Probability as velocity · Epistemic Physics

In elementary probability, numbers live on the unit interval $[0, 1]$ , with 0 meaning impossible and 1 meaning certain.

On August 12, 2026, I wondered: what happens if we slide the neutral balance point to 0 and stretch the interval from $- 1$ to $+ 1$ ?

Suddenly, negative opposition lives on the left, positive support lives on the right, and probability begins to look like a physical velocity.

Centering the probability

Define a centered random variable $B_{E} = 2 B - 1$ . Its expectation value is our epistemic velocity, denoted $v_{E}$ :

The subscript $E$ stands for “epistemic” (related to knowledge or belief).

Notice what happened: a neutral mean ( $μ = 0.5$ ) has $v_{E} = 0$ , sitting comfortably at rest. Absolute refutation is $v_{E} = - 1$ , and absolute confirmation is $v_{E} = + 1$ .

\begin{aligned} v_{E} & = E [2 B - 1] \\ = 2 μ - 1 = \frac{α - β}{α + β} . \end{aligned}

Total evidence as time, balance as position

Now look at the numerator and denominator of $v_{E}$ . They are simply the difference and sum of our two Beta parameters:

The sum $τ_{E} = α + β$ represents total accumulated experience, ticking forward like epistemic time.

The difference $x_{E} = α - β$ represents net balance, measuring displacement along an epistemic spatial axis.

The ratio $v_{E} = x_{E} / τ_{E}$ gives an average velocity from the origin to this evidence state! Along a changing trajectory, the instantaneous velocity would instead be $d x_{E} / d τ_{E}$ .

\begin{aligned} τ_{E} & = α + β, \\ x_{E} & = α - β, \\ v_{E} & = x_{E} / τ_{E} . \end{aligned}

Your turn

Add only support trials ( $Y$ ) in the lab. Watch the belief point climb upward while hugging the right boundary of the cone. Notice that the half-unit prior keeps it safely inside the cone, preventing it from touching the speed of light.

Interactive · 01

The same evidence, another view

A Beta distribution describes uncertainty about a probability. The vertical scale adapts and is labeled.

Support Y44

Opposition N20

μ 0.68mean probability v_E 0.37centered mean τ_E 65with prior m_E 60.41cone normγ_E 1.08Lorentz factor φ_E 0.39rapidity

Drag the belief dot to change both channels, or use the sliders. Focus the dot and use arrow keys for smaller moves; hold Shift for larger moves. The cone scale stays fixed.

Try

Symmetric prior

$α = Y + n_{0} / 2 = 44.5, β = N + n_{0} / 2 = 20.5$

Einstein’s light cone appears

When positive support and opposing evidence are both held ( $α > 0, β > 0$ ), net balance cannot exceed total evidence: $| x_{E} | < τ_{E}$ .

When you plot total evidence $τ_{E}$ vertically and net balance $x_{E}$ horizontally, every proper Beta belief is trapped inside the future light cone $| x_{E} | < τ_{E}$ . In this geometry, certainty is the speed of light ( $c_{E} = 1$ ). Just as no physical massive particle can reach the speed of light, no belief backed by finite evidence can ever reach absolute certainty ( $| v_{E} | = 1$ ).

What about negative evidence? Evidence does not have to be strictly nonnegative. When we retract mistaken claims, discount flawed data, or encounter negative increments (un-observations), or when negative quasi-probabilities appear as bookkeeping artifacts (analogous to Wigner functions in quantum mechanics), states can even probe the boundary or venture into spacelike regions ( $| v_{E} | > 1$ ). But for any settled belief with positive parameters, the light cone holds you firmly within.

τ_{E} > | x_{E} | ⟺ | v_{E} | < 1