Beliefs in motion, from evidence currents to little universes

Imagine a little boat on a river. Its captain believes the current flows west. Each new observation tugs east. A captain carrying a thin notebook changes course quickly; one carrying decades of westward observations takes longer. Then the river itself changes direction.

What is moving here? The stream of observations? The captain’s probability estimate? The accumulated record? These are three different things, and keeping them visible gives us a surprisingly useful kind of navigation.

This is the opening expedition in Bayesian relativity: exploring the Lorentz geometry hidden in positive Beta weights. We will follow currents, try a photon rocket made of retractions, compare two ways of measuring a record’s path, and finish inside a miniature expanding universe.

Put the boat on the map

Suppose an unknown success probability $P$ has a $\mathrm{Beta}(\alpha ,\beta )$ distribution, with . The weights include the prior: for example, $\alpha =Y+\frac{1}{2}$ and $\beta =N+\frac{1}{2}$ after $Y$ independent successes and $N$ independent failures with a Jeffreys prior. Here “independent” matters: repeated copies of one report belong to the same lineage, rather than becoming fresh trials.

Define four coordinates:

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

The total weight $\tau$ is our evidence clock, $x$ is signed support, and $v$ is the centered mean probability: $\mathbb{E}[P]=(1+v)/2$. It is not the Beta mode. The quantity $m$ measures a combination of amount and balance, including the prior. Its mass-like name comes from the exact identity

$${\tau }^2-x^2=m^2.$$

This is the same algebra as the energy-momentum relation in special relativity with light speed set to one. A spacetime reading also works geometrically, but has a different role: a record draws a path through $(\tau ,x)$, called its worldline. Energy-momentum vectors and positions share this cone algebra without becoming interchangeable physical objects. David Tong’s relativity lectures develop the physical counterparts.

A probability alone tells us which ray through the origin contains the belief. The total weight locates the belief along that ray. At mean $0.9$, $\mathrm{Beta}(9,1)$ and $\mathrm{Beta}(9000,1000)$ point in the same direction, but one fresh trial moves them very differently. Their probability variances also differ:

$$\mathrm{Var}(P)=\frac{1-v^2}{4(\tau +1)}.$$

More accumulated weight usually means less malleability under a fixed new packet. A large balanced record can represent substantial conflict; a tiny balanced record can represent near-vacuity.

Two arrows: where you point and what arrives

An incoming packet adds $d\alpha ,d\beta \ge 0$. Its total amount and bias are

$$d\tau =d\alpha +d\beta ,u=\frac{d\alpha -d\beta }{d\tau }.$$

A Bernoulli source with success probability $p$ has expected bias $u=2p-1$. Individual successes and failures have biases $+1$ and $-1$; a smooth current describes their average or a mixed packet.

The two arrows have different definitions:

Subtract the old centered mean from the new one:

$$\mathrm{\Delta }v=\frac{(u-v)\mathrm{\Delta }\tau }{\tau +\mathrm{\Delta }\tau }.$$

For a smooth stream this becomes

$$\frac{dv}{d\tau }=\frac{u-v}{\tau }.$$

The difference $u-v$ is the innovation, the gap between expectation and incoming evidence. The equation resembles a drag law: the current pulls the belief toward itself, with mobility $1/\tau$. In this chosen mechanical reading, the total weight supplies inertia.

Predict what happens when the source has mean $80\mathrm{\%}$, while your estimate is already $90\mathrm{\%}$. The positive current slows your centered mean down! Agreement about the direction does not imply agreement about the speed.

Bayesian relativity · motion

Ride an evidence current

Reset voyage

Steer the current Regroup the trials

Steer the incoming current with the gold handle, then release packets. Bring the teal boat into the right harbor: centered mean 0.2 to 0.4.

Current u = 0.60 Boat v = -0.60

Release 5 packetsReverse current

Keep 90% of old weights per packet

The gold current pulls; the teal boat follows with a lag. Release packets and watch the gap close.

Current history, exact coordinates, and free controls

Current u = -0.6Current u = 0.0Current u = 0.6Add 1 packetAdd 10 packets

● Belief v━ Arriving bias u

Mean probability

20.0%

Centered mean v

-0.600

Total weight τ

10.00

Next current gap u − v

1.200

Each deterministic packet has total weight 1 and support fraction (1 + u)/2. These are smooth weighted updates, not simulated random flips. Discounting applies to the whole state, including its initial weights. The horizontal clock counts packets; τ can shrink when discounting is switched on. Limit: 120 packets.

Start with “Steer the current.” Send ten packets east, then reverse the current. Repeat with discounting switched on. Gold is the arriving bias; teal is the belief. Later, return to “Regroup the trials” for a different kind of voyage.

For constant $u$, integration gives

$$v(\tau )=u+\frac{x_0-u{\tau }_0}{\tau }.$$

The smooth worldline is straight even while its secant direction swings toward the current. Curvature of that smooth path, $du/d\tau \ne 0$, instead describes a changing source. An observed sequence still zigzags between individual outcomes, even when generated by a perfectly stationary coin.

That distinction rescues an important statistical idea. Under the independent, fixed-probability Bernoulli model, the totals are sufficient for learning $p$: every sequence with the same totals has the same parameter-dependent likelihood. Sufficiency is a statement about the sampling model, not a claim that the observed path is straight. If we want to test a changing-source model, order may become informative. Compare a record that alternates heads and tails with one that abruptly switches from a long heads run to a long tails run. Berkeley’s sufficiency notes explain the exact statistical contract.

A single trial, seen from the boat

The rapidity coordinate

$$\varphi =\mathrm{artanh}(v)=\frac{1}{2}\log \frac{\alpha }{\beta }$$

stretches the finite probability interval into an infinite line. Define $\gamma =(1-v^2{)}^{-1/2}=\tau /m$. Then

$$\tau =m\cosh \varphi ,x=m\sinh \varphi .$$

A supporting trial has increment $(\mathrm{\Delta }\tau ,\mathrm{\Delta }x)=(1,1)$, while an opposing trial has $(1,-1)$. Each is null, meaning its own squared interval is zero. In the belief’s Lorentz rest frame, their energy-like weights are $e^{-\varphi }$ and $e^{\varphi }$. The familiar redshift/blueshift algebra appears directly.

Here are the finite changes, with a supporting trial denoted by $+$:

A null trial increases endpoint mass whenever the starting state is interior. Its zero own interval does not make the mass of the combined state unchanged. A single success adds support that interacts algebraically with the opposition already present.

For large $\alpha ,\beta$, the rapidity kicks approximate $e^{-\varphi }/m$ and $-e^{\varphi }/m$. Their magnitude ratio then approaches $e^{2\varphi }=\alpha /\beta$. For finite trials, the exact ratio is a ratio of logarithms. At $(30,10)$ it is about $2.91$, rather than $3$.

Two other ratios equal $\alpha /\beta$ exactly: the opposing-to-supporting mass-squared gain, and the magnitudes of the changes in mean probability. An opposing observation can therefore have a larger effect without assuming a psychological aversion to opposition. Whether people actually behave this way is a separate experimental question.

Averaging the small-trial rapidity approximation over a source gives

$$\mathbb{E}[\mathrm{\Delta }\varphi ]\approx \frac{{\gamma }^2(u-v)}{\tau }.$$

The approximation disappears for the finite mean update, whose conditional expectation is exactly $(u-v)/(\tau +1)$. Nonlinearity is why averaging rapidity and taking the rapidity of an average need different care.

Retraction as a photon rocket

Positive increments cannot preserve $m$ for an interior state. A transformation that changes rapidity while preserving $\alpha \beta$ must reduce one weight as it increases the other. That opens a new modeling choice: allow retractions.

Suppose $n$ units of opposition are found to be duplicates or invalid measurements. Removing them sends $(\alpha ,\beta )$ to $(\alpha ,\beta -n)$, with . Then

$$\mathrm{\Delta }\varphi =\frac{1}{2}\log \frac{\beta }{\beta -n},\frac{m_i}{m_f}=\sqrt{\frac{\beta }{\beta -n}}=e^{\mathrm{\Delta }\varphi }.$$

This is the exponential form of the photon-rocket mass relation. In the evidence model, retraction is the exhaust: the centered mean rises while total weight and mass fall. The state remains a proper Beta distribution as long as both weights remain positive. A past-directed null increment need not create an invalid endpoint.

The equation can describe careful correction or selective erasure, depending on which lineage justified the subtraction. Geometry alone cannot distinguish them. Nor does this equation imply that every fast-changing person must forget: a rapidly changing incoming source or a large new packet can also move a belief. The rocket interpretation has a specific engine and a specified fuel.

Memory buys responsiveness, and responsiveness admits noise

Let $t$ count external time, with evidence arriving at unit rate. Choose exponential discounting of the whole state at rate :

$$\dot{x}=u-\lambda x,\dot{\tau }=1-\lambda \tau .$$

Dots mean derivatives with respect to $t$. Here $\tau$ is effective retained weight, so it may shrink. This is a discounted model, rather than the ordinary posterior for a fixed coin with every past trial retained forever. Direct differentiation gives

$$\dot{v}=\frac{u-v}{\tau },\tau ⟶\frac{1}{\lambda }.$$

The same relaxation rule survives, but its mobility stays positive. At equilibrium weight, $\dot{v}=\lambda (u-v)$, a first-order low-pass filter: slow changes pass through more strongly than rapid fluctuations. For an oscillating current of angular frequency $\omega$, the amplitude gain is

$$|H(\omega )|=\frac{\lambda }{\sqrt{{\lambda }^2+{\omega }^2}}.$$

This connects the boat to ordinary exponential smoothing. NIST’s account explains how progressively older observations receive progressively smaller weights.

For a source drifting locally as $u(t)=u_0+kt$, the gap $c=u-v$ obeys $\dot{c}=k-c/\tau$. During a sufficiently long approximately linear episode, the equilibrium-weight filter approaches $c=k/\lambda$. Without forgetting, with $\tau =t+{\tau }_0$, the exact solution during that episode is

$$c(\tau )=\frac{k\tau }{2}+\frac{C}{\tau },$$

where $C$ is set by the starting condition. Since $u$ must remain between $-1$ and $1$, a nonzero linear drift cannot continue forever. The growing-lag expression describes a finite episode; it cannot establish an infinite-lag horizon. If the source eventually settles, the no-forgetting mean can still catch up. A Rindler or Unruh interpretation would need additional dynamics and an operational notion of horizon or temperature.

The noise price can be stated exactly for a discrete model. Discount both weights by $q\in (0,1)$ and then add one independent trial $Z\in \{-1,1\}$ with mean $u$. At stationary weight ${\tau }_{\ast }=1/(1-q)$,

$$v_{j+1}=q{v}_j+(1-q)Z_j,{\mathrm{Var}}_{\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{b}\mathrm{l}\mathrm{e}}(v)=\frac{(1-u^2)(1-q)}{1+q}.$$

The subscript means variation across repeated independent histories, not uncertainty inside one Beta distribution. Faster forgetting increases this jitter. This is an exact exponentially weighted moving-average calculation, consistent with NIST’s EWMA variance formula.

The Beta’s own centered variance is $(1-v^2)/({\tau }_{\ast }+1)$. Its ensemble average divided by the ensemble variance of $v$ is $2q/(2-q)$, approaching two for very slow forgetting. That factor is a property of this filter and its reporting convention. It invites a fluctuation-dissipation analogy, while stopping short of a thermodynamic temperature law.

There is another price for resisting a current. Add pure support at rate $r$ alongside a unit-rate source. Holding requires

$$\frac{u+r}{1+r}=v⟹r=\frac{v-u}{1-v}.$$

Maintaining a mean near one against a lower-bias current demands a rapidly increasing rate of genuinely new supporting evidence. Copied testimony cannot supply that independent stream.

Two twins, one endpoint, different packet clocks

Return to the demo and choose “Regroup the trials.” Both records start at $\mathrm{Beta}(\frac{1}{2},\frac{1}{2})$ and end after 24 successes and 16 failures. Their endpoint is fixed. Now replace the atomic zigzag by straight chords joining packet boundaries.

A packet containing $a$ successes and $b$ failures has chord length

$$\mathrm{\Delta }s=\sqrt{(a+b{)}^2-(a-b{)}^2}=2\sqrt{ab}.$$

A single trial contributes zero. A mixed pair contributes two. Summing these lengths gives the coarse-grained packet length $S$. It is the Lorentz proper length of the chosen polygonal path. Grouping replaces parts of the original zigzag with different paths; it does not change the proper length of an unchanged physical trajectory merely by relabeling its timestamps.

This makes the twin comparison precise. The trial-resolution path has accumulated length zero after its initial state. The straight mixed-packet path can have positive length, despite sharing its endpoint. The choice of resolved path is doing real mathematical work.

For future-directed increments, the reverse triangle inequality gives

$$m_{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}\ge m_0+\sum _{j}2\sqrt{a_j{b}_j}=m_0+S.$$

The initial prior mass $m_0$ belongs in this accounting. Equality holds when the prior vector and all nonzero packet vectors share the same direction. The excess $m_{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}-m_0-S$ records how much additional endpoint mass arises from combining differently directed pieces at this resolution.

We might call $S$ “conflict imported inside packets” and the excess “conflict arising between packets.” Those descriptions depend on the chosen partition. They include a prior contribution and do not automatically diagnose emotional conflict.

Try sorting the record into pure runs. Small packets mostly become null, except where a packet straddles the switch between successes and failures. The length therefore depends on packet composition and alignment as well as size. For independent trials with success probability $p$, a pair is mixed with probability $2p(1-p)$, so its expected length is $4p(1-p)$. A shuffled record conditioned on fixed totals has a slightly different finite-population probability.

Try a delivery of your own: arrange the same eight trials into packets with a requested clock length. Each seam changes the represented path while the endpoint stays put.

Same cargo · different clocks

The packet-clock courier

Reset cargo

Deliver these eight trials with packet length 0. Tap two trial tiles to swap them; tap a seam to join or split packets.

+trial 1 Split │

+trial 2 Split │

+trial 3 Split │

+trial 4 Split │

−trial 5 Split │

−trial 6 Split │

−trial 7 Split │

−trial 8

Choose a tile, then another. Seams define the arriving packets.

Packet 11+ / 0−length 0.00

Packet 21+ / 0−length 0.00

Packet 31+ / 0−length 0.00

Packet 41+ / 0−length 0.00

Packet 50+ / 1−length 0.00

Packet 60+ / 1−length 0.00

Packet 70+ / 1−length 0.00

Packet 80+ / 1−length 0.00

Packet length 0.00Endpoint mass 9.00

Deliver this routeSplit every trialOne whole packet

Why the clock changes · free exploration

Each straight packet contributes 2√(support × opposition). The eight underlying atomic trials still form a null zigzag with length zero. Changing the seams replaces stretches of that path with straight chords. A Beta(½,½) prior contributes mass 1; the final Beta(4½,4½) has mass 9. Thus 1 + packet length ≤ 9. The goal changes only your invitation; every arrangement remains available.

Try length 0Try length 4Try length 8

What history remembers

The endpoint retains amount and balance; the path retains order, packet boundaries, retractions, and changing rates. A ledger of tiny tracelets can additionally preserve lineage, allowing a correction to travel back to the measurements it invalidates.

A full history of $v$ indexed by known $\tau$ already determines $x=v\tau$. There is no extra hidden position variable to recover. Loss occurs when we discard the clock or compress the history to its final point. A small probability change could mean a weak new packet or a very heavy existing record; a probability-only log cannot generally tell which.

These distinctions suggest practical tools: change-point detectors that inspect the path, trackers whose memory matches the world’s timescale, and belief maps that display amount alongside direction. They also make experiments sharper. A claim about “acceleration” must say whether it means changing source bias, changing mean, changing rapidity, or changing the rate at which trials arrive.

A small Milne universe

There is one more coordinate transformation worth taking home. Substitute $\tau =m\cosh \varphi$ and $x=m\sinh \varphi$ into the metric:

$$d{\tau }^2-d{x}^2=d{m}^2-m^{2}d{\varphi }^2.$$

This is the two-dimensional Milne form of flat spacetime. Constant-rapidity rays play the role of comoving observers; constant-$m$ hyperbolas supply spatial slices. Along a slice, distance is $m|d\varphi |$, so fixed rapidity separations grow in that coordinate description as $m$ increases. Milne coordinates offer an expanding description of a region that remains flat. This cosmology analysis explains the physical coordinate construction.

The semicircle angle $\theta$, defined by $v=\sin \theta$, connects to rapidity through

$$\sin \theta =\tanh \varphi ,d\theta =\mathrm{sech}\varphi d\varphi .$$

Here $\mathrm{sech}\varphi =1/\cosh \varphi$. The Jeffreys measure is uniform in $\theta$, hence has rapidity density $\mathrm{sech}\varphi /\pi$. It is not translation-invariant in $\varphi$. Uniform rapidity measure would instead be proportional to $d\varphi$, which cannot normalize over the whole real line.

That leaves two beautiful geometries side by side: Fisher distance supplies the finite semicircle, while Lorentz rapidity supplies an unbounded direction coordinate. Their connection is exact. Giving the little universe a physical expansion law would be a further adventure.

Further reading

• A belief is more than a number: Beta shapes, trial amount, and malleability.
• David Tong, Dynamics and Relativity: the Lorentz geometry behind the dictionary.
• Will Fithian, Sufficiency: what an endpoint can retain under a specified sampling model.
• NIST, Single Exponential Smoothing and EWMA Control Charts: memory, responsiveness, and ensemble variance.