Evidence atoms: generators, echoes, and a strange mirror

Imagine a tiny instrument with two sockets. Feed observations into one socket; predictions emerge from the other. Place a hundred such instruments in a room, let them listen to one another, and something interesting happens: yesterday’s prediction can return tomorrow disguised as fresh evidence.

This is the beginning of an evidence chemistry. Its atoms have internal states, its bonds carry shared assumptions, and its reactions include learning, copying, and reinterpretation. To build it carefully, we need to know what one atom can do, which transformations preserve its geometry, and what information disappears when we summarize a whole history with two numbers.

A distribution with an output socket

Our atom is a proper Beta distribution, $p\sim \mathrm{Beta}(\alpha ,\beta )$, with positive weights $\alpha$ and $\beta$. The unknown $p$ is the success probability of a Bernoulli trial, a trial with two possible outcomes. These weights include the prior. With prior weights $({\alpha }_0,{\beta }_0)$ and observed counts $(Y,N)$, they become $({\alpha }_0+Y,{\beta }_0+N)$.

Its predictive probability for the next success is its mean:

$$\mu =\frac{\alpha }{\alpha +\beta }.$$

Here are two ways to generate an entire sequence:

Both machines assign the same probability to every finite ordered sequence. If a particular sequence contains $Y$ successes and $N$ failures, that probability is

$$Pr(\text{that sequence})=\frac{B(\alpha +Y,\beta +N)}{B(\alpha ,\beta )},$$

where $B(a,b)={\int }_0^1{p}^{a-1}(1-p{)}^{b-1}dp$ is the Beta function. A count probability additionally needs the number of ways to arrange those outcomes.

The urn representation is a beautiful consequence of exchangeability: rearranging a sequence leaves its probability unchanged. De Finetti’s theorem says an infinite exchangeable binary sequence has a representation as a mixture of independent, identically distributed Bernoulli trials. The mixing distribution need not be Beta; choosing Beta gives the reinforcing urn above. Jim Pitman’s exchangeability lectures develop this connection.

Try the smallest example. Starting from Beta(1, 1), two successes have probability $(1/2)(2/3)=1/3$. Two independent fair coin flips would give $1/4$. The generated trials are independent conditional on the hidden rate, while uncertainty about that shared rate creates marginal dependence.

This distinction matters for talking atoms. An agent may generate a prediction from its current distribution without discovering anything new about the world. Feeding that prediction back as an independent observation creates a reinforcement rule. Whether that rule is appropriate depends on what the emitted message actually measures.

Open the workbench

Four coordinates make the atom’s geometry visible:

$$\tau =\alpha +\beta ,x=\alpha -\beta ,m=2\sqrt{\alpha \beta },\varphi =\frac{1}{2}\log \frac{\alpha }{\beta }.$$

Here $\tau$ is prior-inclusive trial amount, $x$ is net support, $m$ is the cone’s mass coordinate, and $\varphi$ is rapidity. They obey ${\tau }^2-x^2=m^2$. The centered mean is $v=x/\tau =2\mu -1=\tanh \varphi$.

Predict before clicking: which move preserves mass? Which preserves the mean? Can a supporting trial change mass even though its own step lies along a lightlike direction?

Moves and lineage

Choose the move that keeps a promise

Reset workbench

An atom starts at Beta(3, 1). Choose a tool that meets the mission, then reveal its move. All three tools are available; the geometry decides which promise survives.

Mission 1 of 3

Change the mean while keeping mass fixed

Trialα → α + 1 Boostα × e⁰·⁴, β × e⁻⁰·⁴ Dilationα × 2, β × 2

Reveal this moveRetry missionNext mission

Free workbench: all geometric moves

+ support + opposition Boost +0.4 Boost −0.4 Halve weights Double weights Invert

Start with Beta(3, 1). Predict which quantity each move preserves.

Mean μ

0.750 → 0.750

Mass m

3.464 → 3.464

Amount τ

4.000 → 4.000

All coordinates

Quantity	Previous	Current
Support α	3.000	3.000
Opposition β	1.000	1.000
Mean μ	0.750	0.750
Total τ	4.000	4.000
Mass m	3.464	3.464
Rapidity φ	0.549	0.549
Variance of p	0.037	0.037

Open the separate lineage ledger

One observation, how many messages?

This separate ledger starts with a uniform Beta(1, 1) prior and two independent original trials. Source numbers identify original trials, even when several messages repeat them. Click a source to echo it.

New supporting trial New opposing trial

+ Trial 1 ↻ 1 reports− Trial 2 ↻ 1 reports

Count every message

2 reports + prior

Beta(2, 2)

Mean 0.500 · variance 0.050

Count original trials

2 distinct trials + prior

Beta(2, 2)

Mean 0.500 · variance 0.050

Maximum 40 reports. This ledger knows exact ancestry and assumes independent original Bernoulli trials. Real source dependence can survive deduplication. The workbench moves are chosen operations on a Beta state, with no claim that each describes ordinary learning.

The second half of the workbench has its own ledger. Add independent original trials, then click their cards to echo them. Watch how a repeated supporting report can pull the naive mean away from the lineage-aware mean. Neither ledger uses the transformed atom above as its prior.

Four moves, four different questions

Add a trial: what happened?

A supporting trial of weight sends $(\alpha ,\beta )$ to $(\alpha +w,\beta )$. An opposing trial changes the other coordinate. In the cone these increments are $(\mathrm{\Delta }\tau ,\mathrm{\Delta }x)=(w,\pm w)$, so their own squared interval is zero.

The atom receiving them still gains mass:

$$\mathrm{\Delta }{m}^2=4\beta w\text{for support},\mathrm{\Delta }{m}^2=4\alpha w\text{for opposition}.$$

A lightlike increment and a zero change of endpoint mass are different statements. Opposing directions combine into an interior, timelike state.

Positive additions compose freely: add three trials, then two, and you have added five. Inverses require a history and sufficient remaining weight. Removing a known observation can undo its update; arbitrary subtraction is a different modeling choice. The positivity of proper Beta parameters limits both. Integer raw counts give discrete steps, while weighted evidence permits continuous steps.

Boost: how is support reweighted?

Choose a real number $\delta$ and set

$$(\alpha ,\beta )⟼(e^{\delta }\alpha ,e^{-\delta }\beta ).$$

Their product stays fixed, so $m$ stays fixed. Their ratio changes, giving $\varphi \mapsto \varphi +\delta$. The point moves along a constant-mass hyperbola.

Compare this with ordinary Bayes at the level of a single hypothesis probability:

$$\underset{\text{boosted Beta mean odds}}{\underbrace{\frac{{\mu }^{\prime }}{1-{\mu }^{\prime }}=e^{2\delta }\frac{\mu }{1-\mu }}},\underset{\text{Bayes, likelihood ratio }L}{\underbrace{\frac{P(H\mid D)}{1-P(H\mid D)}=L\frac{P(H)}{1-P(H)}}}.$$

Choosing $L=e^{2\delta }$ makes these odds updates match. Bayes’ rule for a scalar hypothesis probability does not specify a new pair of Beta weights. Preserving $m$ is an additional choice that supplies that missing degree of freedom. Updating a Beta distribution with Bernoulli observations instead uses the additive move above.

Dilate: how much weight does this state carry?

For , multiply both weights by $c$. The mean and rapidity stay fixed; $\tau$ and $m$ scale by $c$. The variance of the unknown rate is

$$\mathrm{Var}(p)=\frac{\mu (1-\mu )}{\tau +1}.$$

At fixed mean, reducing trial amount increases this variance and makes a single incoming trial more influential. That is one precise reading of increased malleability.

There is a subtle link to tempering, raising a density to a power and renormalizing. Relative to ordinary $dp$, the Beta density has exponents $\alpha -1$ and $\beta -1$. Taking its power $c$ produces weights $1+c(\alpha -1)$ and $1+c(\beta -1)$ when these remain positive.

Pure dilation appears with a different carrier measure, the reference against which density is expressed:

$$d\nu (p)=\frac{dp}{p(1-p)},\frac{dP}{d\nu }\propto p^{\alpha }(1-p{)}^{\beta }.$$

Taking the power $c$ of this density gives Beta$(c\alpha ,c\beta )$ after normalization. The carrier measure is infinite, although the resulting proper Beta probability measure is finite. It is also invariant under translating log odds. Specifying the carrier makes the tempering connection exact.

Invert: visit the mirror atom

Now try the stranger move:

$$(\tau ,x)⟼\frac{(\tau ,x)}{m^2},(\alpha ,\beta )⟼(\frac{1}{4\beta },\frac{1}{4\alpha }).$$

The mean stays fixed and $m\mapsto 1/m$. Apply it twice and you return home. Beta(30, 10), with mean $3/4$, becomes Beta$(1/40,1/120)$, also with mean $3/4$.

Picture what that means. The first distribution concentrates around an interior rate. The second is U-shaped, with density rising near both endpoints. The two share an average prediction, while making profoundly different claims about the hidden rate. Small trial amount alone does not mean a flat distribution or evenly spread ignorance.

Inversion has an exact geometric definition. A practical learning procedure that should perform this move remains an open question. Could it serve as a deliberate reset that retains a mean prediction? The next observation would have extraordinary influence. That makes it an interesting operation to investigate, with a very specific statistical consequence.

What kind of symmetry is this?

Independent positive reweightings $(\alpha ,\beta )\mapsto (a\alpha ,b\beta )$ split neatly into dilation and boost:

$$c=\sqrt{ab},\delta =\frac{1}{2}\log (a/b).$$

Together with exchanging support and opposition, these describe the linear automorphisms of the positive cone. They preserve the cone; only the boosts and exchange preserve its metric exactly. Learning translations preserve the flat metric where defined, but their negative inverses need not remain in the positive cone.

Inversion belongs to conformal geometry, which preserves the metric up to a position-dependent scale. Combining inversion, translation, and inversion generates special conformal maps on their admissible domains. This finite family sits inside the richer local conformal freedom of two dimensions. It does not supply a conformal field theory of beliefs by itself: a field theory needs fields, dynamics, and a reason those dynamics respect the symmetry. David Tong’s discussion of conformal symmetry provides the physics background.

Echoes have a calculable price

Suppose two reporters each bring $n$ Bernoulli observations at a common fixed rate $p$. A fraction $\rho$ of each report consists of the same original trials. Let $C$ count successes in the shared block and $I_1,I_2$ count successes in the two independent blocks. Assume those three blocks are mutually independent, with $\rho n$ an integer.

The naive pooled estimate is $(2C+I_1+I_2)/(2n)$. Its actual sampling variance is

For , the variance is $1+\rho$ times the independent-trial calculation. This is an exact sampling result for the stated shared-block model, rather than a universal ratio between Bayesian posterior variances.

Two useful counts now separate:

$$\underset{\text{effective size of the naive weighted estimator}}{\underbrace{n_{\mathrm{eff}}=\frac{2n}{1+\rho }}},\underset{\text{original trials after deduplication}}{\underbrace{n_{\mathrm{distinct}}=(2-\rho )n}}.$$

With $n=100$ and $\rho =1/2$, there are 150 distinct trials, but the naive estimator has effective size $400/3$. It overweights the shared block. Once original trials are available, counting each once produces a different, more efficient estimator under these assumptions.

Put the lineage rule to work in an archive puzzle. Collect six independent roots using two bundles, then allow an extra overlapping bundle. Predict whether the combined Beta should change when the underlying roots stay the same.

The archive puzzle

Assemble a theory from shared roots

Clear the assembly

Bring all six original trials to the table using at most 2 archives. Tap archive cards to pack or unpack them. Reports with the same letter share one root.

A+ supportmissing

B+ supportmissing

C− opposemissing

D− opposemissing

E+ supportmissing

F− opposemissing

Field journalA · B · CTap to packReviewB · C · DTap to packSecond surveyD · E · FTap to packNorth archiveA · C · ETap to packSouth archiveB · D · FTap to packBriefingA · FTap to pack

One shared prior
Beta(1, 1)+0/6 independent roots
in 0/2 archives

Seal the ledger

0 reports contain 0 distinct trials and 0 repeated appearances. The prior enters once; every root enters once.

Try overlap, then inspect the bonds

Increase the budget to three and include overlapping archives. Do you get a different final Beta after collecting the same six roots?

Two-archive challengeThree-archive exploration

These bonds record known shared ancestry. All six original Bernoulli trials are assumed independent. This assembly is set union, with no assumed physical attraction and no inference of independence merely from different labels.

Can you hear an echo without reading it?

Copied messages often arrive in clusters. A Hawkes process models this by letting one arrival temporarily increase the rate of later arrivals. With baseline rate ${\lambda }_0$, branching ratio , and exponential triggering kernel $b\omega e^{-\omega t}$, the stationary mean rate is

$$\bar{\lambda }=\frac{{\lambda }_0}{1-b}.$$

The characteristic covariance decay time is $1/[\omega (1-b)]$. Near $b=1$, activity becomes more amplified and persistent. These results follow from the specified self-exciting process. Hawkes’ original paper develops its spectra and covariance.

The optical analogy is enticing: attention supplies a pump, a receptive population supplies amplification, and recirculating messages supply feedback. A self-exciting cascade can resemble an amplifier approaching threshold. It does not establish laser coherence or literal stimulated emission.

Photon experiments use $g^{(2)}$, a normalized pair-arrival correlation, to measure bunching. Related timing statistics can flag clustered testimony. But shared schedules, breaking news, and fluctuating external rates can cluster independent reports too. A useful copying detector would compare timing with observed lineage and those alternatives. Bunching alone neither identifies a copying fraction nor reveals a wave phase.

What two numbers leave out

A weighted binary trial has cone increment $w(1,\pm 1)$. In a chosen directional extension, a unit direction $\hat{n}$ gives $w(1,\hat{n})$, again lightlike because $1-\Vert \hat{n}{\Vert }^2=0$. Assigning category directions is part of that construction; a general categorical model does not automatically arrive with a unique spatial embedding.

Weight and direction still cannot answer: Where did this trial come from?

A tracelet can carry an original-trial identity and its lineage through later reports. Think of the geometric point as a library catalog entry and lineage as the borrowing history attached to it. Mathematicians sometimes call extra state attached over a base point a fiber. This is a useful design picture for provenance, framing, and other internal variables.

Renaming trial 17 to trial 92 should change no prediction. Merging two independent trials or splitting one trial into invented copies should change what can legitimately be pooled. This relabeling invariance resembles a gauge idea, but identifying duplicates is an equivalence-class operation. Calling it gauge fixing adds no theorem about physical gauge fields. Lineage-aware inference already has concrete value without that extra structure.

Framing suggests another experiment. A polarizer transmits an intensity fraction ${\cos }^2\theta$ at angle $\theta$ to incoming linear polarization. Two crossed polarizers block transmission; inserting one at ${45}^{\circ }$ between them passes one quarter of the light emerging from the first. MIT’s polarization lecture explains the optical law.

Could an intermediate reframing similarly open a route between otherwise incompatible interpretations? That is a testable behavioral hypothesis. Predicting the same cosine-squared law requires a specified state-and-projection model plus data. Noncommuting framing operations and phase-weighted amplitudes require additional state beyond one ordinary Beta distribution.

Toward an evidence chemistry

Imagine several propositions connected by shared premises. Changing one premise shifts several beliefs; an analogy reveals a new route between interpretations. The chemistry vocabulary becomes productive: beliefs as atoms, common evidence as bonds, theories as molecules, and arguments as possible catalysts.

Shared sources can induce dependence, but the number of shared sources is no universal bond strength. Mutual information, which measures statistical dependence, is one candidate once a joint probability model exists. Two Beta marginals alone cannot determine it.

For transport, suppose each communication hop retains a fraction $r$, with , of an influence signal. After $d$ hops,

$${\ell }_d=r^d{\ell }_0=e^{-d/\xi }{\ell }_0,\xi =-\frac{1}{\log r}.$$

Here $\ell$ could be a log-likelihood ratio and $\xi$ is a decay length measured in hops. Attenuating $\ell$ toward zero makes the likelihood ratio approach one, the neutral value. Multiplying the likelihood ratio itself repeatedly by $r$ would instead push it toward zero and eventually reverse its meaning.

Exponential decay resembles screening in physics. A Yukawa potential also has dimension-dependent spatial factors and follows from a specific field equation, so this communication rule establishes a screening analogy rather than a particle mediator mass. Likewise, a network’s interaction range and its collective correlation length need not coincide.

Still, a fertile question remains: when a medium preserves messages better, does it also support more elaborate shared theories and more distant coordination? Copying fidelity, trust, network structure, and attention could pull in different directions. Measuring them separately would turn the chemistry picture into an investigable model.

A distribution is a compact atom with a predictive interface. A lineage ledger tells us which encounters were new. Transformations reveal what stays fixed and what changes. With those pieces on the bench, the next adventure is to choose interactions and discover what kinds of collective behavior they actually generate.

Further reading

• Pitman’s probability lectures: exchangeability, de Finetti representation, and Pólya urns.
• Hawkes on self-exciting point processes: the mathematical source for arrival cascades.
• Tong on conformal symmetry: the geometric setting beyond boosts and dilations.
• One trial, many echoes: more on lineage-aware pooling.