a repartitioning and a unitary transformation, respectively.
3. Find states identified by these constraints.

5. Sectors unite and strengthen Born rule derivations

• To illustrate, consider the two-branch case:
  $$ (a{\color{orange}|\!\uparrow\rangle} + b {\color{Magenta}|\!\downarrow\rangle})|A\rangle |E\rangle \to a{\color{orange}|\!\uparrow\rangle |A_\uparrow \rangle |E_\uparrow\rangle } + b {\color{Magenta}|\!\downarrow \rangle|A_\downarrow \rangle |E_\downarrow \rangle} $$ The RHS specifies a microstate \(S\) with macrostates \(\color{orange}X_1\) and \(\color{Magenta}X_2\).
• Apply a unitary transformation that swaps the two branches:
  $$ (a{\color{Magenta}|\!\downarrow\rangle} + b {\color{orange}|\!\uparrow\rangle})|A'\rangle |E'\rangle \to a {\color{Magenta}|\!\downarrow \rangle|A_\downarrow \rangle |E_\downarrow \rangle}  + b  {\color{orange}|\!\uparrow\rangle |A_\uparrow \rangle |E_\uparrow\rangle} $$ The RHS specifies a microstate \(S'\) with macrostates \(\color{Magenta}X_1'\) and \(\color{orange}X_2'\).
  • When \(a=b\), \(S'=S\).
  • And the macrostates repartition the originals; i.e., there is a repartitioning such that \({\color{Magenta}X_{1}'} = {\color{Magenta}X_{2'}}\) and \({\color{orange}X_{2}'} = {\color{orange}X_{1'}}\).
• Combining with OSP: $$\textit{ch}({\color{orange}X_1}\mid S) = \textit{ch}({\color{Magenta}X_{1}'}\mid S') = \textit{ch}({\color{Magenta}X_{2'}} \mid S) = \textit{ch}({\color{Magenta}X_{2} } \mid S) $$

5. Sectors unite and strengthen Born rule derivations

• Existing approaches:
  • Wallace (2012) uses rules for rational preference orderings
  • Sebens and Carroll (2018) use self-locating uncertainty
  • McQueen and Vaidman (2019) use symmetry and locality
• Each only deviates slightly from the core. Our approach is closest to McQueen and Vaidman's, but we replace locality with isolation.
• We've explicitly purged any reference to the principle of indifference or self-locating uncertainty; we only need OSP.

What next?

• Two sectors for orthodoxy and Everett... but wasn't Everett motivated by theoretical unity?
  • Is the need for two sectors anathema to an Everettian approach?
  • Or is it a natural accommodation of lessons from stat mech into an Everettian worldview?
• What other physical theories might yield a chance derivation like the modified box factory? Short (2023) gets a start!
• What should we do when the modified box factory recipe fails to solve the reference class problem?

Thank you!