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Bayes' theorem solves a precise part of the scientific confirmation problem: it gives a coherent relation between prior commitments, evidential likelihoods and updated probabilities. It does not select those commitments from no premises, and it should not be identified with every meaning of confirmation.

For background information $K$, a hypothesis $H$ and newly considered evidence $E$, let $p=P(H\mid K)$, $a=P(E\mid H,K)$ and $b=P(E\mid \neg H,K)$. With a positive denominator,The first formula requires a positive denominator; the odds formula additionally assumes $0<p<1$ and $b>0$. The second equation separates prior odds from the Bayes factor. For $0<p<1$, the evidence raises the probability of $H$ exactly when $a>b$. It is not enough that $E$ is likely under $H$; it must be more likely than under the relevant alternative mixture. This guards against an inverse-probability error in evaluating scientific tests.

For example, let the prior be 0.01 and the two likelihoods be 0.8 and 0.08. The Bayes factor is ten, but the posterior probability is only $10/109\simeq 0.0917$. The evidence strongly favours $H$ relative to its previous odds without making $H$ probable in an absolute sense. Evidence can therefore confirm a hypothesis while leaving a rational investigator doubtful. The comparison also depends on which alternatives are included in $\neg H$ and how their probabilities are distributed.

The raven paradox shows another benefit. Logically equivalent claims receive identical probabilities in the same probabilistic model, but this does not force every investigation of a white object to provide the same evidence about ravens. If an object is selected because it is already known to be a shoe, observing that it is not a raven may have equal likelihood under the rival hypotheses and provide a Bayes factor of one. Random selection from a specified population can give different likelihoods. The sampling procedure and background information must be represented, rather than letting a grammatical positive-instance rule determine evidential support.^[ref][ref]

The problem of old evidence is a genuine limit of the simplest updating account. If $E$ already belongs to $K$, conditioning on it again produces no increase. Yet an old measurement can be important support for a newly formulated theory. One may assess support relative to a specified earlier or counterfactual background, or model learning a previously unrecognized connection between theory and evidence. These are substantive additions. Simply applying the same conditioning rule to an already certain fact does not represent the new intellectual achievement.^[ref]

Other limitations concern model construction and priors. A hypothesis assigned zero prior probability remains at zero after ordinary conditioning on evidence of positive total probability. A badly specified likelihood can generate confident but unreliable results. Different reasonable priors need not disappear under finite or non-discriminating evidence. Nor does the theorem supply a noncircular solution to the problem of induction: the model's connection to future physical events remains an empirical commitment.

Thus the Bayesian framework gives a rigorous account of probability change, comparison of alternatives and some apparent paradoxes. These are real achievements. It is incomplete as a general recipe for scientific judgment unless the hypotheses, backgrounds, likelihoods and forms of learning are also defended. A mathematical rule of coherent updating can organize evidential reasoning without establishing that every coherent input represents nature correctly.

Old evidence challenges a simple account of confirmation as actual conditional updating, but does not refute every probabilistic treatment of the raven paradox. The strongest response distinguishes support from newly acquired information and makes the sampling procedure explicit.

Let $H$ say that every raven is black. Its contrapositive says that every nonblack thing is a nonraven. Since these statements are logically equivalent, evidence should support them equally. Combining this equivalence requirement with the unrestricted rule that every positive instance confirms a universal generalization appears to make discovering a white shoe support $H$. The tension is between that result and the judgement that such a discovery is irrelevant.^[ref]

Bayesian confirmation replaces an automatic instance rule with a comparison of likelihoods. For background $K$, positive prior probability for both $H$ and its alternative, and evidence $E$ of positive probability,The equivalence follows directly from Bayes' theorem. A statement's grammatical appearance as a positive instance does not determine either likelihood.

Consider an intentionally simple finite model. There are $r$ ravens and $M$ nonblack objects, with these counts fixed under both hypotheses. Under $H$ none of the nonblack objects is a raven; under the sole alternative $H_1$, exactly one is. Selecting uniformly among the nonblack objects and discovering a nonraven has likelihoods $1$ and $1-1/M$. Selecting uniformly among the ravens and discovering a black one has likelihoods $1$ and $1-1/r$. With equal prior probabilities, the respective posterior probabilities of $H$ areIf $M\gg r$, the first discovery supplies far less support. For $M=10^6$ and $r=100$, the increases over $1/2$ are approximately $2.5\times 10^{-7}$ and $0.00251$. This illustrates a possible resolution of the intuitive disparity, not a theorem that every nonblack nonraven always confirms $H$. More complex alternatives can reverse the comparison, and examining something already known to be a shoe need supply no relevant new information.^[ref]

Now suppose $E$ is already in $K$. Then $P(E\mid K)=1$, and ordinary conditioning givesThere is no further increase from learning the same fact again. Yet an already established observation can become an important reason to accept a subsequently developed theory. Clark Glymour's problem of old evidence exposes this mismatch between actual updating and scientific assessments of evidential support.^[ref]

One response evaluates the observation relative to an earlier or counterfactual background that does not contain it. That can represent how strongly a theory accounts for the observation compared with alternatives. It must specify that background rather than pretend to remove knowledge uniquely from a person's present beliefs. Another response concerns learning that a theory successfully accounts for the observation. That discovery can be new even when the observation is old. Modelling it requires attention to limited reasoning and changing hypotheses; an ideal agent already knowing every logical consequence cannot simply learn an entailment it previously overlooked.

Neither manoeuvre licenses counting the same evidence twice, or ignoring that parameters were selected to fit it. Neither establishes a unique complete theory of confirmation. Nevertheless, the old-evidence objection concerns the interpretation and completeness of a probabilistic framework; it does not invalidate the likelihood calculation for genuinely new sampling information.

The preferable solution combines a background-sensitive account of support with rejection of unrestricted instance confirmation. It preserves logical equivalence while allowing a white shoe to provide tiny support, no support or contrary support under different justified models. This is more explanatory than assigning confirmation solely from logical form, although assessing old evidence requires additional work beyond routine conditioning.