Posterior Approval Probability

• Posterior Approval Probability (PAP) is a family of measures that assigns the posterior mass to an approval region, used in binary classification, trial design, and adaptive experiments.
• It adapts to prior shifts by updating calibrated posteriors, ensuring that changes in class priors are reflected in the approval decision criteria.
• In sequential and reinforcement learning contexts, PAP compares posterior credibility against preset safety thresholds to guide deployment decisions.

Posterior Approval Probability (PAP) denotes a posterior or posterior-derived probability attached to an approval event. In the cited literature, that event may be a binary class label being appropriate for a case, a classifier’s approval probability after class-prior shift, a learned controller’s deployment capability exceeding a safety threshold, a Bayesian trial’s decision rule being met, or the posterior credibility of $H_1$ after a significant result (Davis, 2020). The quantity is therefore not a single invariant formula; it is a family of quantities defined by a posterior distribution, an approval set or approval threshold, and a decision rule.

1. Terminological scope and recurring structure

In the cited works, the term appears in several closely related senses. This suggests a common template: a posterior distribution is formed for a latent quantity, and PAP is the posterior mass assigned to an approval region.

Context	Approval quantity	Representative form
Binary decisioning	$p(x)=P(y=1\mid x)$	approve if $p(x)\ge \tau$
Prior shift in classification	updated approval posterior	$p_{\text{new}}(A\mid x)$ from old posteriors and new priors
Deployment validation	capability above threshold	$\Pr(C\ge c_0\mid \text{data})$
Bayesian A/B testing	posterior support for $H_1$	$\Pr(\theta\in(\delta_L,\delta_U)\mid y)$
Significance testing	posterior credibility of $H_1$ or $H_0$	$P(H_1\mid \text{sig})$, $p(x)=P(y=1\mid x)$0

For practical classification systems, PAP is the posterior approval probability $p(x)=P(y=1\mid x)$1 that a case $p(x)=P(y=1\mid x)$2 is approval-worthy rather than denial-worthy (Ferrer et al., 2024). Under class-prior shift, the same expression is updated to reflect new priors without changing class-conditional likelihoods (Davis, 2020). In deployment validation for learned landing controllers, PAP is the posterior probability that the true deployment capability exceeds a required threshold $p(x)=P(y=1\mid x)$3 under finite rollout evidence (Jiang et al., 26 May 2026). In Bayesian A/B tests, the decision statistic is $p(x)=P(y=1\mid x)$4, and approval occurs when that posterior probability exceeds a threshold $p(x)=P(y=1\mid x)$5 (Hagar et al., 2023). In the significance-testing setting, PAP becomes the posterior credibility of approving $p(x)=P(y=1\mid x)$6 after a significant result, or of approving $p(x)=P(y=1\mid x)$7 after a non-significant result (Schad et al., 2019).

2. Classification, approval scoring, and prior shift

For binary classification, the approval posterior is $p(x)=P(y=1\mid x)$8, where $p(x)=P(y=1\mid x)$9 means that approval is appropriate and $p(x)\ge \tau$0 that denial is appropriate. Bayes decision theory selects the action that maximizes expected utility, yielding the utility-dependent threshold

$p(x)\ge \tau$1

Approve if $p(x)\ge \tau$2. Under $p(x)\ge \tau$3–$p(x)\ge \tau$4 loss, $p(x)\ge \tau$5; more costly false approvals push $p(x)\ge \tau$6 upward, and more costly false denials push $p(x)\ge \tau$7 downward (Ferrer et al., 2024).

When the class priors change but $p(x)\ge \tau$8 remains invariant, PAP can be updated directly from old calibrated posteriors. If the original classifier outputs $p(x)\ge \tau$9 and the original priors are $p_{\text{new}}(A\mid x)$0, then recovered likelihood surrogates satisfy

$p_{\text{new}}(A\mid x)$1

With new priors $p_{\text{new}}(A\mid x)$2,

$p_{\text{new}}(A\mid x)$3

In the binary approval case,

$p_{\text{new}}(A\mid x)$4

The odds update is multiplicative,

$p_{\text{new}}(A\mid x)$5

and in logits it becomes an additive shift: $p_{\text{new}}(A\mid x)$6 The direct update is $p_{\text{new}}(A\mid x)$7 per $p_{\text{new}}(A\mid x)$8, while the paper’s eigenvector formulation provides a constructive uniqueness guarantee: by Perron–Frobenius, the positive eigenvector associated with the maximal eigenvalue of $p_{\text{new}}(A\mid x)$9 recovers the likelihood vector up to a common scale (Davis, 2020).

The validity conditions are restrictive. The method is valid under prior shift only; if $\Pr(C\ge c_0\mid \text{data})$0 drifts, the update may be biased. Division by zero must be avoided by enforcing floors such as $\Pr(C\ge c_0\mid \text{data})$1, and if a class had $\Pr(C\ge c_0\mid \text{data})$2 originally but $\Pr(C\ge c_0\mid \text{data})$3 now, likelihood recovery is not identifiable from posteriors alone. The inputs $\Pr(C\ge c_0\mid \text{data})$4 must also be calibrated estimates of $\Pr(C\ge c_0\mid \text{data})$5; otherwise calibration should precede prior adjustment (Davis, 2020).

3. Deployment readiness under finite rollout validation

In deployment-oriented reinforcement-learning validation, PAP is defined for a policy’s capability rather than for a class label. Let $\Pr(C\ge c_0\mid \text{data})$6 denote operating conditions drawn from $\Pr(C\ge c_0\mid \text{data})$7, let $\Pr(C\ge c_0\mid \text{data})$8 be the safe-touchdown event, and define the binary rollout outcome $\Pr(C\ge c_0\mid \text{data})$9 if and only if $H_1$0. The capability of policy $H_1$1 is

$H_1$2

also denoted $H_1$3. With $H_1$4 successes in $H_1$5 i.i.d. validation rollouts and a conjugate prior $H_1$6, the posterior is

$H_1$7

PAP is then

$H_1$8

where $H_1$9 and $\Pr(\theta\in(\delta_L,\delta_U)\mid y)$0 (Jiang et al., 26 May 2026).

This formulation makes approval explicitly threshold-based. The paper uses $\Pr(\theta\in(\delta_L,\delta_U)\mid y)$1, approval threshold $\Pr(\theta\in(\delta_L,\delta_U)\mid y)$2, rejection threshold $\Pr(\theta\in(\delta_L,\delta_U)\mid y)$3, prior $\Pr(\theta\in(\delta_L,\delta_U)\mid y)$4, minimum evidence $\Pr(\theta\in(\delta_L,\delta_U)\mid y)$5, and maximum horizon $\Pr(\theta\in(\delta_L,\delta_U)\mid y)$6. After each rollout or batch, one computes

$\Pr(\theta\in(\delta_L,\delta_U)\mid y)$7

and applies

$\Pr(\theta\in(\delta_L,\delta_U)\mid y)$8

The posterior deployment risk is the complementary tail probability,

$\Pr(\theta\in(\delta_L,\delta_U)\mid y)$9

A lower credible bound $H_1$0 yields the equivalent approval condition $H_1$1 when $H_1$2 (Jiang et al., 26 May 2026).

The empirical results emphasize the difference between empirical success frequency and posterior approval. In the reported PPO/SAC experiments, empirical success and reward can be overconfident indicators under finite evidence. PPO-10M achieved empirical success $H_1$3 out of $H_1$4 but yielded $H_1$5, so it was not approved under the posterior rule. SAC-2M achieved empirical success $H_1$6 out of $H_1$7, yielding $H_1$8, and was approved. The framework is conditional on the assumed operating distribution $H_1$9, assumes i.i.d. rollout generation, and does not provide a worst-case or formal certification guarantee (Jiang et al., 26 May 2026).

4. Bayesian clinical trials, assurance, and fixed-sample design

In Bayesian clinical trials, approval often depends on a posterior summary $H_0$0 such as a posterior probability or posterior predictive probability. A generic rule is: approve if $H_0$1. Examples include $H_0$2, a posterior predictive probability of success in a future stage, or a posterior tail probability transformed to $H_0$3. The conditional operating characteristic is

$H_0$4

and the marginal approval probability under a design prior $H_0$5 is the assurance

$H_0$6

The cited work models the sampling distribution $H_0$7 of posterior decision summaries by Beta or logit-Normal families, calibrated from a small set of simulated scenarios. Under the Beta model,

$H_0$8

Reported results include bias $H_0$9 for type-I-error prediction when training on $P(H_1\mid \text{sig})$0 sample sizes, and absolute bias and RMSE both $P(H_1\mid \text{sig})$1 for power prediction when training on $P(H_1\mid \text{sig})$2 scenarios. In the COVID-19 assurance example, achieving $P(H_1\mid \text{sig})$3 assurance required $P(H_1\mid \text{sig})$4 under an optimistic prior and $P(H_1\mid \text{sig})$5 under a conservative prior, with computation taking approximately $P(H_1\mid \text{sig})$6 minutes on a laptop (Golchi et al., 2023).

In Bayesian A/B tests, the approval statistic is

$P(H_1\mid \text{sig})$7

where $P(H_1\mid \text{sig})$8 may encode superiority, non-inferiority, or practical equivalence. Approval occurs if

$P(H_1\mid \text{sig})$9

The design framework uses prior predictive operating characteristics: power is the probability of approval under a design prior placing all mass in a “green region” $p(x)=P(y=1\mid x)$00, and type I error is the probability of approval under a design prior placing all mass in a “red region” contiguous with $p(x)=P(y=1\mid x)$01. The paper’s scalable algorithm calibrates $p(x)=P(y=1\mid x)$02 and $p(x)=P(y=1\mid x)$03 using two thoroughly explored sample sizes, a low-dimensional mapping of MLEs, and randomized Sobol’ sequences. In its ordinal example, the algorithm returns $p(x)=P(y=1\mid x)$04 and $p(x)=P(y=1\mid x)$05 (Hagar et al., 2023).

A distinctive feature of this literature is the use of design priors and operating-characteristic calibration rather than posterior probabilities alone. This connects PAP to power, type I error, and false discovery rate control, while preserving a posterior decision statistic at the analysis stage (Hagar et al., 2023).

5. Sequential monitoring, predictive approval, and adaptive experimentation

For sequential designs, PAP is indexed by interim look. With planned looks $p(x)=P(y=1\mid x)$06 and cumulative sample sizes $p(x)=P(y=1\mid x)$07, the posterior approval probability at look $p(x)=P(y=1\mid x)$08 is

$p(x)=P(y=1\mid x)$09

and the posterior predictive probability of eventual approval is

$p(x)=P(y=1\mid x)$10

Stop for success if $p(x)=P(y=1\mid x)$11, stop for futility if $p(x)=P(y=1\mid x)$12, and analogously use $p(x)=P(y=1\mid x)$13 and $p(x)=P(y=1\mid x)$14 for predictive success and predictive futility. The central theoretical result is that, under Bernstein–von Mises and MLE asymptotics, the logits of $p(x)=P(y=1\mid x)$15 and $p(x)=P(y=1\mid x)$16 are asymptotically linear in $p(x)=P(y=1\mid x)$17. The proposed SeqSSD procedure therefore estimates operating characteristics and sample sizes using simulations at only two sample sizes. In the PLATINUM-CAN example, the recommended first-look size for rate ratio $p(x)=P(y=1\mid x)$18 was $p(x)=P(y=1\mid x)$19 for Pocock-like thresholds and $p(x)=P(y=1\mid x)$20 for OBF-like thresholds; in the decaffeinated-coffee example, the recommended first-look size was $p(x)=P(y=1\mid x)$21 under the predictive approach and $p(x)=P(y=1\mid x)$22 under the conditional approach (Hagar et al., 1 Apr 2025).

A related literature replaces full Monte Carlo predictive-probability computation with closed-form approximations. Under canonical group-sequential assumptions with information fraction $p(x)=P(y=1\mid x)$23, one-sided interim $p(x)=P(y=1\mid x)$24-value $p(x)=P(y=1\mid x)$25, and final boundary $p(x)=P(y=1\mid x)$26, the approximation is

$p(x)=P(y=1\mid x)$27

For Bayesian primary analyses with interim posterior probability $p(x)=P(y=1\mid x)$28 and final threshold $p(x)=P(y=1\mid x)$29,

$p(x)=P(y=1\mid x)$30

Across dichotomous, time-to-event, and ordinal settings, the reported agreement between approximate and imputation-based predictive probabilities exceeded $p(x)=P(y=1\mid x)$31 at approximately $p(x)=P(y=1\mid x)$32 enrolled, exceeded $p(x)=P(y=1\mid x)$33 for dichotomous and ordinal endpoints at that stage, and was at least $p(x)=P(y=1\mid x)$34 at approximately $p(x)=P(y=1\mid x)$35 enrolled (Marion et al., 2024).

In Bayesian response-adaptive randomization with binary endpoints, PAP is the posterior probability that a treatment satisfies an approval criterion such as superiority against control, threshold exceedance against a fixed benchmark, or being best among $p(x)=P(y=1\mid x)$36 arms: $p(x)=P(y=1\mid x)$37 The cited work develops an exact recursion for these probabilities, shows that the exact algorithm is often the fastest even for up to $p(x)=P(y=1\mid x)$38 treatment arms, and reports that Gaussian approximations can lead to significant power loss and Type I error rate inflation. The practical guidance is explicit: prefer exact computation when $p(x)=P(y=1\mid x)$39, prefer repeated sampling when $p(x)=P(y=1\mid x)$40, and use Gaussian approximations cautiously, particularly for testing in small or imbalanced settings (Kaddaj et al., 2024).

6. Hypothesis approval after significance and sample-size effects

In the hypothesis-testing literature, PAP is the posterior credibility of a hypothesis after observing a binary test outcome. If $p(x)=P(y=1\mid x)$41, $p(x)=P(y=1\mid x)$42, and $p(x)=P(y=1\mid x)$43, then

$p(x)=P(y=1\mid x)$44

$p(x)=P(y=1\mid x)$45

and

$p(x)=P(y=1\mid x)$46

The paper’s central claim is that a statistically significant result does not in general imply a low posterior probability of $p(x)=P(y=1\mid x)$47. With $p(x)=P(y=1\mid x)$48, $p(x)=P(y=1\mid x)$49, and $p(x)=P(y=1\mid x)$50, one obtains $p(x)=P(y=1\mid x)$51; with low power $p(x)=P(y=1\mid x)$52, the same prior yields $p(x)=P(y=1\mid x)$53. Replication is multiplicative on the odds scale: $p(x)=P(y=1\mid x)$54 with $p(x)=P(y=1\mid x)$55 and $p(x)=P(y=1\mid x)$56 (Schad et al., 2019).

A more general sequential analysis of posterior support studies the approval event

$p(x)=P(y=1\mid x)$57

Under the true parameter $p(x)=P(y=1\mid x)$58, the posterior $p(x)=P(y=1\mid x)$59 is a submartingale and approval probabilities for $p(x)=P(y=1\mid x)$60 converge to $p(x)=P(y=1\mid x)$61. Under $p(x)=P(y=1\mid x)$62, $p(x)=P(y=1\mid x)$63 almost surely, but it need not be monotonically decreasing; the paper gives Bernoulli examples in which the expectation goes down, then up, then down again, and other examples with multiple modes. In exponential families,

$p(x)=P(y=1\mid x)$64

so the expectation is eventually strictly decreasing. Log-concavity and unimodality are established for Bernoulli data with a Uniform prior, for Normal data with a Normal prior in stated regimes, and for Exponential data with an Exponential prior (Hart et al., 2022).

A separate calibration approach starts from a reported $p(x)=P(y=1\mid x)$65-value and maps it to an approximate posterior probability using the Adaptive Robust Lower Bound. For $p(x)=P(y=1\mid x)$66, the Sellke–Bayarri–Berger lower bound is

$p(x)=P(y=1\mid x)$67

The adaptive lower bound multiplies this by

$p(x)=P(y=1\mid x)$68

giving posterior odds

$p(x)=P(y=1\mid x)$69

Under equal prior odds,

$p(x)=P(y=1\mid x)$70

The reported properties are large-sample consistency, information consistency in the canonical Normal case, and asymptotic behavior aligned with posterior probabilities while avoiding small-sample problems of BIC and information-consistency failures of certain $p(x)=P(y=1\mid x)$71-priors (Pericchi et al., 2017).

7. Calibration, scoring rules, and posterior quality

Because PAP is often used directly for decisions, downstream inference, or human interpretation, its numerical quality matters independently of any approval threshold. The cited decision-theoretic account argues that expected proper scoring rules, not calibration-only metrics, should be the primary measure of posterior quality. For binary posteriors, canonical examples are log loss

$p(x)=P(y=1\mid x)$72

and Brier score

$p(x)=P(y=1\mid x)$73

Expected PSRs assess both calibration and discrimination. Calibration metrics such as ECE measure only one aspect of posterior quality and should play no role in the assessment of posterior quality; they may instead be used as diagnostic tools during system development. The paper introduces calibration loss,

$p(x)=P(y=1\mid x)$74

with relative calibration loss

$p(x)=P(y=1\mid x)$75

as a PSR-based diagnostic tied to realized improvement under post-hoc calibration (Ferrer et al., 2024).

This emphasis on calibration is consistent with prior-shift adaptation and SVM-based approval scoring. The prior-shift formula requires $p(x)=P(y=1\mid x)$76 to be calibrated estimates of $p(x)=P(y=1\mid x)$77; if not, calibration such as Platt scaling or temperature scaling should be applied before prior updating (Davis, 2020). In SVMs, the raw margin $p(x)=P(y=1\mid x)$78 is not a probability and must be mapped to $p(x)=P(y=1\mid x)$79 by calibration. Two standard mappings are Platt scaling,

$p(x)=P(y=1\mid x)$80

and isotonic regression $p(x)=P(y=1\mid x)$81, with $p(x)=P(y=1\mid x)$82 fit by the Pool-Adjacent-Violators algorithm. The tutorial also describes an implied-posterior construction obtained by reweighting class penalties across many SVMs and then calibrating the aggregated score by isotonic regression (Nalbantov et al., 2019).

Several recurrent misconceptions are therefore explicitly rejected in the cited literature. A small $p(x)=P(y=1\mid x)$83-value is not $p(x)=P(y=1\mid x)$84 (Schad et al., 2019). ECE is not a principled posterior-quality metric (Ferrer et al., 2024). Prior-shift correction is not valid under class-conditional drift (Davis, 2020). Deployment PAP under finite rollouts is not a formal certification guarantee (Jiang et al., 26 May 2026). These constraints do not diminish the importance of PAP; rather, they define the conditions under which a posterior approval probability is interpretable as a decision statistic.