= Solution
<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,
$$
P(H\mid E,K)=\frac{pa}{pa+(1-p)b},\qquad
\frac{P(H\mid E,K)}{P(\neg H\mid E,K)}=\frac{p}{1-p}\frac ab.
$$
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 $10/109\simeq0.0917$. 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.<Studies in the Logic of Confirmation I by Carl Hempel>[Section 5: equivalence and the paradox of positive-instance confirmation]{ref}<How Bayesian Confirmation Theory Handles the Paradox of the Ravens by Branden Fitelson and James Hawthorne>[Analysis of dependence on background knowledge, sampling assumptions and alternative hypotheses]{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.<Why I Am Not a Bayesian by Clark Glymour>[Chapter 3 of Theory and Evidence: old evidence and newly considered theory]{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.
\Image[/past-exam-of-the-philosophy-of-science-supplementary-subject-of-the-university-of-oxford/2016-confirmation-and-underdetermination.png]
{title=Posterior probability depends on prior odds, while finite observations leave alternative continuations}
{description=Original mathematical illustrations with explicitly chosen probabilities and functions, not historical or experimental data.}
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.
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