Foundations: probability as degree of belief

Every decision framework in this section rests on one question: what is a probability? Kolmogorov's 1933 axioms answer it mathematically — $P$ is a measure on a $\sigma$-algebra of events with $P(\Omega)=1$ and countable additivity — but they say nothing about where the numbers come from. Two philosophies fill that gap.

Frequentist: $P(A)$ is the long-run relative frequency of $A$ in repeated trials. This works beautifully for a die or a large book of homogeneous auto policies, but it is silent about one-off propositions — "this reserve will develop adversely," "this drug will work for this patient," "it will rain tomorrow." There is no repeated trial to average over.

Bayesian (subjective/personalist): $P(A)$ is a degree of belief held by a particular decision-maker, constrained only by the requirement that the beliefs be coherent — that is, they must not expose the holder to a Dutch book, a set of bets that produces a guaranteed loss regardless of outcome. De Finetti's coherence theorem (1937) shows that a set of belief-based betting quotients avoids a Dutch book if and only if those quotients satisfy the Kolmogorov axioms. This is the deepest justification for probability in decision-making: it is not merely a convenient calculus, it is the unique calculus of belief that cannot be exploited by a clever bookmaker.

Coherence has an operational meaning. If you say "I believe there is a 30% chance of a large loss this year," and someone offers you a bet that pays if the number is calibrated, refusing to honour the implied odds — or holding beliefs about related events that are logically inconsistent (e.g., $P(A)+P(A^c) \ne 1$) — is a mistake a rational agent should be able to recognise and correct, just as an arithmetic error is a mistake. This puts subjective probability and objective (frequency-based) probability on the same logical footing: both are probabilities in Kolmogorov's sense, and a Bayesian updates a subjective prior toward frequency data as evidence accumulates (this is the content of Bayes' theorem and, asymptotically, of the Bernstein–von Mises theorem).

Why this matters for the Actuarium Decision Engine (ADE). Every input to the ADE — the prior $p(s)$, the likelihood $L[e\mid s]$, the reference/credibility distribution $q(s)$ — is a probability in exactly this coherent sense. When there is abundant, relevant, homogeneous data (a large stable book of policies), the frequentist and Bayesian numbers converge and the distinction is academic. When data is thin, structurally different, or one-off (a new product line, a novel treatment, a single firm's expansion decision), the only rigorous way to reason is with coherent subjective probabilities updated by whatever evidence exists — which is precisely what ADE's Bayesian layer, $p(s\mid e)\propto L[e\mid s]p(s)$, formalises.

Ambiguity — the sharp edge of coherence. Coherence requires only that your degrees of belief add up consistently; it does not require you to have a single number for every event. The Ellsberg paradox (see "Ambiguity and robustness") shows real decision-makers often refuse to assign one, and prefer known odds to unknown ones. ADE accommodates this without abandoning coherence, by allowing a set of distributions (the $\varepsilon$-contamination model) rather than insisting on one — a controlled, quantified departure from full Bayesianism rather than an incoherent one.

Limits. Coherence tells you your beliefs must be internally consistent; it does not tell you they are right. Two equally coherent actuaries can hold very different priors about the same emerging class of business, and both are being "rational" in de Finetti's sense. This is why ADE always reports how sensitive the recommendation is to the prior (via the sensitivity/break-even panel) — coherence is necessary, not sufficient, for a good decision, and eliciting a defensible prior (from data, expert judgement, or both) is itself part of professional practice, governed in insurance by standards such as ASOP 23 (data quality) and ASOP 41 (communications).