Subjectivism — also called subjective Bayesianism or personalism — holds that probability is not a physical property of the world like mass or charge, nor a long-run frequency of repeatable events, but rather a measure of an individual agent's uncertainty. On this view, the statement "the probability of rain tomorrow is 0.7" does not describe a fact about the atmosphere; it describes a fact about someone's state of mind. Two rational agents with different backgrounds and experiences may assign different probabilities to the same proposition, and neither need be wrong — provided each agent's credences are internally coherent.
This interpretation of probability is the philosophical foundation on which Bayesian statistics rests. Every time a Bayesian statistician specifies a prior distribution, she is expressing a subjective judgment. The posterior is then the rational consequence of combining that judgment with the data.
Historical Origins
While the mathematical foundations of probability were laid in the seventeenth and eighteenth centuries by Pascal, Fermat, Bernoulli, and Laplace, the explicit articulation of subjectivism as a philosophical position belongs to the twentieth century. Three figures stand out as its architects.
Frank Ramsey argues in "Truth and Probability" that probabilities are degrees of belief, measurable through preferences over gambles. He shows that rational preferences imply beliefs satisfying the probability axioms — the first Dutch book argument.
Bruno de Finetti develops the subjective interpretation independently and more radically. His 1937 paper "La prevision" proves the representation theorem: exchangeable sequences behave as if drawn from a distribution with a prior, even though de Finetti denies that the "true parameter" exists objectively.
L. J. Savage publishes The Foundations of Statistics, providing a complete axiomatic derivation of both subjective probability and utility from rational preference axioms, unifying personal probability with decision theory.
Dennis Lindley, I. J. Good, and others develop the statistical methodology flowing from subjective foundations. Lindley's work on Bayesian inference and Good's weight-of-evidence framework become standard tools.
The Core Doctrine
Subjectivism can be distilled into three claims. First, probabilities are credences: they quantify an agent's degree of belief in a proposition. Second, the only normative requirement is coherence: credences must satisfy the probability axioms to avoid sure loss (Dutch book vulnerability). Third, there is no uniquely correct prior: rational agents may disagree in their initial probabilities, and the data will gradually bring their posteriors into agreement.
The third claim — sometimes called the washing-out theorem or merging of opinions — is crucial. It defuses the most common objection to subjectivism: that allowing arbitrary priors makes Bayesian inference arbitrary. Under regularity conditions, any two agents with non-dogmatic priors (priors that assign nonzero probability to all possibilities) who observe the same accumulating data will see their posteriors converge. The subjectivist concedes that priors are personal but argues that this personality is eventually overwhelmed by the evidence.
Condition P₁ and P₂ must be mutually absolutely continuous (neither assigns probability 0 to an event the other considers possible)
Subjectivism is not a monolith. Radical subjectivists like de Finetti hold that any coherent prior is acceptable. Moderate subjectivists accept constraints beyond mere coherence — for instance, that priors should respect known symmetries, physical constraints, or calibration data. Objective Bayesians like Edwin Jaynes argue for a unique rational prior determined by maximum entropy or other principles. Most practicing Bayesian statisticians fall somewhere in the middle, using informative priors when background knowledge is available and weakly informative priors otherwise.
De Finetti's Operationalism
De Finetti offered the most uncompromising version of subjectivism. He argued that the very notion of an "unknown probability" is incoherent — there is no such thing as the "true" probability of a coin landing heads, only your degree of belief. His famous dictum — "Probability does not exist" — was not a denial that the mathematical formalism is useful, but a denial that probabilities are objective features of reality to be discovered.
His representation theorem gave this position mathematical teeth. Consider an infinite exchangeable sequence of binary random variables (like coin flips where you believe the order does not matter). De Finetti proved that your joint probability over such a sequence can always be written as a mixture of i.i.d. Bernoulli sequences — as if there were a "true parameter" θ drawn from a prior distribution. The "as if" is essential: the parameter and the prior emerge from your symmetry judgment (exchangeability), not from any objective feature of the coin.
Criticisms and Responses
The principal objections to subjectivism are well-known. The problem of the priors complains that subjectivism is too permissive: if any coherent prior is acceptable, then a scientist can believe anything she likes before seeing data. The subjectivist responds with the washing-out theorem and by noting that in practice, the prior must be defended to the scientific community — peer review acts as a constraint that the axioms alone do not impose.
The problem of interpersonal disagreement asks how to adjudicate between scientists with different priors who reach different posteriors. The subjectivist can point out that frequentist methods also involve subjective choices (significance levels, stopping rules, model specifications) but disguise them as objective standards. At least the Bayesian framework makes all assumptions explicit and quantifiable.
The problem of objective chance asks how subjectivism accounts for physical probabilities — the half-life of a radioactive atom, for instance, seems to be an objective feature of nature, not a personal opinion. David Lewis's Principal Principle addresses this: your credences about events governed by known objective chances should match those chances. Subjectivists can accept this as a rationality constraint without abandoning the view that credences are fundamental.
"There are no such things as true or objective values of probability. All assignments of probability are, and must be, personal." — Dennis Lindley, Understanding Uncertainty (2006)
Subjectivism in Contemporary Practice
Modern Bayesian statistics is overwhelmingly subjectivist in practice, even when its practitioners do not identify as philosophical subjectivists. The choice of prior, the specification of the likelihood model, and the decision of which parameters to estimate are all subjective judgments. What Bayesian methods provide is not objectivity but transparency: every assumption is encoded in the model and can be examined, challenged, and tested through prior predictive checks, sensitivity analyses, and posterior predictive checks.
This transparent subjectivity has proven more robust and honest than the pseudo-objectivity of classical statistics, where subjective choices are hidden in the selection of test statistics, significance thresholds, and stopping rules. As George Box observed, "all models are wrong, but some are useful" — and the Bayesian framework, grounded in subjectivism, provides the most principled way of navigating among useful models.
Example: A Climate Scientist's Subjective Prior
A climate scientist is modeling the Earth's climate sensitivity — how much global temperature rises if atmospheric CO₂ doubles. Before running her new ocean-temperature analysis, she must specify a prior distribution over the sensitivity parameter. This is inherently subjective.
Choosing the Prior
Based on decades of research, paleoclimate data, and previous studies, she believes the sensitivity most likely falls between 2°C and 4.5°C, with a central estimate of 3°C. She encodes this as a log-normal distribution:
Median ≈ 3.0°C, 90% interval ≈ [1.8°C, 5.2°C]
A colleague who weights paleoclimate evidence differently might choose a wider prior with a higher median. A third scientist skeptical of high-sensitivity scenarios might use a tighter distribution. All three are legitimate — they represent honestly held beliefs informed by different assessments of the same body of evidence.
Data Overwhelms the Prior
After incorporating the new ocean temperature data (10,000 observations), all three scientists' posteriors converge to nearly identical distributions centered around 3.2°C. The data were strong enough to overwhelm the differences in their priors.
Critics call the prior "arbitrary." Subjectivists reply that it is transparent. Every assumption is explicitly encoded and can be scrutinized, debated, and tested via sensitivity analysis. Compare this with a frequentist analysis that silently embeds subjective choices in the model specification, test selection, and significance threshold — but calls the result "objective." As the subjectivist sees it, honesty about assumptions is not a weakness; it is the foundation of trustworthy science.