✧ An interactive exploration

The Relativity of Belief: Bayesian Probability as Velocity, Evidence as Spacetime

A scientist weighs new evidence. A population evolves. An astronaut combines velocities. In the right coordinates, all three follow the same mathematical rule.

Epistemic Physics represents beliefs with Beta distributions: curves over the possible values of an unknown probability $B$ . We write $B \sim Beta (α, β)$ . The positive weights $α$ and $β$ track support and opposition, including the prior, our starting weights before observations.

The curve's mean is $μ = α / (α + β)$ . Center that mean probability into a velocity, with 50% at rest and the two outcomes in opposite directions. The total trial weight determines how readily a new trial can move the belief. Together, they give beliefs a light-cone geometry. Follow this connection into evolution, thermodynamics, and light.

v_{E} = 2 μ - 1

The mean probability

μ

becomes an epistemic velocity

v_{E}

between −1 and +1. Try steering it in the light cone.

Watch it beat the peeking habit Start with the fundamentals

Discovered August 12, 2026 · An open notebook by Miraia Chiou ↗

No advanced physics degree needed: bring your curiosity and undergraduate math.

Interactive · 01

Support Y44

Opposition N20

μ 0.68mean probability v_E 0.37centered mean τ_E 65with prior

Drag the belief dot or adjust support and opposition to steer it. Both sides start with a half-unit Jeffreys prior (n₀ = 1).

FIG. 01 One light cone, many connections. Move the evidence and watch probability become velocity.

Belief and relativity share a rule for change

What do Bayes and Einstein have in common?

You're 75% confident that a hypothesis is true. A new observation is three times as likely if it's true as if it's false. How far should that probability move?

Your starting odds, the probability of the hypothesis divided by the probability of its alternative, are 3:1. Bayes' rule multiplies them by the observation's likelihood ratio, here 3, giving 9:1. That's 90% confidence.

Now recenter and treat probability as a velocity, $v_{E} = 2 μ - 1$ . You've moved from 0.5 to 0.8. Combine two velocities of $0.5 c$ using Einstein's rule, where $c$ is the speed of light, and you get $0.8 c$ . The same calculation has walked into a physics textbook!

And here comes Darwin: a variant occupying 75% of a population, with three times its competitor's reproductive fitness, reaches 90% in the next generation. The common coordinate is rapidity, the inverse hyperbolic tangent of epistemic velocity. Its half-log-odds expression turns multiplication of odds into addition of rapidity. And a sum has a practical superpower a product of p-values lacks: you can audit it mid-flight, as often as you like.

Steer all three stories to the same dock in the playground, follow the derivation in A little π between belief and relativity, then take it into living cells and evolution.

Does it work in practice?

Same data. Same peeks. 27× fewer false alarms.

You changed something: a recipe, a headline, the color of a button. Does the new version truly win? The standard tool is an A/B test ↗: show both versions, count which one wins more head-to-head rounds, and ask a p-value, a score for how surprising your tally would be if nothing had really changed. The fine print says to fix the number of trials in advance and check the score exactly once. Nobody waits. Checking the dashboard early feels harmless, yet each peek hands chance another try at fooling you. Peek every ten trials and a change that does nothing gets crowned a winner 36% of the time: the famous peeking problem ↗.

Epistemic Physics dissolves it by keeping the whole evidence ledger instead of one number. On the ledger, every trial adds a step of rapidity, so the verdict line is a fixed rung that one guarantee covers at every peek, forever, and real improvements are still caught at the pace of the strictest classical schedule. Race the rules yourself:

A measured benchmark

Race a hundred impatient labs

A hundred labs run the same test: each trial is one head-to-head round that the new version either wins or loses, and every lab peeks at its tally after every ten trials. All three verdict rules watch the same trials; only the rule differs. Choose the hidden truth, then run the race.

Peek at p-values p < 0.05, every look

0 false alarms · 100 labs

Rationed peeks classical fix: 5% split over 100 looks

0 false alarms · 100 labs

Evidence ledger epistemic physics: cross E ≥ 20

0 false alarms · 100 labs

false alarm real difference found still running

Odds multiply while rapidities add, so the verdict line E ≥ 20 is one fixed rung in rapidity: when nothing really differs, an evidence path ever climbs above it with probability at most 1 in 20, however long you watch and however often you peek. The exact rates are sums over every path in the evidence cone; only the hundred sample labs are random.

Follow the proof in When the numbers refuse to fit, take the single-path version for a spin in the playground, or go deep with the mathematics of safe anytime-valid inference ↗.

One coordinate, many worlds

What if these are all speaking rapidity?

A chess rating, an acid's pH, a neuron's voltage, a decoder's message. Center their probabilities or shares into an epistemic velocity, and the same coordinate keeps appearing. Define epistemic rapidity from that velocity:

ϕ_{E} = artanh v_{E}

The inverse hyperbolic tangent turns the bounded velocity into an unbounded coordinate. Using $v_{E} = 2 μ - 1$ , where $μ$ is a probability or share, we recover half the log odds:

ϕ_{E} = \frac{1}{2} \ln \frac{μ}{1 - μ}

Read it in reverse and $μ = σ (2 ϕ_{E})$ gives the familiar sigmoid, while $v_{E} = \tanh ϕ_{E}$ gives its velocity. Every row below offers another way into this geometry.

21 ways to meet $ϕ_{E} = artanh v_{E}$ . Here $μ$ is a probability or share, and $σ (x) = 1 / (1 + e^{- x})$ .
Field	Familiar law	Rapidity $ϕ_{E}$
Statistics	Logistic regression $μ = σ (ℓ)$	$ℓ / 2$
Coding theory	LLR messages and the tanh rule $μ = σ (L)$	$L / 2$
Chess	Elo expected score $μ = \frac{1}{1 + 10^{- Δ_{E l o} / 400}}$	$\frac{\ln 10}{800} Δ_{E l o}$
Psychometrics	Rasch model $μ = σ (a - d)$	$(a - d) / 2$
Chemistry	Henderson-Hasselbalch $\frac{μ}{1 - μ} = 10^{pH - p K_{a}}$	$\frac{\ln 10}{2} (pH - p K_{a})$
Biochemistry	Hill occupancy $μ = \frac{[L]^{n}}{K^{n} + [L]^{n}}$	$\frac{n}{2} \ln \frac{[L]}{K}$
Enzymology	Michaelis-Menten fractional rate $μ = \frac{[S]}{K_{m} + [S]}$	$\frac{1}{2} \ln \frac{[S]}{K_{m}}$
Electrophysiology	Boltzmann gating $μ = σ (\frac{V - V_{1 / 2}}{k})$	$\frac{V - V_{1 / 2}}{2 k}$
Membrane biology	Nernst equilibrium $\frac{c_{o u t}}{c_{i n}} = e^{z F E / (R T)}$	$\frac{z F E}{2 R T}$
Statistical mechanics	Spin-½ magnetization $⟨ s ⟩ = \tanh (\frac{μ_{m a g} B}{k_{B} T})$	$\frac{μ_{m a g} B}{k_{B} T}$
Quantum statistics	Fermi-Dirac occupancy $μ = σ (\frac{μ_{c h e m} - ε}{k_{B} T})$	$\frac{μ_{c h e m} - ε}{2 k_{B} T}$
Prediction markets	LMSR price $μ = σ (\frac{q_{Y} - q_{N}}{b})$	$\frac{q_{Y} - q_{N}}{2 b}$
Machine learning	Binary softmax, temperature T $μ = σ (Δ z / T)$	$Δ z / (2 T)$
Evolution	Replicator solution $μ (t) = σ (s t + 2 ϕ_{0})$	$s t / 2 + ϕ_{0}$
Decision theory	Kelly, even-money stake $f^{*} = 2 μ - 1$	$artanh f^{*}$
Special relativity	Velocity and boosts $u / c = \tanh ϕ_{E}$	$artanh (u / c)$
Bayesian inference	Likelihood-ratio update $O^{'} = O Λ$	$ϕ_{E}^{'} = ϕ_{E} + \frac{1}{2} \ln Λ$
Interacting spins	A zero-field Ising bond $⟨ s_{1} s_{2} ⟩ = \tanh (\frac{J}{k_{B} T})$	$\frac{J}{k_{B} T}$
Stochastic thermodynamics	Forward and backward hopping $μ = \frac{k_{+}}{k_{+} + k_{-}}$	$\frac{1}{2} \ln \frac{k_{+}}{k_{-}} = \frac{A}{2}$
Sensor geometry	Share of the stronger eigenvalue $μ = \frac{λ_{+}}{λ_{+} + λ_{-}}$	$\frac{1}{2} \ln κ$
Quantum state spectra	A qubit’s larger eigenvalue $λ_{+} = \frac{1 + \| r \|}{2}$	$\frac{1}{2} \ln \frac{λ_{+}}{λ_{-}}$

Different units, matching positions. A 200-point Elo advantage and a half-unit pH shift both land at about 76%. Could a lesson about chess teach us something about a chemical sensor? Move the shared coordinate and try the translation.

Matching equations invite matching experiments. Multiplying odds adds rapidity. Near the middle, a small push changes the fraction quickly; near either end, the response softens. Follow these patterns across the table and see what else they predict.

A rapidity translator

Turn one dial. Visit several worlds.

Start with centered velocity, then straighten its bounded range into rapidity. Drag the gold point sideways: a chess advantage, a chemical balance, and an electrical push move together. Choose other worlds, or type into a card to drive them all.

Share or probability μDrag or tap ↔

Rapidity

ϕ_{E}

Rapidity φ_E: 0.550
Share μ: 75.0%
Centered mean v_E: 0.501

World 1

ϕ_{E} = \frac{\ln 10}{800} Δ_{Elo}

Rating advantage Elo points

Expected score 75.0%

World 2

ϕ_{E} = \frac{\ln 10}{2} (pH - p K_{a})

pH − pKₐ pH units

Deprotonated share 75.0%

World 3

ϕ_{E} = \frac{V - V_{1 / 2}}{2 k}

Voltage scale k Voltage above midpoint mV

Open share 75.0%

ϕ_{E} = artanh v_{E}

v_{E} = \tanh ϕ_{E} = 2 μ - 1

μ = σ (2 ϕ_{E})

Every card shares the gold point. Epistemic rapidity φ_E is the inverse hyperbolic tangent of the centered mean velocity. The sigmoid σ then translates that rapidity into a share between 0 and 1. Arrow keys move the graph, and Enter applies a typed value.

Explore the derivations and possibilities in One rapidity, many worlds

Choose your next rabbit hole

Follow a connection.

All articles

Start with something familiar. Change coordinates. See what else comes into view.

Information theory · Thermodynamics

A wrong prediction, a magnet, a shared geometry.

KL divergence measures the extra cost of predicting with the wrong probabilities. Follow it into the entropy of a magnet, where the Lorentz factor appears as curvature.

A coin, a magnet, and a learning machine Probability · Optics · Quantum theory

What happens when beliefs interfere?

Write evidence as amplitudes and their interference reveals Bhattacharyya overlap, a measure of how closely two probability distributions agree. Next stop: polarized light and qubits.

When evidence puts on sunglasses Biology · Evolution · Relativity

Natural selection takes a relativistic turn.

The same coordinate that describes a Lorentz boost also turns relative fitness into a shift in a population’s odds. Follow this rapidity coordinate into evolution, cell receptors, and pH.

A cell that measures in rapidity

Keep exploring: springs and sensor networks, agreement and frustrated magnets, or signed evidence and electric currents.

The central synthesis

Three fields, one geometry.

Epistemology · Statistics · Relativity

For centuries, the philosophy of knowledge, the mathematics of statistics, and the physics of spacetime were taught in separate departments. Here is how they snap together in my framework:

Epistemology

The Nature of Knowledge

How do we justify what we believe? A central question in epistemology is distinguishing complete ignorance from balanced disagreement.

If you know nothing about an exoplanet, you have zero evidence (vacuity). If two telescope teams recorded 500 signals for and 500 signals against, you have a mountain of clashing observations (conflict).

Scalar probability assigns 50% to both cases, collapsing ignorance and contradiction into the exact same number. In Epistemic Physics, vacuity sits near the origin, while conflict climbs high into epistemic time. We also track lineage and tracelets so that copied reports cannot pretend to be independent evidence.

Statistics

The Beta Engine

Instead of a single mean probability $μ \in [0, 1]$ , we model our belief as a distribution over possible probabilities:

B \sim Beta (α, β)

Support trials $Y$ and opposition trials $N$ combine with a neutral prior concentration $n_{0}$ to give $α = Y + n_{0} / 2$ and $β = N + n_{0} / 2$ .

The expected probability is the distribution mean:

μ = E [B] = \frac{α}{α + β}

At a fixed mean, the spread contracts as total evidence grows. A surprising observation can widen it, giving belief inertia and shaken certainty a precise mathematical form.

Special Relativity

The Spacetime Geometry

Take the two Beta parameters and define sum and difference coordinates:

τ_{E} = α + β, x_{E} = α - β

Total evidence $τ_{E}$ behaves as forward-ticking epistemic time. Net balance $x_{E}$ behaves as epistemic position.

Recenter the probability from $[0, 1]$ to $[- 1, 1]$ :

v_{E} = 2 μ - 1 = \frac{x_{E}}{τ_{E}} = Velocity

Certainty is the speed of light ( $c_{E} = 1$ ). The cumulative evidence state has a mass-like cone norm:

m_{E}^{2} = τ_{E}^{2} - x_{E}^{2} = 4 α β

The field dictionary

One cone, several readings.

Connect the essential equations of belief with spacetime, energy, momentum, and mass.

Y, N

Raw Trials · Bernoulli Photons

Counts of observed support ( $Y$ ) and opposition ( $N$ ). A single new support trial adds one unit to both time and position. Its step has cone norm $(Δ τ_{E})^{2} - (Δ x_{E})^{2} = 0$ : a lightlike arrow, like the direction of a photon.

τ_{E} = α + β

Prior-inclusive Epistemic Time

The total volume of evidence, including the prior concentration $n_{0}$ . Every unit trial ticks this clock forward. It is the upward coordinate of the accumulated evidence state.

x_{E} = α - β

Epistemic Position

Net support minus opposition. Positive values lean toward confirmation; negative values lean toward refutation. Together with epistemic time, it locates the current state in the evidence cone.

E_{packet} = ε Δ τ_{E}

Energy Carried by an Exchange

A new packet changes the ledger by $Δ τ_{E}$ ; both channels add to this incoming amount. Choose $ε > 0$ as the energy carried per trial, in units with $c = 1$ . Fully receiving the packet changes the recipient's energy by $Δ E = E_{packet}$ .

p_{packet} = ε Δ x_{E}

Momentum Carried by an Exchange

The packet's support minus opposition gives $Δ x_{E}$ . With the same calibration, its momentum changes the recipient's momentum by $Δ p = p_{packet}$ . Its local direction is $Δ x_{E} / Δ τ_{E}$ . The cumulative $x_{E}$ records the notebook's balance so far. Try a collision and follow the exchange.

v_{E} = \frac{x_{E}}{τ_{E}} = 2 μ - 1

Epistemic Velocity · Centered Mean

Expected probability recentered to $[- 1, 1]$ . Neutral balance is at rest ( $v_{E} = 0$ ). This is the direction from the origin to the accumulated evidence state. The endpoints $\pm 1$ represent unattainable absolute certainty (the speed of light).

m_{E} = 2 \sqrt{α β}

Epistemic Rest Mass · Internal Conflict

The invariant interval $\sqrt{τ_{E}^{2} - x_{E}^{2}}$ . Opposing contributions give the combined state a positive invariant mass. Together with velocity, this scale describes how readily a new trial can move a belief.

γ_{E} = \frac{1}{\sqrt{1 - v_{E}^{2}}}

Lorentz Factor · Bernoulli Fisher Information

Relativistic dilation factor. In probability coordinates, $γ_{E} = \frac{1}{2 \sqrt{μ (1 - μ)}}$ , which is exactly half the square root of Bernoulli Fisher information $I (μ)$ .

ϕ_{E} = artanh (v_{E}) = \frac{1}{2} \ln \frac{α}{β}

Mean-state Rapidity · Half Log-Odds

This is the rapidity of the centered mean. The whole random belief has its own coordinate, $Φ_{E} = artanh (B_{E})$ . In relativity, collinear boosts add via rapidities. For two competing hypotheses, Bayesian updates multiply their odds by likelihood ratios, so half-log-odds shift additively: Einstein's velocity addition is Bayes' rule for these odds updates! Beta trial counts pool by addition, with their shared prior retained once.

Lineage

Source Lineage & Tracelets

A tracelet is an elementary unit of evidence. Lineage tracks where evidence originated, ensuring that duplicate transmissions of the same underlying event are not mistakenly counted as independent trials.

New here?

Start with the fundamentals.

Explore Epistemic Physics from trials and probability to spacetime, mass, energy, and momentum. Short explanations and interactive experiments make this a good place to begin.

Start here

Interactive playground

A playground for evidence.

Play with the evidence directly and watch the geometry respond. Test how beliefs resist acceleration, see how rapidities add, or explore what a single average leaves out.

Open the playground

An open research notebook

Epistemic Physics is an evolving inquiry bridging epistemology, Bayesian statistics, and relativity. Whether you want to see how Bayes' rule becomes relativistic velocity addition, follow evidence into a chemistry cell, or ask what a spring knows about belief, there is always another question to explore.

About this research