Tensions in Practice: Concentrated prior belief in tension with rapid revision¶
Two explicit starting beliefs for a contingent hypothesis
H is a contingent explanation, not a logical certainty. Consider prior probabilities 1/100 and 1/10 for H. Evidence E has probability 9/10 under H and 1/10 under Not H, so it multiplies the odds by nine. The smaller prior becomes 1/12; the larger becomes 1/2. Both can update, but being above zero does not make their responsiveness the same.
Respect a strong rarity judgment
Keep most initial probability on the established alternative.
Allow faster revision toward H
Reserve more starting probability for the contingent alternative.
Why these aims pull against each other
More prior mass for H must come from Not H. It increases responsiveness to supportive evidence but weakens the starting rarity claim; desired responsiveness alone does not justify the prior.
Choose an arrangement to see what changes and what remains difficult.
Both rows form a probability distribution before and after updating. Likelihoods remain fixed; the changed prior allocation explains the different posterior.
What this choice protects
What it costs
When it fits
Compare the arrangements
Use the stronger rarity prior
Start with H = 1/100, then apply the specified likelihoods.
| Before | P(E | row) | After | |
|---|---|---|---|
| H | 1/100 | 9/10 | 1/12 |
| Not H | 99/100 | 1/10 | 11/12 |
- What it protects
- The prior encodes strong evidence or judgment that H is unusual while remaining revisable.
- What it costs
- This ninefold odds update still leaves H at only 1/12.
- When it fits
- Fits a defensible strong rarity judgment; it must not be chosen merely to suppress an unwelcome explanation.
Illustration note: Posterior mass for H is 0.009/(0.009 + 0.099). Its small value is not failure of Bayes’ rule.
Reserve more probability for H
Start with H = 1/10, reducing Not H from 99/100 to 9/10, and apply the same likelihoods.
| Before | P(E | row) | After | |
|---|---|---|---|
| H | 1/10 | 9/10 | 1/2 |
| Not H | 9/10 | 1/10 | 1/2 |
- What it protects
- The same evidence can bring H to equal odds.
- What it costs
- More initial mass is taken from the established alternative, potentially misrepresenting a genuinely rare hypothesis.
- When it fits
- Fits a defensible less-concentrated prior or an explicitly labeled sensitivity analysis.
Illustration note: Posterior mass is 0.09/(0.09 + 0.09). A convenient posterior does not establish that this prior is correct.
What this illustration does—and does not—establish
The source supplies the structural tension; the invented example makes one relation inspectable. Costs and conditions are part of each arrangement, not exceptions to a universal recommendation.
- All probabilities are invented, positive and mutually consistent. No evidence here selects the correct prior.
- The likelihood ratio is finite; neither update reaches certainty. Repeated evidence would require its own dependence assumptions.
- Posterior belief is distinct from an action threshold. This comparison does not prescribe a decision.
Source entries
Cromwell's Rule
The canonical tension motivates this comparison. The setting, finite values and arrangements are declared editorial illustrations, not measured findings.
Small credence versus effectively-zero in practice (scalar)
"Strictly positive" is the mathematical fix, but a credence of $10^{-40}$ is operationally indistinguishable from zero and will never be moved by any evidence a real agent can gather, so the rule's guarantee that "evidence can act" is hollow at extreme-but-nonzero priors. The failure mode is satisfying the letter of the rule (nonzero) while violating its spirit (revisable in practice), so a belief is technically open but practically sterile. Diagnostic: ask how much evidence it would take to move the credence to a decision-relevant level; if no attainable evidence suffices, the prior is functionally on the boundary despite being formally off it.
The source operation
Cromwell's rule is the structural injunction never to assign a prior probability of exactly 0 or exactly 1 to a contingent proposition, because Bayesian updating cannot move a probability away from those endpoints. If *P(H) = 0*, then for any evidence *E* the posterior *P(H|E) = 0* as well, so the proposition is permanently unfalsifiable from below; if *P(H) = 1*, the proposition is permanently unrevisable. Closed beliefs are evidence-sterile: no observation, however striking, can disturb them.
Flooring the prior versus distorting calibration (measurement)
Smoothing keeps unseen events nonzero, but every flooring scheme redistributes mass and thereby *biases* the estimates it touches — add-one smoothing notoriously over-weights rare events.