A belief has a shape · Epistemic Physics

Suppose two astronomers both report that a distant exoplanet has a 50% chance of hosting water.

The first astronomer reached 50% after taking two quick spectral scans: one positive, one negative ( $Y = 1, N = 1$ ). The second astronomer reached 50% after analyzing two thousand painstaking observations: one thousand positive, one thousand negative ( $Y = 1000, N = 1000$ ).

Their quoted averages agree completely. Yet their epistemic states could not be more different.

A distribution over probabilities

To capture this difference, we treat the uncertain probability itself as a random variable $B \in [0, 1]$ governed by a Beta distribution.

The parameters $α$ and $β$ act as shape knobs. The expected value, or mean probability $μ$ (mu), provides the familiar single-number summary:

\begin{aligned} B & \sim Beta (α, β), \\ μ & = E [B] = \frac{α}{α + β} . \end{aligned}

Spread at a fixed mean

If you multiply both $α$ and $β$ by 100, the ratio $μ$ does not budge. But look at what happens to the variance:

The denominator contains $τ_{E} + 1 = α + β + 1$ . As total evidence accumulates, the distribution contracts into a tight, needle-like spike around $μ$ .

Variance measures how widely the possible probabilities are distributed. In this comparison, the mean stays fixed: increasing total concentration narrows the curve and reduces its response to the same incoming trial. That response is its malleability. Across different means, variance alone does not tell us how easily a belief will move.

\begin{aligned} Var (B) & = \frac{μ (1 - μ)}{τ_{E} + 1} \\ = \frac{α β}{(α + β)^{2} (α + β + 1)} . \end{aligned}

Your turn

Thicken the second notebook using the slider while keeping both sides balanced. Watch the center stay fixed at 0.50 while the uncertainty band narrows.

Two notebooks, one experiment

Same answer. Different amounts of evidence.

Balanced trials in each column of the thicker notebook 40

A little evidence

μ = 0.500

τ_{E} = 5

A lot of evidence

μ = 0.500

τ_{E} = 81

Both curves use the same axes. Thicken the second notebook: the center stays put while its uncertainty contracts.

Reading probability density curves

Because the Beta curve is a probability density, its peak height can easily exceed 1. What matters is the area beneath the curve between any two points: it gives the probability of that interval. The area over the full range from 0 to 1 always totals 1.

When you see a tall peak in the plot, do not confuse it with an impossible probability greater than 1: it simply means that probability mass has concentrated into a narrow band of plausible values.