Subjectivity of Priors

A foundational critique of bayesian-apologetics: the “garbage in, garbage out” problem. ¹

The Problem

Bayes’ Theorem requires:

• P(H): The prior probability of the hypothesis
• P(E|H): The likelihood of the evidence given the hypothesis

When H involves a supernatural entity (God), assigning numerical values to these variables is entirely speculative. One cannot statistically model the psychology, intentions, or behaviors of a divine being.

Example: Swinburne’s Priors

richard-swinburne assigns a high prior probability to the idea that if God exists, He would want to become incarnate, suffer, and provide atonement. This is a theological assumption about divine psychology — not an empirical claim that can be assigned a probability. ¹

Consequence

No matter how rigorous the subsequent mathematics, if the inputs are based on subjective theological assumptions rather than empirical data, the output is ultimately meaningless. This is why the same Bayesian framework can be used to “prove” diametrically opposed conclusions — the math simply reflects the priors chosen.

Relation to the post-hoc probability fallacy

A distinct but frequently co-travelling failure mode is the post-hoc-probability-fallacy: even granting a chosen prior, computing the probability of an outcome after it has been observed — and reading the small result as evidence against chance — is invalid (“the probability of X occurring, given that X has been observed to occur” is unity). Design-style probability arguments typically need both moves: an unjustified measure over possibilities to supply the space, and post-hoc reckoning to convert it into an improbability figure. The safeguards used in empirical fields — preregistration, blinding, held-out data — exist precisely because post-hoc computations lose their evidential meaning. ²

Why “neutral” priors cannot be rescued by indifference

The problem is not merely that priors are hard to guess; the most natural rescue — deriving them from ignorance via the principle of indifference — is internally inconsistent. Norton’s Chapter 10 shows that “completely neutral support” cannot be defined: the conditions neutrality imposes collide on von Mises’ wine-and-water case, forcing contradictory assignments to the same events (1/3 = 2/3). The priors problem is thus a constraint on the formalism itself, not merely a failure of nerve. ³

See Also

• bayesian-apologetics
• dwindling-probabilities
• humes-argument-against-miracles
• agrippas-trilemma — The regress problem explains why priors must be chosen somewhere: justification cannot be infinite
• post-hoc-probability-fallacy — The complementary set-up failure: post-hoc test specification
• fine-tuning-argument — Priors over possible universes: the “measure problem” in physical guise
• material-theory-of-induction — the neutral-support impossibility result: why indifference cannot supply the missing prior

Footnotes

1. raw/articles/bayesian-analysis-miracles-critique-2026.md ↩ ↩²

2. raw/articles/mathscholar-bailey-post-hoc-probability-2020.md ↩

3. raw/papers/norton-2021-material-theory-ch10-why-not-bayes.md ↩