The hidden springs of a belief

Miraia S. Chiou

Survey · Statistics

The hidden springs of a belief

Find Gaussian springs inside a Beta belief, then listen with complex rotating arrows.

2026-09-08 · 17 min read · 2 interactive demos

Imagine opening a weather sensor and finding a spring inside. Open the spring, and you find a cloud of smaller springs. Turn on the radio, and the cloud starts drawing circles.

That is approximately our itinerary. A Beta belief about a probability can become a hyperbolic landscape, a collection of hidden Gaussian wells, and a complex pattern of rotating arrows. These are exact changes of mathematical viewpoint. Epistemic Physics invites us to ask what we could build by moving between them.

Our sensor answers one question: will a seed germinate? The unknown success probability is $p$ . Supporting and opposing trials have produced a Beta distribution,

\begin{aligned} p & \sim Beta (α, β), \\ α & = Y + \frac{1}{2}, \\ β & = N + \frac{1}{2} . \end{aligned}

Here $Y$ counts germinations, $N$ counts failures, and the starting halves give a total prior amount of one. We could use other positive shapes; this starting point makes the before-and-after experiments particularly clear.

Keep the whole uncertain $p$ with us. We are about to change the coordinate of every possible probability, rather than replace the distribution with one summary number.

Stretch probability into a landscape

Probability lives between zero and one. Rapidity stretches that interval onto the whole real line:

\begin{aligned} r & = \frac{1}{2} \log \frac{p}{1 - p}, \\ p & = \frac{1 + \tanh r}{2} . \end{aligned}

The logarithm is natural, and $\tanh$ is the hyperbolic tangent. A probability of one half becomes $r = 0$ . Probabilities approaching one travel arbitrarily far right; probabilities approaching zero travel arbitrarily far left.

Define the shape amount $A = α + β$ and polarity $P = α - β$ . The density in this new coordinate is

f_{r} (r) = \frac{2^{1 - A}}{B (α, β)} \exp (P r) \cosh^{- A} r .

The Beta function $B$ supplies normalization. The hyperbolic cosine is $\cosh r = (e^{r} + e^{- r}) / 2$ . To obtain this density, multiply the Beta density by the change-of-coordinate factor $d p / d r = 2 p (1 - p)$ . That extra factor is why the exponent contains $A$ , including the prior amount.

A useful landscape is the potential, the negative logarithm of density with an irrelevant constant removed:

V (r) = A \log \cosh r - P r .

Low ground has high density. Its minimum occurs at

r_{*} = \frac{1}{2} \log \frac{α}{β} .

This is the mode of the density in rapidity. Mapping it back gives the Beta mean $α / A$ . Density modes depend on the coordinate used to measure density; the mean of random rapidity is another quantity again.

Near the minimum, the landscape looks like a spring’s bowl. Far away, it behaves differently. Its slope and curvature are

\begin{aligned} V^{'} (r) & = A \tanh r - P, \\ V^{″} (r) & = A {sech}^{2} r, \end{aligned}

where $sech r = 1 / \cosh r$ . Curvature measures how quickly the slope changes. An ordinary quadratic spring grows progressively steeper. Here the rightward slope approaches $2 β$ , while the leftward slope approaches $- 2 α$ .

Choose to read $- V^{'}$ as a restoring force, and we have a force that saturates. Close to the favored state, the belief responds elastically; far away, the force approaches a fixed magnitude. The two tails separately remember the opposing and supporting shape amounts.

Open the trapdoor beneath the bowl

The Pólya-Gamma identity introduces a positive hidden variable $ω$ :

\cosh^{- A} r = E_{ω \sim PG (A, 0)} [e^{- 2 ω r^{2}}] .

The symbol $E$ means an average, and $PG$ names the Pólya-Gamma distribution. The identity converts the hyperbolic factor into an average of quadratic exponential factors. It underlies the Bayesian logistic inference method of Polson, Scott, and Windle.

Now include the tilt $P r$ and complete the square:

\begin{aligned} P r - 2 ω r^{2} \\ = - 2 ω {(r - \frac{P}{4 ω})}^{2} \\ + \frac{P^{2}}{8 ω} . \end{aligned}

Hold $ω$ fixed and the rapidity distribution becomes Gaussian, the familiar bell curve:

r ∣ ω \sim N (\frac{P}{4 ω}, \frac{1}{4 ω}) .

The vertical bar means “given.” The two arguments of $N$ are its mean and variance. The spring’s precision, the reciprocal of variance, is $4 ω$ . Increasing it narrows the bell and pulls its center toward zero when $P$ stays fixed.

There is a lovely bookkeeping detail inside that completed square. The normalized Gaussian contributes a factor $\sqrt{π / (2 ω)} \exp (P^{2} / (8 ω))$ when integrated over $r$ . Consequently, the hidden variable’s marginal weight in this joint construction is

\begin{aligned} q (ω) & \propto PG (ω; A, 0) \\ \times \sqrt{\frac{π}{2 ω}} \exp (\frac{P^{2}}{8 ω}) . \end{aligned}

Here $PG (ω; A, 0)$ denotes a density evaluated at $ω$ . These are the weights for mixing normalized Gaussian bells. The simpler raw Pólya-Gamma weights average the unnormalized exponential factors in the identity above.

The other conditional distribution is especially convenient:

ω ∣ r \sim PG (A, 2 r) .

An alternating calculation can therefore refresh the hidden spring given rapidity, then refresh rapidity given the spring. A complicated marginal curve becomes two manageable conditional steps.

Predict before touching: move far into a tail. Will the full landscape’s curvature equal the average precision of the hidden springs found there? Pin that mean spring and compare the two wells.

Open the hyperbolic well

Fit a spring to the bend

The sky well gets softer away from its middle. Choose a gold spring that follows its local bend at the rose probe. Then compare that fitted spring with the hidden conditional spring.

Pick a spring from the rack. Gold will touch the same point with the same slope; its precision controls the bend.

Full Beta rapidity potentialLocal trial springMovable rapidity probe

Inspect curvatures, free controls, and the hidden recipe

α = 2.5, β = 2.5, including one Jeffreys half-count per channel.

Gold precision: 5.000

Probe rapidity r: 1.50
Sky curvature at r: 0.904
Mean precision given r: 3.017
Gold Gaussian center: 0.000

At r = 1.50, the full well has curvature 0.904. The hidden spring has mean precision 3.017 when conditioned on that r. Changing a hidden variable and bending the complete curve are different operations.

Watch the infinite spring recipe converge

Multiplying the first N Gamma factors gives an upper bound for the hyperbolic factor. A bound on the remaining factors gives the lower end.

Lower bound: 1.3747e-2
Exact hyperbolic factor: 1.3881e-2
Upper bound: 1.6003e-2

\begin{aligned} V (r) & = A \log \cosh r - P r, \\ r ∣ ω & \sim N (\frac{P}{4 ω}, \frac{1}{4 ω}) \end{aligned}

Open the hidden-spring experiment in the playground.

Why hiding the springs makes the landscape softer

The average hidden precision at a specified rapidity is

4 E [ω ∣ r] = A \frac{\tanh r}{r},

with value $A$ at $r = 0$ . Compare it with the full curvature $A {sech}^{2} r$ . Both agree at the origin; they separate as we move away.

We can explain the difference exactly. Differentiate the conditional mean while remembering that the spring weights themselves change with $r$ :

\begin{aligned} \frac{d}{d r} E [ω ∣ r] \\ = - 4 r Var (ω ∣ r), \\ V^{″} (r) = 4 E [ω ∣ r] \\ - 16 r^{2} Var (ω ∣ r) . \end{aligned}

The variance measures the spread of possible hidden stiffnesses. Its contribution subtracts from the average precision.

Picture testing a mattress while its internal springs can rearrange. Holding the arrangement fixed measures one stiffness. Allowing the arrangement to respond makes the mattress softer. Here that rearrangement is a change in which hidden values receive weight when we condition on a different rapidity.

For $A = 7$ and $r = 2$ , the mean hidden precision is about $3.374$ , while the marginal curvature is about $0.495$ . Their difference records substantial freedom beneath the surface.

This suggests an EP interpretation worth pursuing: a belief’s malleability may partly reflect unresolved internal arrangements. Even when each conditional mechanism is simple, averaging over possible mechanisms can produce a flexible macroscopic response.

Give neighboring beliefs connecting springs

Suppose two gardens have related growing conditions. Each has a rapidity $r_{1}$ or $r_{2}$ describing uncertain germination probability. Choose a coupling that favors similar rapidities:

\begin{aligned} V_{pair} & = V_{1} (r_{1}) + V_{2} (r_{2}) \\ + \frac{λ}{2} (r_{1} - r_{2})^{2} . \end{aligned}

The coupling strength $λ \geq 0$ controls how strongly the gardens inform one another. This is a chosen statistical model of their relationship. Separate trial lineages still record which observations belong to each garden.

After introducing one hidden spring per garden, the equations for the conditional mean become

\begin{aligned} (4 ω_{1} + λ) r_{1} - λ r_{2} & = P_{1}, \\ - λ r_{1} + (4 ω_{2} + λ) r_{2} & = P_{2} . \end{aligned}

Two coupled linear equations! The full conditional distribution is a two-dimensional Gaussian. The diagonal terms resist motion; the off-diagonal terms transmit it between gardens.

Now imagine a continent of sensors, a tissue of cells, or a scientific community whose local models exchange information. Local Beta-shaped landscapes could become a network of fluctuating quadratic constraints. The computational engine would repeatedly solve Gaussian problems while allowing the hidden stiffnesses to change.

A particularly inviting experiment would compare fixed coupling with learned coupling: let two regions share information until their outcomes suggest different environments. The network could discover where a common explanation remains useful and where it should split.

Listen with an imaginary antenna

We have inspected uncertainty by opening a hidden coordinate. Another approach leaves the distribution intact and asks how it responds to a rotating probe.

Attach to each possible rapidity a unit arrow,

e^{i k r} = \cos (k r) + i \sin (k r) .

The real number $k$ is a frequency measured in radians per unit rapidity. The imaginary unit $i$ , with $i^{2} = - 1$ , labels the perpendicular direction of the drawing plane. Averaging the arrows gives the characteristic function,

ϕ (k) = E [e^{i k r}] .

Its real part is the average cosine, its imaginary part the average sine. At $k = 0$ , every arrow points right, so $ϕ (0) = 1$ . Increasing the frequency makes nearby possibilities point in increasingly different directions. Cancellation shortens their average.

The entire function $ϕ$ , across all frequencies, determines the distribution. One chosen frequency acts like a particular listening channel.

For a Beta belief, substitute $r = \frac{1}{2} \log [p / (1 - p)]$ :

\begin{aligned} e^{i k r} & = p^{i k / 2} (1 - p)^{- i k / 2}, \\ ϕ (k) & = \frac{B (α + i k / 2, β - i k / 2)}{B (α, β)} . \end{aligned}

The imaginary shifts follow directly from Euler’s Beta integral. The complex expression is the average of a probe over our ordinary positive distribution. Its two components encode how the arrows reinforce or cancel.

For the starting shapes $α = β = 1 / 2$ , something beautiful happens. The density and its frequency response have matching forms:

\begin{aligned} f_{r} (r) & = \frac{sech r}{π}, \\ ϕ (k) & = sech (\frac{π k}{2}) . \end{aligned}

The hyperbolic secant reappears after the Fourier transformation, with a different scale. The same curve speaks in two coordinate languages.

Predict before touching: balanced trials should gather rapidities closer to zero. At a fixed frequency, do the arrows cancel more or less? Then add mostly supporting trials. Finally, translate the entire distribution without changing its shape.

An ordinary probability, a complex probe

Listen to a belief with rotating arrows

Turn without shrinking. Translate the cloud until the teal arrow reaches the gold target. Keep its length.

Free exploration: trials, examples, and frequency

0 yes, 0 no; α = 0.5, β = 0.5. Trials update the underlying Beta belief. Translation d = 0.00 is then applied to its rapidity, moving the whole distribution without changing its shape.

Probe frequency k 1.00

The density and its probes

The average arrow

Distance to target: 0.382. The gold ring is the target; teal is your average.

Rapidity densityUnit probesComplex averageAdjustable frequency

Translation controls, numbers, and the Fourier rule

Real part, average cosine: 0.3985
Imaginary part, average sine: 0.0000
Arrow length |φ(k)|: 0.3985
Phase angle: 0.0°

Opposite rapidities carry equal weight. Their sine parts cancel, leaving a real average. Translating the whole curve rotates its average through kd, while preserving the arrow length.

\begin{aligned} ϕ (k) & = E [e^{i k r}], \\ ϕ_{r + d} (k) & = e^{i k d} ϕ_{r} (k) \end{aligned}

Open the rotating-arrow experiment in the playground.

At $k = 1$ , the starting distribution has arrow length about $0.399$ . After five yes and five no trials, it is about $0.951$ . After eight yes and two no trials, the complex average is approximately $0.718 + 0.585 i$ . Learning has changed both the distribution’s spread and its directional position in rapidity.

A change of calibration rotates the answer

The translation experiment has an unexpectedly concrete interpretation. Independent positive Gamma amounts can generate a Beta share. A Gamma distribution has a positive shape and rate; at fixed shape, increasing the rate reduces the typical amount.

Let $U_{+}$ and $U_{-}$ have shapes $α, β$ and unit rates. Using positive rates $λ_{+}$ and $λ_{-}$ , define differently calibrated amounts:

\begin{aligned} G_{+} & = \frac{U_{+}}{λ_{+}}, \\ G_{-} & = \frac{U_{-}}{λ_{-}} . \end{aligned}

The rapidity of their share $G_{+} / (G_{+} + G_{-})$ is

\begin{aligned} r & = \frac{1}{2} \log \frac{G_{+}}{G_{-}} \\ = \frac{1}{2} \log \frac{U_{+}}{U_{-}} + \underset{d}{\underset{⏟}{\frac{1}{2} \log \frac{λ_{-}}{λ_{+}}}} . \end{aligned}

Unequal Gamma rates become a simple rapidity translation $d$ . Their normalized share generally leaves the ordinary Beta family, while its rapidity density keeps exactly the same shape, shifted sideways.

Every arrow gains the same extra angle $k d$ :

ϕ_{r + d} (k) = e^{i k d} ϕ_{r} (k) .

The average rotates without changing length. A calibration shift and a change in the distribution’s spread therefore leave different spectral signatures. We could use that distinction to compare sensor recalibration with genuine learning about a probability.

What if the boosts themselves are uncertain?

Rapidity makes successive odds multipliers add. Suppose several independent uncertain rapidity increments $R_{1}, \dots, R_{m}$ accumulate:

R_{total} = R_{1} + \dots + R_{m} .

Their characteristic functions multiply:

ϕ_{total} (k) = \prod_{j = 1}^{m} ϕ_{j} (k) .

Independence is the assumption that permits this factorization. With shared lineages, we would average the product of arrows using the joint distribution instead. When evidence takes different roads explores how to reconstruct that shared structure from the original observations.

Imagine an uncertain calibration travelling through several laboratories, or an uncertain sequence of changes of reference frame. Each stage adds a possible boost. The spectrum keeps track of the resulting cloud without reducing it to one trajectory.

At a small frequency, the arrow reveals familiar statistics:

\begin{aligned} ϕ (k) & = 1 + i k E [R] \\ - \frac{k^{2}}{2} E [R^{2}] + \dots . \end{aligned}

The initial turn records mean rapidity; the initial decrease in its length records variance. Higher frequencies inspect finer structure.

We can now ask a larger question. Could an EP inference system switch representations according to the calculation it needs: trials for updating, hidden springs for coupling, rotating arrows for accumulating uncertain transformations? A sensor network could learn locally in Beta coordinates, negotiate shared conditions through Gaussian conditionals, and carry calibration uncertainty through spectral products. Every switch in that proposal has an explicit mathematical doorway. The adventure is deciding which combinations make a useful machine.

Follow this intoMathematics ↗