Objective Subjective Frequency Probability

Chance, credence, and frequency answer different questions: what a physical process tends to produce, what an agent should expect, and what a record actually contains. Quantum theory supplies numerical probabilities without making these three notions identical. Distinguishing them prevents a disagreement about meaning from being mistaken for a disagreement about a laboratory prediction.

Required background. Born rule: Translate a specified state and measurement into outcome probabilities.

Chance, belief, and recorded frequency

An objective chance $p$ is a feature attributed to a physical situation. A credence $q$ describes an agent’s uncertainty given information. An observed frequency $f_N=K/N$ is a number calculated from data. A propensity interpretation treats chance as a disposition of an experimental arrangement; a frequentist account emphasizes stable long-run statistical behavior. These labels each cover several philosophical positions.

Even an advocate of objective chance can use subjective uncertainty about its value. If a device has an unknown success chance $\theta$, assigning a distribution $\pi(\theta)$ does not imply that the device’s operation depends on an observer’s beliefs.

The interpretation-specific pages ask what occupies each role in a quantum account. In a stochastic collapse model, a physical noise law supplies chance. In Bohmian equilibrium, the configuration distribution supplies statistical weights despite deterministic motion.

Updating uncertainty about a preparation

Suppose a qubit source is characterized only by the unknown probability $\theta$ of the $Z=+1$ outcome. For $0<\theta<1$ and parameters $\alpha,\beta>0$, a beta prior is

$$\pi(\theta)\propto\theta^{\alpha-1}(1-\theta)^{\beta-1}.$$

For conditionally independent outcomes with $k$ successes in $N$ trials, Bayes’ rule gives a beta posterior with parameters $\alpha+k,\beta+N-k$. Its mean is

$$\mathbb E[\theta\mid k,N]=\frac{\alpha+k}{\alpha+\beta+N}.$$

With a uniform prior, 7 successes in 10 trials yield $8/12=2/3$, whereas the observed frequency is $7/10$. Neither number is the other by definition. They converge under suitable repeated-sampling assumptions, but finite data preserve the distinction.

This example estimates one measurement probability. It does not reconstruct a full quantum state: tomography requires an informationally complete measurement scheme, such as measurements of all three Pauli components or one informationally complete POVM.

Typicality and branch weight

A typicality statement identifies a set of states or histories having measure close to one. It is incomplete until the measure is named. “Most trajectories” by counting, phase-space volume, or squared-amplitude weight can refer to different sets.

Similarly, the Everettian number $|c_a|^2$ is first a mathematical weight in a state decomposition. Explaining why it should guide expectations is an additional interpretive step. Typicality and probability in many worlds treat these issues separately.

What comparisons can establish

Equal numerical assignments imply equal predictions only after the preparation, measurement, and data model are matched. Interpretations may share those predictions while explaining them differently. An empirically distinct theory must provide a different distribution for some specified possible experiment; changing vocabulary alone does not create a test.

Subjective probabilities also need not be arbitrary. Coherence constrains their calculus, evidence constrains reasonable updating, and a quantum theory supplies additional constraints among measurement probabilities.

Exercises

Two agents use beta priors $(1,1)$ and $(9,1)$ and observe 5 successes in 10 trials. Find their posterior means and explain what remains different.

Solution

Their means become $6/12=1/2$ and $14/20=0.7$. Both agents saw the same frequency $1/2$; their prior information differs. With sufficiently informative data from the stated stationary model, prior influence decreases. This convergence claim depends on the model, not merely on calling probabilities subjective.

References

• C. A. Fuchs and R. Schack, “Quantum-Bayesian coherence,” Reviews of Modern Physics 85, 1693–1715 (2013), doi:10.1103/RevModPhys.85.1693.
• D. Wallace, “A formal proof of the Born rule from decision-theoretic assumptions” (2009), arXiv:0906.2718.