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A normative Bayesian inference model of retrospective confidence. For two exhaustive alternatives with equal prior probabilities, let $L=\log [p(E\mid H_1)/p(E\mid H_0)]$ be the log-likelihood ratio. Bayes' theorem gives posterior odds $e^L$. Choosing the alternative with the larger posterior yields $P(\text{correct}\mid E)=(1+e^{-|L|}{)}^{-1}$. Unequal priors add prior log odds; choosing a different action changes which posterior probability is appropriate. This calculation does not establish that an observer knows the true likelihoods.

A CUSUM accumulates evidence for a change in a monitored probability distribution. With $W_t$ a log-likelihood ratio increment favoring a specified changed distribution, reset the accumulated score at zero and signal when it exceeds a chosen boundary $h$. For $C_0=0$, $C_t$ is the largest nonnegative sum of successive increments ending at $t$. This permits repeated detection of a change with an unknown onset, rather than a single fixed-start test.

A fast initial response CUSUM starts a CUSUM with a positive head start below its signalling boundary. Before its first reset, it accumulates a fixed-start log-likelihood ratio like a sequential probability ratio test; after resetting to zero it behaves as an ordinary CUSUM. It gives faster initial detection when the process may already be out of control, at the cost of changed initial false-alarm behavior.

For equal prior probabilities and independent and identically distributed observations on a finite alphabet, minimum sum of errors in a simple hypothesis test givesFor nonnegative $a,b$ and $0<s<1$, $\min (a,b)\le a^s{b}^{1-s}$. Applying this termwise and using the independence of random variables to factor the sum yieldsOptimizing $s$, with an infimum if needed, proves $p_n^{\ast }\le \frac{1}{2}{e}^{-nC(P,Q)}$. This short argument gives a bound for every $n$. The theorem identifying the optimal error exponent also needs the converse: no decision rule can achieve a strictly larger limiting exponent. A log-likelihood ratio is a sum of independent contributions, so large deviation principle supplies the relevant converse idea.

Stanford's archive records a technical report by Herman Chernoff, dated November 1951, on asymptotic efficiency of hypothesis tests based on sums of observations.^[ref] This is historical source metadata, rather than a claim to have inspected the report's full proof. The modern theorem cited above connects that efficiency question to the log-likelihood ratio accumulated across repeated observations.

A log-likelihood ratio compares two likelihoods on an additive scale. For independent observations its increments add. Under distributions $P_0,P_1$ with suitable integrability and common support, $E_0[\log (d{P}_1/d{P}_0)]=-D(P_0\Vert P_1)\le 0$ and $E_1[\log (d{P}_1/d{P}_0)]=D(P_1\Vert P_0)\ge 0$, where $D$ is Kullback-Leibler divergence. This sign change motivates likelihood-based sequential monitoring.

Let ${\mathcal{G}}_t=\sigma (X_1,\dots ,X_t)$. Under $P_{{\theta }_{\star }}$, the true probability density function is positive and finite at every observation almost surely. Define the sequential likelihood ratioA zero numerator sets the product to zero thereafter. Thus $M_t$ is a well-defined nonnegative process, and $L_t({\theta }_{\star })=\log M_t$, with $\log 0=-\infty$.

The next observation is independent of ${\mathcal{G}}_{s-1}$, while ${\hat{\theta }}_{s-1}$ is measurable with respect to that sigma-algebra. Its conditional expectation is thereforeConsequently $\mathbb{E}(M_s\mid {\mathcal{G}}_{s-1})\le M_{s-1}$; induction also gives $\mathbb{E}{M}_s\le 1$. Thus $(M_t)$ is a nonnegative supermartingale for the filtration $({\mathcal{G}}_t)$. It need not be a martingale: an alternative distribution can put mass outside the support of the true distribution.

The maximal inequality for a nonnegative supermartingale, the corresponding form of the Ville inequality, givesThe strict inequality in the definition of the confidence set correctly becomes a non-strict crossing event.

There is a minor definitional omission in the source: for false candidates $\theta$, the printed sum of log-likelihood ratios can involve $0/0$ or both infinite signs. To define the entire confidence set without a positivity assumption, use the logarithm of the ratio of the two full products, ${\prod }_s{p}_{{\hat{\theta }}_{s-1}}(X_s)$ and ${\prod }_s{p}_{\theta }(X_s)$, taking $0/a=0$ for $a>0$, $a/0=+\infty$ for $a>0$, and, for example, $0/0=1$. This is a measurable extension of the printed formula wherever that formula is defined. The choice at $0/0$ cannot affect coverage of the truth, since its full denominator is positive almost surely. Alternatively, strict positivity of every density on the common support removes the omission.

Use a uniform independent DNA background, so each nucleotide has probability $1/4$. Conditional on length five, its sequence probability is $4^{-5}=1/1024$. The requested score, often called log-odds in sequence analysis, is a log-likelihood ratio against this background:Using natural logarithms instead gives $\ln (27/4)\approx 1.90954$. Identically distributed independent positions do not, by themselves, require equal base frequencies. If the intended background instead has specified frequencies $q_A,q_T,q_G,q_C$, the general answer isThe numerical value above uses the conventional equal-frequency random DNA model.

Taking a logarithm only after multiplying probabilities does not prevent floating-point underflow: the product may already have rounded to zero, and its logarithm is then $-\infty$. Accumulate logarithms before forming tiny products; where alternative paths must be added, use log-sum-exp in the forward algorithm or rescale its intermediate probabilities.

For the same uniform background and base-two log-likelihood ratio, replace every emission probability $b_i(x)$ by $e_i(x)={\log }_2[b_i(x)/(1/4)]$. The background specifies independent letters, not a competing hidden-state process. Its single emitting state has continuation factor one when scoring a fixed-length word, so each model transition contributes $s_{ij}={\log }_2{a}_{ij}$; dividing every transition by $1/4$ would incorrectly count the background twice.

The complete transition-score diagram isEvery omitted transition has score $-\infty$. Silent Start and End supply no emission score. The complete table of emission scores isFor the unique nonzero path identified in (b), add the five emission scores and all transition scores:This equals (c). For a sequence with several nonzero hidden paths, the full log-likelihood ratio is the log-sum-exp of their path scores; selecting only the largest score would instead compute a best-path approximation.

For a specified change, use the log-likelihood-ratio score. A log-likelihood ratio accumulates comparable evidence for the alternative hypothesis against the null hypothesis, includes the changing expected number of deaths, and yields the standard CUSUM designed to detect that alternative. Under the null its expected increment is negative; under the target increase it is positive.

By contrast, raw $O-E$ has zero null drift and does not specify which increase is being sought. Its noise scale changes with expected exposure, since a Poisson count with mean $E$ has variance $E$. A reset-at-zero chart with this zero-drift score can wander upwards through chance variation, and its threshold has no direct likelihood-evidence interpretation. An $O-E$ chart can be redesigned with an allowance and appropriately calibrated limits, but unadjusted $O-E$ is not the same as the Poisson likelihood-ratio CUSUM.

Iterating the CUSUM recursion shows thatThus it searches all possible starting times for a recent stretch of evidence favoring an increase. If adding an increment makes the current sum negative, that accumulated evidence favors the null; retaining it would force a later real deterioration first to repay a deficit accumulated during healthy years. Starting a new candidate segment gives zero instead. Resetting at zero makes detection responsive to a change with unknown onset. Negative values would be permissible for a fixed-start signed log-likelihood ratio, but that is a different statistic from this one-sided tabular CUSUM.

The source plot's sharp rise in the final year crosses the chosen control boundary. Trigger a prompt independent investigation of the mortality signal. First verify records, deaths, follow-up and expected-count calculations, then review case mix, changes in referral or recording, and plausible clinical explanations. The response should include appropriate clinical governance and safeguards while the cause is assessed; a statistical signal is not, by itself, proof of misconduct or a particular cause.

The stated null average run length of $111$ years makes a signal unusual under the calibrated null, while the alternative value of $5.2$ years indicates sensitivity to a sustained doubling under the specified model. Neither number gives the posterior probability that a doubling has occurred, and $1/111$ is not automatically the false-alarm probability for this particular ten-year history. Likewise the threshold is not a standalone fixed-test significance level.

The earlier value $X_{1993}=0$ says only that, at that point, no recent accumulated positive log-likelihood ratio survived the resets. It is not evidence that the earlier death rate was certainly normal, and does not cancel the later signal. The zero reset is deliberately part of the change-detection design; the 1994–1996 increments can independently justify an alarm. The full earlier clinical history remains relevant to investigation, even though a negative earlier accumulated score is not carried into the new CUSUM segment.

A classical sequential probability ratio test addresses a fixed-start choice between two simple statistical hypotheses. It sums log-likelihood ratio increments until an upper boundary accepts the alternative or a lower boundary accepts the null; it then terminates that test. A CUSUM instead monitors indefinitely for a change at an unknown time. Its reset at zero discards sustained evidence from a previously satisfactory period, so a late change need not first undo all earlier negative scores. An upper boundary $h$ signals when the accumulated change evidence is sufficiently large.

A fast initial response CUSUM takes $0<X_0<h$, often $X_0=h/2$, in place of a zero start. Until its first reset or signal, it acts like a fixed-start sequential probability ratio test with a lower boundary at $-X_0$ and an upper boundary at $h-X_0$ for the subsequent cumulative increments. Upon reaching the lower boundary it resets and continues as an ordinary CUSUM. The head start gives faster detection if a change is already present when monitoring begins or restarts. It is useful after an intervention, a shutdown or a previous alarm when immediate renewed surveillance is desired. Its initial false-alarm behavior differs from the zero-start scheme, so the head start and signalling boundary must be calibrated together; the preceding numerical choice is a design convention, not a universal optimum.

Use the binomial distribution for independent outcomes in each nonoverlapping batch, and let the CUSUM accumulate the log-likelihood ratio favoring a specified change. The two numbered calculations below treat the batch score and the initial-response design separately.

A common computational target is plausible: retrospective confidence evaluates whether an already selected answer is correct. In a Bayesian inference model this can be written $P(\text{selected answer correct}\mid \text{available evidence, action})$. The evidence might be a sensory signal in perception or retrieved information in memory retrieval. A shared target does not require the same evidence representation, parameters or neural implementation in both domains.

A basic signal detection theory account illustrates the common structure. Two alternatives generate different distributions of internal evidence, the decision selects an alternative, and confidence reflects evidence relative to the decision boundary. With equal prior probabilities and two alternatives, let $L$ be the log-likelihood ratio for the observed evidence. For the alternative with the larger posterior probability, confidence as posterior correctness probability givesThis is a normative consequence of the assumed model, rather than a universal empirical law. The same algebra can describe old/new recognition or a perceptual discrimination only when their assumed likelihoods, priors and decision rule justify it. An old/new discrimination also has a different response structure from comparing two simultaneously presented objects.

A different common account is self-consistency theory of confidence: confidence tracks how consistently a choice is supported across sampled representations. Asher Koriat found predictions of this account, previously applied to knowledge judgments, also held in perceptual judgments.^[ref] Its distinctive implication is that confidently repeated errors are possible when misleading evidence consistently favours the wrong choice. Reliable internal agreement need not entail external correctness. This differs from simply identifying subjective confidence with an objectively calibrated posterior.

Important limitations remain. Recognition can be supported by familiarity and recollection, while a perceptual choice may depend on a currently visible stimulus. The content and timing of evidence differ. Confidence can incorporate an additional retrieval attempt, response fluency, or evidence acquired after the initial decision. A common computation can therefore give different levels of metacognitive efficiency across tasks without requiring a wholly different function for confidence. Conversely, similar average rating curves do not prove a single mechanism: quite different latent processes can produce them.

Second-order inference about decisions makes the relation explicit by distinguishing the information supporting the decision from the partially correlated information supporting its evaluation.^[ref] A model in which confidence has exactly the same evidence as the choice is a stronger hypothesis than a model with a shared inferential goal and separate noisy samples. Error detection and later changes of mind are useful tests because they constrain what further information confidence can use.

Strong evidence for a common mechanism would include quantitative predictions across domains, transfer to a new domain after controlled training, and common effects of selective interventions after matching first-order performance. A shared brain region or a cross-task correlation alone is insufficient: task demands and measurement reliability can produce either apparent commonality or apparent dissociation. The most defensible common basis is a family of evidence-based evaluations of correctness, while the claim of one identical process for memory and perception remains substantially stronger.

For independent Poisson observations, comparing mean ${\lambda }_{0t}$ with $\theta {\lambda }_{0t}$ gives the displayed log-likelihood ratio. Updating a CUSUM with these scores targets the specified multiplicative increase. Its expected score is negative under the null and positive under the alternative when $\theta >1$, so resetting at zero discards earlier evidence against an increase.

A sequential probability ratio test accumulates the log-likelihood ratio for two specified simple statistical hypotheses until it reaches an upper or lower boundary. An upper crossing favors the alternative, a lower crossing favors the null, and observations continue between them. Error-calibrated boundaries are approximately $\log [(1-\beta )/\alpha ]$ and $\log [\beta /(1-\alpha )]$; discrete overshoot means these approximations need calibration.