Self-Locating Probabilities

• Self-locating probabilities are a formal measure of the uncertainty an observer has about their indexical position within systems featuring indistinguishable copies.
• They play a critical role in quantum mechanics and cosmology, clarifying phenomena like Everettian branching and duplicated observer scenarios.
• Applications include deriving the Born rule and informing debates on the physical versus epistemic status of probabilities in multiverse and deterministic models.

Self-locating probabilities quantify an agent’s uncertainty about their own “indexical” situation—i.e., about which instance (or “copy”) of physically indistinguishable observers they are—within a broader physical or cosmological context. Such probabilities play a foundational role in quantum mechanics, cosmology, and formal epistemology, particularly in theories admitting multiple actual instantiations of otherwise identical observers, as well as in the interpretation of probability in deterministic frameworks like Everettian (Many-Worlds) quantum mechanics. Their analysis requires attention to both physical symmetries, epistemic constraints, formal updating procedures, and foundational debates about the ontological and pragmatic status of probabilities.

1. Definition and Core Scenarios

Self-locating probabilities arise when an agent is required to assign degrees of belief over possibilities that differ only in the agent’s location, identity, or temporal instance, rather than in the underlying physical state of the universe. Two major types are distinguished:

• Pure self-locating uncertainty (PSL): The uncertainty is exclusively about “which am I?” among subjectively indistinguishable and physically isomorphic observer-instances within a single possible world.
• Superficial self-locating uncertainty (SSL): The uncertainty arises via ignorance about some physical fact (e.g., what time it is, or which branch was selected in a random experiment), so there is a process or structure that can be used to ground credences.

Contexts where self-locating probabilities are essential include:

Scenario	Type	Remarks
Everettian branching	PSL/SSL hybrid	Quantum branches post-decoherence
Duplicated observers	PSL	E.g., “Duplicating Dr. Evil”
Cosmological multiverse	PSL/SSL hybrid	Many identical “pocket universes” or Boltzmann brains
Sleeping Beauty-like puzzles	PSL/SSL hybrid	Duplication + amnesia or protocol-based self-location
Observer-moments in time	SSL	Ignorance about time index given “self”

In all cases, the agent’s uncertainty remains even with complete knowledge of the global physical or algorithmic state, due to their inability to determine their own indexical "location."

2. The Everettian Context: Decoherence, Branching, and the Epistemic Separability Principle

In Everettian quantum mechanics, the universal wavefunction evolves deterministically without collapse, resulting in a branching structure after decoherence:

• After a quantum measurement plus decoherence, the global wavefunction can be written as a sum over orthogonal “branches,” each corresponding to a different distinct outcome, with its own copy of the observer-entangled with the measurement result and environment.
• During the post-decoherence, pre-observation interval, the observer knows the total wavefunction but is “self-locating uncertain” about which branch they inhabit (Sebens et al., 2014, Carroll et al., 2014).

Key formal mechanism:

The Epistemic Separability Principle (ESP) posits that the credence one should assign to being any one of several observers with identical experiences depends solely on the observer–detector subsystem’s reduced density matrix, and is independent of the environment:

$P(O_i | \Psi) = P(O_i | \widehat{\rho}_{AD}),$

where $O_i$ indexes the measurement outcomes and $\widehat{\rho}_{AD}$ is the reduced density matrix for observer plus detector.

This principle prevents environmental or distant “branching” events from artificially influencing local probabilistic assignments. The only way probabilities can rationally differ between branches is if the reduced state of the observer–detector subsystem differs (Sebens et al., 2014, Carroll et al., 2014, Tappenden, 2017).

3. Derivation of the Born Rule from Self-Locating Uncertainty

Self-locating probabilities in quantum mechanics are used to derive the Born rule:

• Equal-amplitude branching: If the universal state can be written as a sum of $N$ orthogonal branches of equal modulus, ESP and self-locating symmetry dictate that each branch receives probability $1/N$.
• General case: For branches with amplitudes $\alpha_i$, the probability assigned is $|\alpha_i|^2$, justified by:
  • Unitary transformations on the environment allowing the decomposition of unequal-amplitude branches into equal-amplitude ones, to which the above symmetry applies.
  • Normalization over all branches (Sebens et al., 2014, Carroll et al., 2014, Ridley, 2021, Ridley, 2 Oct 2025, Saunders, 11 May 2025).

General formula:

$P(O_i | \Psi) = |\langle O_i | \Psi \rangle|^2$

For cosmological multiverses with multiple observer copies (possibly located in different regions or branches),

$$P(i|U) = \frac{w_i}{\sum_j w_j}, \quad w_i = |\alpha_i|^2$$

where each $w_i$ represents the amplitude-squared “weight” of copy $i$ (Sebens et al., 2014, Carroll et al., 2014).

Illustrative example:

For a quantum Sleeping Beauty scenario in which duplication occurs only for one measurement outcome, the overall credence is distributed by self-locating principles spanning both quantum and classical duplication—quantitatively, via the normalization of branch weights plus indifference across duplicates inside a branch (Sebens et al., 2014).

4. Updating, Indifference, and Protocol Dependence

Updating self-locating probabilities involves both technical challenges and conceptual distinctions:

• Naive conditioning (simply restricting probabilities to observed events in the “naive space”) is often unjustified—protocols for generating observations must be considered.
• CAR (Coarsening At Random) condition: Naive conditioning is only warranted if the probability of the observation given the underlying world is independent of which world within the observed event obtained. This is a rare condition; most protocols (like Monty Hall) violate CAR (Grunwald et al., 2014).
• Generalized updating:
  • Jeffrey conditioning is justified under a generalized CAR, where the reassignment of probabilities over partitions is protocol-invariant.
  • Minimizing relative entropy (MRE) updating rarely produces “correct” self-locating probabilities unless the constraints are “Jeffrey-like”—such cases are exceptional.
• Indifference and Bertrands’s paradox: Attempts to resolve pure self-locating probabilities via the principle of indifference are undermined by the arbitrariness of partitions and the absence of a physical process to privilege one partition over another. This is especially acute in PSL contexts (Adlam, 9 Sep 2024).

5. Modal, Logical, and Structural Formalisms

Several structural and logical frameworks capture and extend the notion of self-locating probability:

• Modal logics of qualitative probability (“belief as willingness to bet”): Belief is modeled by thresholds on probability (e.g., $B\varphi$ holds if $P(\varphi) > c$), and knowledge entails probability 1. Soundness and completeness of these frameworks (notably for $c = 1/2$) can be established with appropriate axioms (e.g., Scott’s condition), and properties such as introspection and strong commitment (Eijck et al., 2014).
• Probabilistic entailment in enriched propositional logic: Probabilities are assigned to formulas or valuations; sound and weakly complete axiom systems exist for expressing self-locating constraints and reasoning about probabilistic indexical claims, especially within finite or algorithmically manageable spaces (Rasga et al., 2016).
• Subclassical probability in generalized event structures: When event structures transcend classical or even quantum logic (e.g., via overlapping Boolean contexts), probability assignments obey additivity within each context, but global self-locating assignments may remain underdetermined, mirroring the PSL ambiguity (Svozil, 2015).

6. Controversies and Critical Perspectives

The assignment and interpretation of self-locating probabilities are subject to foundational controversies:

• Ontological status: There is debate over whether self-locating probabilities—especially PSL credences—are objectively mandated by physical or rational principles, or if they merely codify subjective “caring measures” (Adlam, 9 Sep 2024). Work by Kent (Kent, 2014) stresses ambiguities in branch definition and the legitimacy of “pre-observation splitting” of observers, challenging the cogency of self-locating uncertainty as a basis for probabilities.
• Non-uniqueness issues: In the absence of an external stochastic mechanism, protocols such as indifference or branch counting are underdetermined. Bertrand’s paradox is cited as demonstrating that in continuous spaces, there is no objective way to partition pure self-locating uncertainty (Adlam, 9 Sep 2024).
• Scope of scientific inference: The reliance on self-locating probabilities in cosmology, anthropics, and simulation arguments has been critiqued; if self-locating assignments are unconstrained, predictions about “typicality” in multiverses, the likelihood of Boltzmann brains, or the simulation hypothesis lack objective grounding (Adlam, 9 Sep 2024, Chen, 2020).

7. Extensions and Applications

The technical apparatus of self-locating probabilities extends to various foundational and applied domains:

• Time-symmetric and retrocausal quantum frameworks: Probabilities are associated with “fractions” of wavefunction or history-space, such as in Keldysh contour-based, fixed-point, or two-state-vector approaches (Ridley, 2021, Ridley, 2 Oct 2025). The measure of existence associated with a branch or history determines the observer’s rational credence, reproducing the Born rule:

$$m(h) = \frac{\Delta \Psi_h}{\sum_{h'} \Delta \Psi_{h'}}$$

for history $h$ characterized by fixed points in time.

• Physical probability and locality postulates: The distinction between physical (ontic) and epistemic probability clarifies that, in no-collapse quantum theories, microstate-counting arguments (with a critical locality postulate) imply that equiamplitude expansions support equiprobable microstates, thus recovering the Born rule and providing a frequentist grounding for self-locating probabilities (Saunders, 11 May 2025).
• Cosmological and temporal self-location: When analyzing the arrow of time or observer moments distributed across cosmological histories, self-locating probabilities are essential in formalizing updating and in discussing empirical underdetermination inherent in e.g., the Past Hypothesis and Fluctuation Hypothesis debates (Chen, 2020).

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Self-locating probabilities form a central conceptual and technical tool in current foundational research, providing the key to reconciling the deterministic dynamics of quantum theory (and certain cosmological models) with the observed appearance and structure of uncertainty. Their precise formulation, however, depends critically on physical symmetries, updating mechanisms, the status of indexical claims, and ongoing debates in the physics and philosophy of probability.