--- title: "Beliefs in motion, from evidence currents to little universes" author: "Miraia S. Chiou" creator: "Miraia Chiou" published: "2026-09-18" identifier: "epics-bsr-000" source: "https://epistemicphysics.com/articles/beliefs-in-motion/" citation: "https://epistemicphysics.com/cite/?work=epics-bsr-000" --- # Beliefs in motion, from evidence currents to little universes Ride an evidence current, feel the cost of changing course, and discover why two identical Beta beliefs can carry different histories. [Read the illustrated article](https://epistemicphysics.com/articles/beliefs-in-motion/) 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 $\operatorname{Beta}(\alpha,\beta)$ distribution, with $\alpha,\beta>0$. The weights include the prior: for example, $\alpha=Y+\tfrac12$ and $\beta=N+\tfrac12$ 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: $$ \tau=\alpha+\beta,\qquad x=\alpha-\beta, \qquad v=\frac{x}{\tau},\qquad m=2\sqrt{\alpha\beta}. $$ 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](https://www.damtp.cam.ac.uk/user/tong/relativity/) 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 $0.9$, $\operatorname{Beta}(9,1)$ and $\operatorname{Beta}(9000,1000)$ point in the same direction, but one fresh trial moves them very differently. Their probability variances also differ: $$ \operatorname{Var}(P)=\frac{1-v^2}{4(\tau+1)}. $$ 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 $d\alpha,d\beta\ge0$. Its total amount and bias are $$ d\tau=d\alpha+d\beta,\qquad u=\frac{d\alpha-d\beta}{d\tau}. $$ 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: | The belief’s direction | The incoming direction | | ------------------------------------------ | -------------------------------------------- | | $v=x/\tau$ | $u=dx/d\tau$ | | Slope from the origin to the current state | Slope of the arriving segment | | Centered posterior mean | Current packet bias, or expected source bias | Subtract the old centered mean from the new one: $$ \Delta v=\frac{(u-v)\,\Delta\tau}{\tau+\Delta\tau}. $$ 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 $80\%$, while your estimate is already $90\%$. The positive current slows your centered mean down! Agreement about the direction does not imply agreement about the speed. [Ride an evidence current](https://epistemicphysics.com/articles/beliefs-in-motion/#bsr-motion) 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, $du/d\tau\ne0$, 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](https://www.stat.berkeley.edu/~wfithian/courses/stat210a/sufficiency.html) explain the exact statistical contract. ## A single trial, seen from the boat The rapidity coordinate $$ \phi=\operatorname{artanh}(v)=\frac12\log\frac\alpha\beta $$ stretches the finite probability interval into an infinite line. Define $\gamma=(1-v^2)^{-1/2}=\tau/m$. Then $$ \tau=m\cosh\phi,\qquad x=m\sinh\phi. $$ A supporting trial has increment $(\Delta\tau,\Delta x)=(1,1)$, 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 $e^{-\phi}$ and $e^{\phi}$. The familiar redshift/blueshift algebra appears directly. Here are the finite changes, with a supporting trial denoted by $+$: $$ \begin{aligned} \Delta\phi_+&=\tfrac12\log(1+1/\alpha), &\Delta(m^2)_+&=4\beta=2me^{-\phi},\\ \Delta\phi_-&=-\tfrac12\log(1+1/\beta), &\Delta(m^2)_-&=4\alpha=2me^{\phi}. \end{aligned} $$ 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 $e^{-\phi}/m$ and $-e^{\phi}/m$. Their magnitude ratio then approaches $e^{2\phi}=\alpha/\beta$. 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 $$ \mathbb E[\Delta\phi]\approx\frac{\gamma^2(u-v)}{\tau}. $$ 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 $0\le n<\beta$. Then $$ \Delta\phi=\frac12\log\frac\beta{\beta-n}, \qquad \frac{m_i}{m_f}=\sqrt{\frac\beta{\beta-n}}=e^{\Delta\phi}. $$ 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 $\lambda>0$: $$ \dot x=u-\lambda x,\qquad \dot\tau=1-\lambda\tau. $$ 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 $$ \dot v=\frac{u-v}{\tau},\qquad \tau\longrightarrow\frac1\lambda. $$ 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](https://www.itl.nist.gov/div898/handbook/pmc/section4/pmc431.htm) 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 $\tau_*=1/(1-q)$, $$ v_{j+1}=qv_j+(1-q)Z_j,\qquad \operatorname{Var}_{\rm ensemble}(v)=\frac{(1-u^2)(1-q)}{1+q}. $$ 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](https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc314.htm). The Beta’s own centered variance is $(1-v^2)/(\tau_*+1)$. 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 $u