Keep the evidence · Epistemic Physics

Imagine running a photon detector in an astronomy lab. Every time a photon strikes the sensor, you make a mark in the detection column. Every time a cycle passes without a signal, you mark the non-detection column.

Standard data pipelines immediately divide the two counts to report a single percentage: “70% probability.” Before you do that, pause: keep the two columns intact. A single probability throws away how much work you did to get there.

Two counts, one starting point

Let $Y$ denote positive support trials (such as detected signals) and $N$ denote opposing trials (such as non-detections).

In standard empirical counts, $Y$ and $N$ start as positive tallies of observed evidence. But evidence does not have to be strictly nonnegative: we can also encounter negative evidence, such as retractions or discounting flawed data, where evidence increments are subtracted.

Before switching on the detector, we also hold prior assumptions. In Bayesian statistics, we describe this baseline with a symmetric prior concentration $n_{0}$ , split equally between the two sides. The prior yields our shape parameters $α$ (alpha) and $β$ (beta):

\begin{aligned} α & = Y + n_{0} / 2, \\ β & = N + n_{0} / 2. \end{aligned}

Your turn

Begin with an empty notebook ( $Y = 0, N = 0$ ) and add a single support trial ( $Y = 1$ ). Notice how the half-unit of prior opposition prevents your belief from leaping to 100% certainty.

Interactive · 01

The same evidence, another view

The density diverges at both endpoints. The curve is sampled only inside (0, 1).

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

Support Y0

Opposition N0

μ 0.50mean probability v_E 0.00centered mean τ_E 1with prior m_E 1.00cone normγ_E 1.00Lorentz factor φ_E 0.00rapidity

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

$α = Y + n_{0} / 2 = 0.5, β = N + n_{0} / 2 = 0.5$

Raw trials versus prior concentration

It is essential to distinguish raw empirical counts from prior-inclusive parameters.

The raw observed trials total $t_{E} = Y + N$ . But the total concentration governing our belief is $τ_{E} = Y + N + n_{0}$ .

The Greek letter $τ$ (tau) will soon play the role of time in our spacetime geometry. Every time a new observation arrives, $τ_{E}$ ticks forward by exactly one unit.

\begin{aligned} t_{E} & = Y + N, \\ τ_{E} & = α + β \\ = Y + N + n_{0} . \end{aligned}

The Jeffreys prior as an uncommitted baseline

What should $n_{0}$ be before any trials arrive?

A natural first choice is $n_{0} = 1$ , giving $α_{0} = 1 / 2$ and $β_{0} = 1 / 2$ . This is the Jeffreys prior: an uncommitted baseline that remains invariant under reparameterization. You could also choose a uniform prior with $n_{0} = 2$ (one unit per side).

Notice that the prior is not an empirical observation made by our detector: it is the mathematical scaffolding that prevents infinite certainty before we even begin.