Pre-Probabilities: Foundations and Applications

• Pre-probabilities are preliminary assignments or weights that require normalization, updating, or extension to form final operational probabilities.
• They manifest in diverse fields such as quantum cosmology, Bayesian inference, prequential forecasting, and logical probability, each addressing unique challenges like self-location uncertainty and prior sensitivity.
• The proper calibration and transformation of pre-probabilities are crucial for accurate predictions, robust model comparisons, and coherent decision-making in various scientific domains.

Pre-probabilities are quantities that precede fully operational probabilities, but the term is not univocal across disciplines. In quantum cosmology it denotes raw, unnormalized weights assigned to observations before explicit normalization; in Bayesian uncertainty quantification it denotes prior probabilities specified before the current dataset is observed; in prequential imprecise probability it denotes interval forecasts for the next outcome; in logic it denotes partial belief assignments or generic priors fixed before extension to a full probability on sentences; and in the de Finetti coherence tradition it is naturally realized as previsions of conditional random quantities. A common schema is that a pre-probability is not yet the final observational probability, but rather an antecedent object from which such probabilities are derived by normalization, updating, extension, or coherence constraints (Page, 2017, Zhang et al., 2017, Persiau et al., 2023, Hutter et al., 2012, Hasse, 2014, Sanfilippo et al., 2019).

1. Core meanings and recurring structure

Across the literature, “pre-probability” marks an intermediate level between raw formal structure and an operational probability assignment. In some settings the intermediate object is numerical but unnormalized; in others it is a prior, a partial belief assignment, or a set-valued forecast. What unifies these uses is that the object constrains or generates later probabilities without itself yet being the final normalized probability of interest.

Domain	Pre-probability object	Operational step
Quantum cosmology	$w_j=\langle A_j\rangle=\mathrm{Tr}(\rho A_j)$ or $P_j=\langle A_j\rangle$	Normalize by $\sum_k \mathrm{Tr}(\rho A_k)$
Bayesian multimodel UQ	$p(M_k)$ and $p(\theta_k\mid M_k)$	Update with $D$ to obtain posterior model and parameter probabilities
Prequential imprecise forecasting	$I_k=[\underline p_k,\overline p_k]$	Induce $\underline E_I,\overline E_I$ and test via superfarthingales
Logic on sentences	Partial beliefs $p_0$ or generic priors	Extend to a full $\mu:\mathrm{Sent}(L)\to[0,1]$

This taxonomy suggests two broad families. One family treats pre-probabilities as raw weights that must still be normalized. The other treats them as prior or partial assignments that must still be updated or extended. The distinction matters because different mathematical constraints dominate: positivity and normalization in the first family, coherence and extendability in the second.

2. Quantum pre-probabilities: raw weights before Born-style normalization

In Don Page’s cosmological framework, standard Born probabilities

$P_j=\langle A_j\rangle$0

are adequate for small, local experiments but not for many cosmological purposes. The central problem is that in a vast or infinite universe an observation may occur in many places and times, so no projector fixed independently of the quantum state can in general encode self-locating uncertainty. Page’s example is two identical observers measuring $P_j=\langle A_j\rangle$1 of an electron, one seeing up and the other down: before self-location is resolved, the probability of “spin up” must lie strictly between $P_j=\langle A_j\rangle$2 and $P_j=\langle A_j\rangle$3, yet a single projector cannot yield that intermediate value when both outcomes occur somewhere with certainty. The same setting motivates concern about measure ambiguities, typicality, and Boltzmann brain domination (Page, 2017).

The proposed generalization, “Sensible Quantum Mechanics” (SQM), assigns to each observation $P_j=\langle A_j\rangle$4 a positive awareness operator $P_j=\langle A_j\rangle$5 and defines the pre-probability

$P_j=\langle A_j\rangle$6

The corresponding observational probability is obtained only after explicit normalization,

$P_j=\langle A_j\rangle$7

The set $P_j=\langle A_j\rangle$8 need not be a POVM: the operators are not required to be projectors, mutually orthogonal, complete, or to sum to the identity. The framework is deterministic in the sense that all observations with positive measure occur; the normalized $P_j=\langle A_j\rangle$9 are measures of existence for conscious perceptions and are intended for Bayesian model comparison rather than collapse dynamics.

The spacetime-integral construction makes the pre-probabilities explicitly cosmological. If the universal state contains semiclassical spacetimes $\sum_k \mathrm{Tr}(\rho A_k)$0 with expectation weights $\sum_k \mathrm{Tr}(\rho A_k)$1, if $\sum_k \mathrm{Tr}(\rho A_k)$2 is the expectation value of the localized projector corresponding to observation $\sum_k \mathrm{Tr}(\rho A_k)$3 at spacetime point $\sum_k \mathrm{Tr}(\rho A_k)$4 in $\sum_k \mathrm{Tr}(\rho A_k)$5, if $\sum_k \mathrm{Tr}(\rho A_k)$6 is a location-dependent weight, and if $\sum_k \mathrm{Tr}(\rho A_k)$7 is an intrinsic weight for the efficacy of the corresponding matter configuration in producing consciousness, then

$\sum_k \mathrm{Tr}(\rho A_k)$8

and

$\sum_k \mathrm{Tr}(\rho A_k)$9

This permits region weighting, intrinsic weighting of different observer-types, and explicit control of late-time divergences. The concrete proposal combines volume averaging on preferred hypersurfaces, $p(M_k)$0, with Agnesi weighting, $p(M_k)$1, so that

$p(M_k)$2

The combined choice is designed to render the spacetime integral finite and suppress late-time vacuum fluctuations that would otherwise dominate by producing Boltzmann brain perceptions.

The toy model with two identical copies exhibits the logic transparently. With equal region weights $p(M_k)$3 and

$p(M_k)$4

the state yields $p(M_k)$5, $p(M_k)$6, and therefore

$p(M_k)$7

The same logic extends to decoherent histories by taking $p(M_k)$8, so that the pre-probabilities are $p(M_k)$9 and ordinary normalization is replaced, when needed, by Page’s global normalization.

A different quantum use of the term appears in Sutherland’s derivation of the Born rule. There the amplitudes $p(\theta_k\mid M_k)$0 function as pre-probabilities because they determine the size of the accessible region of final states for outcome $p(\theta_k\mid M_k)$1. With a uniform density on the disk $p(\theta_k\mid M_k)$2, where $p(\theta_k\mid M_k)$3, the probability of outcome $p(\theta_k\mid M_k)$4 is proportional to disk area and therefore becomes

$p(\theta_k\mid M_k)$5

In this usage the pre-probability is not an unnormalized operator expectation but an amplitude whose modulus determines a measure over available final states (Sutherland, 2020).

3. Bayesian and statistical pre-probabilities: priors before current data

In Bayesian multimodel uncertainty quantification, pre-probabilities are the prior probabilities specified before observing the current dataset $p(\theta_k\mid M_k)$6. They include prior model-form probabilities $p(\theta_k\mid M_k)$7 over candidate distributions and prior parameter probabilities $p(\theta_k\mid M_k)$8 within each model. These enter both model-form inference and parameter inference:

$p(\theta_k\mid M_k)$9

$D$0

$D$1

The same framework defines the multimodel posterior predictive

$D$2

The paper also distinguishes a “pre-prior,” the initial noninformative prior used with historical data $D$3 to construct an informative prior $D$4 for the current analysis. This architecture is explicitly motivated by small-sample regimes, where prior choices strongly affect both model-form and parametric posteriors (Zhang et al., 2017).

The plate buckling study makes the prior sensitivity concrete. With small datasets, strong model-form priors can dominate posterior model probabilities, and informative but inappropriate parameter priors can induce persistent bias. The paper reports that an ABS-A parameter prior led model-form inference to converge to the wrong Gamma model even at $D$5. In propagation, inappropriate priors produced narrow but incorrect probability bounds. For the mean buckling strength, the true value was $D$6, whereas under the ASTM-A7 prior with equal model-form priors the reported $D$7 CDF range was $D$8, excluding the true mean. For the failure probability at threshold $D$9, the true value was $I_k=[\underline p_k,\overline p_k]$0, whereas the ASTM-A7 prior gave $I_k=[\underline p_k,\overline p_k]$1 and the ABS-A prior gave $I_k=[\underline p_k,\overline p_k]$2, both excluding the true value. The study therefore treats pre-probabilities not as harmless preliminaries but as structurally consequential inputs to multimodel inference and propagation.

A more foundational statistical use appears in the least-sensitivity program for scarce data. There pre-probabilities are priors in Bayesian inference, and the proposal is to choose them so that the resulting assignment is least sensitive to stipulated variations of the prior. In the continuous case this leads to minimizing the Fisher information functional

$I_k=[\underline p_k,\overline p_k]$3

subject to the available-information constraints. Writing $I_k=[\underline p_k,\overline p_k]$4 yields the Euler–Lagrange equation

$I_k=[\underline p_k,\overline p_k]$5

with $I_k=[\underline p_k,\overline p_k]$6. In the discrete case the analogous robustness criterion is a Rényi distance of order $I_k=[\underline p_k,\overline p_k]$7 between a distribution and its shifted version. The same paper emphasizes that with abundant data the posterior becomes dominated by the likelihood and sensitivity to the pre-probability diminishes, whereas with scarce data the assignment is driven by the prior (Dimitrov, 2012).

4. Prequential pre-probabilities: interval forecasts and randomness on the fly

In prequential imprecise probability, the pre-probability for the next outcome is an interval forecast $I_k=[\underline p_k,\overline p_k]$8 announced before the binary outcome $I_k=[\underline p_k,\overline p_k]$9 is revealed. An infinite prequential path is therefore

$\underline E_I,\overline E_I$0

The interval $\underline E_I,\overline E_I$1 is interpreted as the set of plausible success probabilities for $\underline E_I,\overline E_I$2. Precise forecasts are the singleton case $\underline E_I,\overline E_I$3. The interval induces coherent lower and upper expectations on gambles $\underline E_I,\overline E_I$4,

$\underline E_I,\overline E_I$5

These operators satisfy boundedness, constant additivity, monotonicity, positive homogeneity, subadditivity, and continuity under pointwise limits (Persiau et al., 2023).

The betting semantics is encoded through capital processes. If the sceptic chooses a gamble $\underline E_I,\overline E_I$6 after observing $\underline E_I,\overline E_I$7, admissibility requires

$\underline E_I,\overline E_I$8

and capital evolves by

$\underline E_I,\overline E_I$9

with a no-borrowing constraint $p_0$0. The linear-stake family

$p_0$1

illustrates the robust constraint

$p_0$2

The abstract version of such admissible capital processes is the prequential superfarthingale, a function $p_0$3 satisfying

$p_0$4

for all finite prequential situations $p_0$5 and rational intervals $p_0$6.

Game-randomness is then defined by boundedness of all lower semicomputable test superfarthingales along a non-degenerate prequential path. Non-degeneracy means that the realized outcome never has worst-case probability $p_0$7 relative to the announced interval. The paper proves that, under non-degenerate recursive rational forecasting systems, this prequential notion coincides with the standard imprecise Martin-Löf randomness notion for a compatible forecasting system. It also establishes calibration-type bounds for computably selected subsequences:

$p_0$8

Thus the interval forecast acts as a one-step pre-probability whose operational meaning is fixed by coherent upper expectations and by the impossibility of computable betting strategies that make capital explode on a game-random path.

5. Logical pre-probabilities: partial beliefs, sentence probabilities, and generic priors

In higher-order logic, a probability on sentences is a function $p_0$9 satisfying validity bounds, complementarity, logical invariance, monotonicity, generalized finite additivity, and the usual conditional-probability rule on sentences. Within this setting, pre-probabilities are partial specifications of belief on finitely many sentences,

$\mu:\mathrm{Sent}(L)\to[0,1]$0

For a finite family $\mu:\mathrm{Sent}(L)\to[0,1]$1 the induced logical atoms are

$\mu:\mathrm{Sent}(L)\to[0,1]$2

The extension problem becomes a linear feasibility problem for the atom masses $\mu:\mathrm{Sent}(L)\to[0,1]$3:

$\mu:\mathrm{Sent}(L)\to[0,1]$4

with the additional requirement $\mu:\mathrm{Sent}(L)\to[0,1]$5 whenever $\mu:\mathrm{Sent}(L)\to[0,1]$6 has no relevant model. These conditions are necessary and sufficient for extendability. The same framework introduces Gaifman probabilities, which treat quantifiers via limits of finite instantiations, and Cournot probabilities, which assign positive mass to any sentence with a separating model. The combination Gaifman+Cournot is what supports confirmation of universal hypotheses and learning in the limit; in particular, if $\mu:\mathrm{Sent}(L)\to[0,1]$7, then

$\mu:\mathrm{Sent}(L)\to[0,1]$8

The “least dogmatic” extension relative to a prior $\mu:\mathrm{Sent}(L)\to[0,1]$9 is the KL-projection characterized by

$P_j=\langle A_j\rangle$00

with the Lagrange multipliers chosen to enforce the original partial constraints (Hutter et al., 2012).

A more radical logical construction identifies pre-probabilities with generic priors determined purely from propositional form. The primitive rules are the Cox/Jaynes product and sum rules, together with permutation symmetry and a negation symmetry that swaps a basic proposition $P_j=\langle A_j\rangle$01 with its negation $P_j=\langle A_j\rangle$02 everywhere. Under minimal assumptions this yields

$P_j=\langle A_j\rangle$03

and therefore a uniform measure over truth assignments. Probabilities then reduce to model counts:

$P_j=\langle A_j\rangle$04

The principle of indifference is thereby recovered as a combinatorial theorem. Under exactly-one assumptions $P_j=\langle A_j\rangle$05, one obtains

$P_j=\langle A_j\rangle$06

Under exhaustivity alone $P_j=\langle A_j\rangle$07, one instead obtains

$P_j=\langle A_j\rangle$08

If one assumes only that there exists a space of possibilities of unknown finite size, represented by $P_j=\langle A_j\rangle$09, the resulting probability for a distinguished label tends to

$P_j=\langle A_j\rangle$10

Here the pre-probability is neither a prior density nor a partial finite constraint set, but a generic prior fixed by syntactic symmetry and normalized model counting (Hasse, 2014).

6. Conditional previsions, iterated conditionals, and terminological boundaries

In the coherence approach derived from de Finetti, pre-probabilities are previsions of conditional and iterated conditional events treated as conditional random quantities. A conditional event $P_j=\langle A_j\rangle$11 is represented as

$P_j=\langle A_j\rangle$12

where $P_j=\langle A_j\rangle$13 is both its assessed conditional probability and its “void” value on $P_j=\langle A_j\rangle$14. More generally, for a random quantity $P_j=\langle A_j\rangle$15,

$P_j=\langle A_j\rangle$16

with $P_j=\langle A_j\rangle$17. Conjunctions and iterated conditionals are again random quantities. If $P_j=\langle A_j\rangle$18 and $P_j=\langle A_j\rangle$19, then the conjunction has prevision $P_j=\langle A_j\rangle$20 constrained by

$P_j=\langle A_j\rangle$21

and the iterated conditional satisfies

$P_j=\langle A_j\rangle$22

with

$P_j=\langle A_j\rangle$23

This formalism avoids Lewis’s triviality because import–export fails: $P_j=\langle A_j\rangle$24 is not in general reducible to $P_j=\langle A_j\rangle$25 or any truth-functional analogue (Sanfilippo et al., 2019).

The same framework distinguishes bare conditional information from strengthened antecedents. If $P_j=\langle A_j\rangle$26, then

$P_j=\langle A_j\rangle$27

so simply learning the conditional does not raise the antecedent probability. But several strengthened antecedents do raise it. The paper proves

$P_j=\langle A_j\rangle$28

and, under explicit background conditions,

$P_j=\langle A_j\rangle$29

It also gives sharp bounds for Affirmation of the Consequent. If $P_j=\langle A_j\rangle$30 and $P_j=\langle A_j\rangle$31, then coherent values $P_j=\langle A_j\rangle$32 satisfy

$P_j=\langle A_j\rangle$33

Finally, the paper interprets “independence” of conditional events as uncorrelation:

$P_j=\langle A_j\rangle$34

A terminological boundary is useful here. Predictive probability in clinical-trial monitoring is not a pre-probability in the senses above, but a posterior probability of future trial success conditional on current data:

$P_j=\langle A_j\rangle$35

In the canonical normal setup the approximation is

$P_j=\langle A_j\rangle$36

or, for Bayesian primary analyses,

$P_j=\langle A_j\rangle$37

This is conceptually downstream from the pre-probability notions surveyed above: it is already a predictive probability, not a raw weight, prior, interval forecast, or prevision (Marion et al., 2024).

Taken together, these usages show that “pre-probability” is a family resemblance term rather than a single mathematical object. In quantum cosmology it is an unnormalized observational weight; in Bayesian statistics it is a prior probability; in prequential forecasting it is a one-step interval constraint; in logic it is a partial or generic prior awaiting extension; and in conditional-event semantics it is a coherent prevision. The unifying idea is always anteriority: a pre-probability is specified before the final probability of observation, prediction, or belief is fixed.