✓ The Alleged Resurrection and Bayesian Variables

A Bayesian Walk-Through of the Resurrection Claim (Steps A–D)

The post below is grounded in the introductory, foundational post ✓ An Intro to Bayesian Analysis which lays out the basics of Bayes’ Theorem. In this follow-up post, we’ll more deeply explore the application of Bayesian probability to the Bible’s claim of the resurrection of Jesus.

✓ An Intro to Bayesian Analysis

Step A — Prior P(H): The initial Probability Assigned to the Hypothesis that Jesus was Resurrected by the Christian God.

Think of the prior as your starting credence before looking at the specific historical evidence. It is built from multiple factors that multiply:

$P(H) = P(\text{theism}) \times P(\text{target Jesus} \mid \text{theism}) \times P(\text{resurrection} \mid \text{that targeting})$

◉ Factors that feed the prior

✓ Natural base rate (biological background): In ordinary biology, dead bodies stay dead. That empirical regularity drives the non-miracle prior high and the miracle prior low from the outset.

✓ Theism prior $P(\text{theism})$ : If you assign some probability to any god existing, $P(H)$ inherits that—but only to the degree independently supported. (If you treat theism as nearly certain, you must also account for the model complexity of specifying which god and what aims it has.)

✓ Targeting/specificity cost $P(\text{target Jesus}\mid\text{theism})$ : Even granting a god, would it choose this act—raising this person, at this time and place, for this message? Specificity multiplies costs: the more narrowly tailored the event, the slimmer the prior.

✓ Act-type rate $P(\text{resurrection}\mid\text{that targeting})$ : Conditional on intending to spotlight Jesus, how often does the god employ a literal bodily resurrection rather than other, clearer options (worldwide sky-writing, direct cognitive broadcast, global daylight phenomena, etc.)?

✓ Comparative miracle base rates: In world religions, miracle reports are abundant (Buddhist relics, Hindu visions, Islamic miracle stories). If you grant priors for Christian miracles, consistency demands adjusting priors for other miracle traditions. This broadens the denominator of competing supernatural claims and further shrinks $P(H)$ for any one miracle.

✓ Historical timing factor: Why this era? If theism is true, what’s the probability the decisive miracle would occur in a pre-scientific context (low literacy, no photography), rather than in a global, modern, verifiable setting? That specificity penalty pushes priors down further.

✓ Anthropological expectation: In human cultures, charismatic founders are regularly elevated into mythic figures after death. This systematic background increases the base rate for “apotheosis narratives,” which must be folded into the prior assessment.

✓ Complexity penalty: Hypotheses that require multiple independent assumptions—existence of God, intent to intervene, intent to do so via resurrection, intent to do so via Jesus—pay an Occam penalty in Bayesian reasoning. Priors shrink multiplicatively.

◉ Interdependence notes for Step A

Coupling between theism and targeting: The more you tilt theism toward a god whose aims already include a Jesus-centered program, the larger $P(\text{target Jesus}\mid\text{theism})$ becomes—but only at the price of baking those aims into your prior (a complexity/“Occam” penalty). In other words, raising targeting by redefining the god increases prior belief because of prior assumption, not because of evidence.
Granularity of the hypothesis: A coarser hypothesis (“some miracle connected to Jesus”) has a larger prior than a very specific one (“a bodily resurrection on the third day in Jerusalem leading to these specific appearances”). Bayes rewards simpler hypotheses with bigger priors.

Step B — Likelihood if true P(E | H): The Probability of Observing the available Evidence, Assuming it is True that (H) Jesus was Resurrected by the Christian God.

This term asks whether the observed evidence is what we’d expect if H were true.

◉ What should be expected under H?

✓ Core fit: Reports of post-mortem appearances, strong conversions, growth of a movement, early creedal formulas, and narratives centering on the event. These raise $P(E \mid H)$ .

✓ But not $1.0$ : Even if H is true, history is messy. Documents are lost, memories diverge, political motives shape accounts, and different groups curate different versions. So $P(E \mid H) < 1$ .

✓ Degree of expected publicity: If the resurrection truly occurred, we might expect widespread independent testimony (multiple villages, Roman records, external chroniclers). The narrow and internally dependent nature of the evidence lowers $P(E \mid H)$ from its maximum.

✓ Durability of evidence: A divine resurrection intended to ground faith might be expected to leave durable traces—public monuments, artifacts, Roman imperial recognition. Their absence tempers $P(E \mid H)$ .

✓ Comparison to false positives: Under H, true resurrections should be rare but decisive. Yet other miracle traditions (Marian apparitions, Buddha relics, Joseph Smith’s golden plates) yield superficially similar evidence profiles. If P(E) under false miracle traditions is similar, that compresses the discriminating power of H.

✓ Expected theological clarity: If God staged the resurrection, we might expect the evidence to settle doctrinal disputes (nature of resurrection body, timeline contradictions). But the actual evidence sparked centuries of disputes—lowering $P(E \mid H)$ relative to what divine intent would predict.

◉ Auxiliary expectations under H (often overlooked)

Clarity/targeting expectations: If an omnipotent agent is aiming at maximal persuasion, we might expect high-clarity signals (multiple independent, public, well-documented observations; physical artifacts; consistent timelines). Shortfalls here temper $P(E \mid H)$ .
Cross-cultural spread: A globally targeted miracle might predict broader, earlier, more uniform uptake; localized and delayed spread slightly lowers $P(E \mid H)[$ relative to a maximally persuasive plan.
Consistency constraints: Fewer contradictions, tighter chronology, and better preservation would be more expected under sustained divine intent.

◉ Interdependence notes for Step B

Dependent testimonies: If testimonies share sources or editorial lines, evidence items are correlated, not independent. Treating them as independent inflates $P(E \mid H)$ .
A simple discount is to use an effective sample size:
$n_{\text{eff}} = \frac{n}{1+(n-1)\rho}$
where $\rho$ is average pairwise dependence. Higher dependence → smaller $n_{\text{eff}}$ → weaker total lift.

Step C — Likelihood if false P(E | ¬H): The Probability that the same Evidence would be Observed if (¬H) Jesus was Not Resurrected by the Christian God.

The non-miracle case is not a blank box; it’s a mixture of live alternatives:

$P(E \mid \neg H) = \sum_j w_j \cdot P(E \mid \text{Alt}_j), \quad \sum_j w_j = 1$

Each alternative receives a weight $w_j$ (its share of plausibility), multiplied by how well that alternative explains the evidence. Adding the contributions gives the total denominator share from the non-miracle side.

◉ A comprehensive (but still schematic) menu of “Alts”

Source & memory dynamics

✓ Legend/memory drift: Oral transmission amplifies and reshapes. Predicts expanding claims, heroic motifs, and retrospective harmonization.
✓ Midrash/Scriptural fulfillment genre: Retelling in the language of scripture can generate event shapes that fit prophecies, raising $P(E \mid \text{Alt})$ without requiring a miracle.
✓ Euhemerization & heroization: Revered teachers gradually accrue divine/heroic attributes.

Experiential/cognitive mechanisms

✓ Bereavement visions and dreams: Common across cultures; produce sincere “appearance” claims.
✓ Group expectancy and contagion: Shared grief, ritual settings, and leadership influence prime vivid experiences.
✓ Cognitive biases: Agency detection, confirmation bias, motivated reasoning—all increase narrative cohesion after the fact.

Strategic/social mechanisms

✓ Fraud/embellishment (not required, but possible): Theft of a body, staged “signs,” purposeful editorial upgrading.
✓ Social/political incentives: Identity formation, boundary-marking, and leadership legitimation make certain stories sticky.
✓ Rivalry and sectarian differentiation: Distinct communities amplify distinctive claims to stand out.

Communications environment

✓ Rumor cascades / selection effects: Striking stories spread; mundane corrections die out.
✓ Document dependence: Later texts borrow frameworks from earlier ones (attenuating independence).

Long-tail aggregate (rare, but many)

✓ Low-probability, high-variety coincidences: Misidentifications, calendar/astronomy coincidences, transcription quirks, archaeological losses. Each is rare; in aggregate they claim a real slice. Think of this as a fat tail of oddities.

Unknowns (Reserve)

✓ Explicit humility budget: Assign 5–15% to mechanisms not listed. This prevents overfitting the analysis to our favorite few Alts and acknowledges model incompleteness.

◉ Interdependence inside the mixture

Co-occurrence: Several Alts can jointly produce E (e.g., grief visions → persuasive testimony → legend growth). The mixture can include compound Alts (pairs/triads) with their own $w_j$ .
Shared pipelines: Many Alts use the same channels (oral networks, liturgy, leadership), so their contributions are positively correlated. When estimating $w_j$ , avoid double-counting overlapping mechanisms.
Data dependence & design effect: If large portions of E are generated within one communication cluster, the effective denominator strength rises less than naïve sums suggest—but it still rises enough to keep $P(E \mid \neg H)$ substantial.

◉ Why the mixture keeps the denominator strong

Most of these mechanisms predict (at least qualitatively) the evidence we have: appearance traditions, growth of a movement, and written narratives decades later. Therefore $P(E \mid \neg H)$ is not small. A large denominator restrains the posterior unless the numerator is enormous.

Step D — Posterior P(H | E): Where a Fair Analysis Lands

Bayes in one line:

$P(H\mid E)=\frac{P(H)\cdot P(E\mid H)}{P(H)\cdot P(E\mid H)+(1-P(H))\cdot P(E\mid \neg H)}$

Tiny prior $P(H)$ × less-than-decisive $P(E\mid H)$ → small numerator.
Substantial $P(E\mid \neg H)$ (from the mixture and dependence-aware accounting) → large denominator.
Result: $P(H\mid E)$ remains small unless you preload the prior with strong theism and a very light targeting penalty.

◉ Credence hygiene (make the result robust)

Sensitivity sweeps: Vary $P(\text{theism})$ , targeting cost, and key $w_j$ . Publish how the posterior moves.
Dependence penalties: Report $n_{\text{eff}}$ for testimonies so readers see how correlation reduces evidential force.
Prior-predictive check: Ask whether H, as you parameterize it, would predict a clearer global signal than we see. If yes, temper $P(E\mid H)$ .
Mixture transparency: List your Alts, weights, and an Unknowns (Reserve) allocation. No denominator engineering.

A “Toy” illustration with an Example Instantiation of the Variables

Using classroom-friendly toy inputs (purely illustrative):

$P(H) = 0.01,\quad P(E \mid H) = 0.85,\quad P(E \mid \neg H) = 0.40$

gives
$P(H\mid E)=\frac{0.01\cdot 0.85}{0.01\cdot 0.85+(1-0.01)\cdot 0.40}=\frac{0.0085}{0.4045}\approx 0.0210$
→ about 2.1% (odds ≈ 1 in 46). Different priors change the number, but the structure constrains how far it can move without very strong, independence-rich evidence.

◉ Take-home

Priors matter: Theism and targeting multiply; specific miracles pay a large specificity cost.
Evidence must discriminate: If the evidence is also likely under natural mixtures, its Bayes factor is modest.
Dependence matters: Correlated testimonies are not many datapoints; they are one noisy pipeline.
Unknowns and long tails keep us honest: A reserve for unmodeled causes and an allowance for rare coincidences protect against overconfidence.

That’s the exhaustive structure. If you want, I can turn these sections into a printable 1-page Bayes checklist (with blanks for $P(H)$ , the $w_j$ mixture, and an $n_{\text{eff}}$ calculator) to accompany the post.

◉ A Narrative Write-Up of the A-D Process

Step A — Prior P(H): How plausible is the resurrection before any evidence?

Before we even look at reports of an empty tomb or appearances, Bayes asks us to consider what we already know about the world. The prior is like our “starting point” before adding new information. And here’s the blunt truth: in biology, once people die, they stay dead. We’ve never recorded a natural resurrection. That natural base rate makes the starting probability vanishingly small.

Of course, the prior can be nudged up if you allow for the existence of God. If you believe that some god exists, then miracles aren’t impossible anymore, and the probability rises a little. But it only rises as far as independent evidence for “a god” justifies. And even if a god exists, Bayes forces another question: would that god really choose to resurrect this particular person, at this time and place, to prove this point? That’s a huge “specificity cost,” because there are infinitely many other ways and moments a god could choose to act. The more narrowly you specify the miracle, the more the prior shrinks.

There are other background penalties too. For example, miracle claims appear in many religions—Hindu visions, Buddhist relics, Islamic miracle stories. If we treat the Christian claim generously, fairness demands we treat those other claims the same way. That lowers the prior for any one miracle in particular. We can also ask: why would a decisive act happen in an age without cameras or global media? If God’s intent was to prove the case clearly, it seems oddly timed. Add in the fact that charismatic leaders across cultures are regularly mythologized into semi-divine figures, and you’ve got anthropological expectations that shrink the prior even further.

When you put all this together, the prior isn’t just small—it’s tiny. That’s not prejudice; it’s Bayes forcing us to start where the weight of background knowledge actually lies.

Step B — Likelihood if true P(E|H): How expected is the evidence if the resurrection really happened?

Now suppose the resurrection really did occur. Would the evidence we have today fit that? To some extent, yes. We’d expect to see reports of appearances, conversions of followers, a religious movement centered on the event, and early creedal traditions. These all make sense if the resurrection truly happened.

But Bayes reminds us that this likelihood can’t be 1.0. Even if Jesus did rise, history is messy. Documents can be lost, memories diverge, politics shape accounts, and narratives get tailored by different communities. So even under the miracle hypothesis, the evidence we have is less clear and less durable than we’d expect if God wanted everyone, everywhere, to know for certain.

Think about it this way: if the resurrection really happened, shouldn’t there have been widespread testimony outside of Christian sources? Roman officials, neighboring villages, even other religions might have reported it. But there isn’t much of that. If the miracle was meant to persuade the whole world, we might also expect physical traces—monuments, artifacts, or Roman records—that never appeared.

And there’s another wrinkle. Many other miracle traditions—Marian apparitions, Buddha relics, Joseph Smith’s golden plates—generate evidence that looks strikingly similar: small groups of believers, strong conversions, written accounts later on. If false miracle traditions produce the same sort of evidence, then the resurrection evidence loses some of its uniqueness. That lowers $P(E \mid H)$ relative to the perfect case.

So yes, the evidence fits the resurrection, but not as tightly or uniquely as we’d expect under an all-powerful, clarity-seeking God.

Step C — Likelihood if false P(E|¬H): Could the same evidence appear without a resurrection?

This is where Bayes becomes especially powerful. The non-miracle case isn’t just one vague “no.” It’s a mixture of many live alternatives that all compete to explain the same evidence. In math form:

$P(E \mid \neg H) = \sum_j w_j \cdot P(E \mid \text{Alt}_j), \quad \sum_j w_j = 1$

That means we take each plausible alternative, give it a weight (its share of plausibility), figure out how well it explains the evidence, and then add them all up. The miracle doesn’t get to hoard the evidence; it has to compete.

And there are a lot of competitors. Stories passed along orally almost always expand and change—legend and memory drift predict exactly the kind of growth we see. Jewish writers often used scripture to “frame” stories so that events fulfilled prophecy, creating a pattern of “midrash” without miracles. Revered leaders are often heroized into divine figures (Euhemerization), so the trend fits Jesus too.

On the psychological side, grief and trauma can create powerful visions and dreams. Groups can fuel each other into shared experiences, and cognitive biases like agency detection and confirmation bias make supernatural stories feel natural.

There are also strategic reasons. Fraud or embellishment is always possible—stolen bodies, staged signs, or upgraded stories. Social and political incentives matter too: resurrection claims gave identity and authority to a fragile movement, making the story extremely “sticky.” Rivalries with other groups may have amplified claims further.

On top of this, rumor cascades and selection effects explain why striking stories survive while boring corrections fade. Later documents also lean heavily on earlier sources, making testimonies less independent than they appear. Add rare but real “long-tail” oddities—like calendar coincidences, transcription quirks, or mistaken identities—and you’ve got even more coverage. Finally, Bayes demands we hold back 5–15% for “unknowns”: things we haven’t thought of but could still explain the evidence.

Put all of this together and the denominator side— $P(E \mid \neg H)$ —is strong. These mechanisms make the evidence not only plausible without a miracle, but fairly likely. That keeps the resurrection from claiming the evidence as its own.

Step D — Posterior P(H|E): Where Bayes lands after the math

Now we put it all together:

$P(H \mid E) = \frac{P(H)\cdot P(E \mid H)}{P(H)\cdot P(E \mid H) + (1-P(H))\cdot P(E \mid \neg H)}$

The numerator is tiny, because the prior was tiny. The denominator is big, because the alternatives explain the evidence well. So the posterior—the updated belief—stays small.

Even if you push theism prior high and keep targeting costs low, the posterior only climbs into the single digits. With toy numbers, for example, if we start with $P(H) = 0.01$ , $P(E \mid H) = 0.85$ , and $P(E \mid \neg H) = 0.40$ , the final probability works out to about 2.1%—odds of about 1 in 46. That’s still very low for a claim as enormous as “God raised someone from the dead.”

Take-home

Bayes’ Theorem forces intellectual honesty. Priors matter, and specific miracles pay a heavy specificity cost. Evidence only helps if it strongly discriminates between miracle and non-miracle—and here, it doesn’t, because the alternatives explain the data well. Testimonies aren’t independent, which shrinks their weight. And by leaving a slice of probability for unknowns and long-tail coincidences, we stay humble and avoid overconfidence.

In plain terms: the resurrection story makes sense under the miracle hypothesis, but it also makes good sense under natural explanations, and the starting probability was already tiny. Bayes makes sure we don’t skip those steps. The result is a fair, structured, and mathematically honest assessment: the resurrection claim, while culturally powerful, remains extraordinarily unlikely.

◉ Disclaimer
There isn’t just one “official” set of priors or list of factors to use in a Bayesian analysis. Bayes’ Theorem gives us the framework, but each person has to decide what goes into it—how likely they think a god exists, how common miracles are, how stories change over time, and so on. Different people will plug in different numbers depending on what they think best matches reality. The important part is being honest about those choices and not pretending the math answers the question all by itself. Bayes is valuable because it makes our assumptions clear, so we can talk about them openly instead of hiding them.

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