A workshop for beliefs

An engineer walks into a workshop carrying a belief.

At the first bench, someone wants to know how stiff it is. At the next, someone attaches a capacitor and asks how quickly it forgets. A roboticist wants to bolt it into a map. A chemical engineer pours it into a beaker. Upstairs, the aerospace team is already asking whether it will still work when the reference frame rotates.

These are excellent questions to ask of a belief.

Epistemic Physics gives us a way to bring them into the same room. Its evidence ledger retains both the direction a belief points and the amount behind it. Its geometry connects those quantities to velocity, inertia, and rapidity. Now we can try something practical: take that geometry apart, lay its pieces on the bench, and see which engineering tools fit.

This field trip follows those ideas into mechanical, electrical, aerospace, chemical, and civil engineering, with a robot wandering between the benches.

Put a belief on the workbench

Suppose a component passes $Y$ trials and fails $N$ trials. We want to estimate its unknown probability $p$ of passing the next trial under the same conditions. A Beta distribution describes our uncertainty over that probability using two positive parameters:

$$\alpha =Y+\frac{1}{2},\beta =N+\frac{1}{2}.$$

The halves are our starting prior. They leave room for both possibilities before any measurements arrive. The ledger’s total, including that prior, is

$${\tau }_{\mathrm{E}}=\alpha +\beta =Y+N+1.$$

Epistemic Physics calls this epistemic time. It measures accumulated trial amount; an ordinary clock measures elapsed time. We will use $t$ for the clock throughout this article.

Three other quantities will accompany us:

$$\mu =\frac{\alpha }{{\tau }_{\mathrm{E}}},v_{\mathrm{E}}=2\mu -1,m_{\mathrm{E}}=2\sqrt{\alpha \beta }.$$

Here $\mu$ is the mean probability, $v_{\mathrm{E}}$ is its centered version running from $-1$ to $1$, and $m_{\mathrm{E}}$ is epistemic mass. If we also write the difference between the channels as $x_{\mathrm{E}}=\alpha -\beta$, the ledger sits inside a light cone:

$${\tau }_{\mathrm{E}}^2-x_{\mathrm{E}}^2=m_{\mathrm{E}}^2.$$

The other useful coordinate is rapidity, half the natural logarithm of the odds:

$$\rho =\frac{1}{2}\ln \frac{\mu }{1-\mu },v_{\mathrm{E}}=\tanh \rho .$$

A little notebook and a huge notebook can point at the same $\mu$. They will respond very differently when the next trial arrives. That brings us to the mechanical bench.

Mechanical engineering: a spring can take a measurement

Imagine a little carriage sliding along a rail. One spring wants it at position $a$; another wants it at position $b$. Their stiffnesses are $k_a$ and $k_b$. A stiff spring pulls harder for the same displacement.

The stored elastic energy is

$$E(x)=\frac{1}{2}{k}_a(x-a{)}^2+\frac{1}{2}{k}_b(x-b{)}^2.$$

At equilibrium, the opposing pulls balance. Differentiating with respect to the carriage position $x$ and setting the result to zero gives

$$x_{\ast }=\frac{k_{a}a+k_{b}b}{k_a+k_b}.$$

Now give an estimation engineer two independent Gaussian measurements of that same position. A Gaussian is the familiar bell-shaped distribution. Let the measurement variances be $s_a^2$ and $s_b^2$. Their precisions, the reciprocals of variance, are $J_a=1/s_a^2$ and $J_b=1/s_b^2$.

Multiplying their Gaussian likelihoods gives the very same quadratic expression in the negative logarithm. The combined estimate is

$$\hat{x}=\frac{J_{a}a+J_{b}b}{J_a+J_b},s_{\ast }^2=\frac{1}{J_a+J_b}.$$

Precision plays the role of stiffness. Each observation brings a preferred position and a strength of pull. The algebra survives the change of workshop; the units change with it. Spring systems and their quadratic energies are developed in MIT’s treatment of spring networks.

For example, let one contribution ask for position $2$ with strength $3$, and the other ask for position $8$ with strength $1$. Both calculations give $(3\times 2+1\times 8)/4=3.5$. The stronger contribution wins more of the compromise. Doubling both strengths keeps that position unchanged; in the Gaussian model it halves the combined variance.

Our Beta notebook has its own response law. Let a new result be $z=1$ for a success or $z=0$ for a failure. Its updated mean is

$${\mu }^{+}=\frac{\alpha +z}{{\tau }_{\mathrm{E}}+1}=\mu +\frac{z-\mu }{{\tau }_{\mathrm{E}}+1}.$$

The superscript $+$ means “after the update.” The new trial pulls toward its outcome, while the accumulated ledger determines how far the mean travels. The response weight $1/({\tau }_{\mathrm{E}}+1)$ is a direct measure of its malleability under this update.

That already looks like an engineer’s correction rule: present estimate, plus an error signal, times a gain. Keep that equation handy. We will meet its relatives at almost every bench.

There are several kinds of resistance here. A spring’s stiffness sets its static displacement under force. A mechanical mass sets its acceleration under force. A ledger’s trial amount sets the change produced by a new trial. Giving each one its own response equation is how we turn the resemblance into something we can build with.

Electrical engineering: put the memory in a circuit

An ordinary resistor-capacitor circuit, or RC circuit, is a tiny machine for remembering the recent past. The capacitor stores charge. The resistor limits how quickly charge moves. For input voltage $V_{\mathrm{in}}$ and capacitor voltage $V$, its equation is

$$RC\frac{dV}{dt}=V_{\mathrm{in}}-V.$$

The product $RC$ is a time constant. Slowly changing inputs get through; rapid oscillations are smoothed. This is a low-pass filter. MIT’s RC primer develops the circuit from current balance.

Our notebook with fading memory has supporting and opposing arrival rates ${\lambda }_{+}$ and ${\lambda }_{-}$. Each old trial fades at rate $\kappa$:

$$\dot{Y}={\lambda }_{+}-\kappa Y,\dot{N}={\lambda }_{-}-\kappa N.$$

A dot means differentiation with respect to ordinary time. Let the total arrival rate be the constant $\lambda ={\lambda }_{+}+{\lambda }_{-}$, and let $u=({\lambda }_{+}-{\lambda }_{-})/\lambda$ describe the incoming balance. Once the total trial amount has settled, it is ${\tau }_{\ast }=n_0+\lambda /\kappa$, where $n_0=1$ is our fixed prior amount. The centered mean then follows

$${\dot{v}}_{\mathrm{E}}=\kappa (gu-v_{\mathrm{E}}),g=\frac{\lambda }{\lambda +\kappa n_0}.$$

The memory time is $1/\kappa$. The extra factor $g$ tells us that the fixed neutral prior keeps pulling toward zero.

We can build that feature into the circuit too. Connect the capacitor to the input through resistance $R_{\mathrm{in}}$, and add a second resistance $R_0$ from the capacitor to ground. Current balance gives

$$\dot{V}=\frac{V_{\mathrm{in}}-V}{R_{\mathrm{in}}C}-\frac{V}{R_{0}C}=\kappa (g{V}_{\mathrm{in}}-V),$$

with

$$\kappa =\frac{1/R_{\mathrm{in}}+1/R_0}{C},g=\frac{R_0}{R_{\mathrm{in}}+R_0}.$$

The second resistor gives the prior a physical address: a leakage path toward neutral.

After one memory time following a constant input, either output has covered about $63\mathrm{\%}$ of the distance to its new settled value. The fraction $g$ controls that destination; $\kappa$ controls how quickly it gets there.

Before touching the controls, predict what very long memory will do to an oscillating signal. Then try very short memory. How much evidence remains to compete with the prior?

A notebook for changing seasons

How much of the past should it keep?

Reset

The environment oscillates. Old observations fade while the neutral prior stays at one. Change the memory and see how much of the signal the belief follows. A matched resistor-capacitor circuit follows the same signal, with its voltage on the same scale.

Memory timescale

1.60 time units

One environmental cycle

8 time units

New evidence per time unit

12.0

Environment / input voltageBeliefCapacitor voltage

Signal amplitude retained59.2%

Delay behind environment1.14 time units

Settled concentration20.20

Very short memoryLargest responseLong memory

The sky dashes sit on the teal curve: the two systems have the same periodic response after their initial adjustment, with the evidence total settled. “Largest response” maximizes the output amplitude for these settings; its nearest slider value is shown. Long memory smooths the signal. Very short memory leaves the prior with a larger share.

A notebook with fading trials

$${\dot{v}}_{\mathrm{E}}=\kappa (gu-v_{\mathrm{E}})$$

Incoming balance u drives the centered mean v_E.

Memory time 1.60 Gain for slow signals 0.950

A capacitor with a ground leak

$$\dot{V}=\kappa (g{V}_{\mathrm{in}}-V)$$

Input voltage V_in drives capacitor voltage V.

Memory time 1.60 Gain for slow signals 0.950

A dot means change per unit time. The forgetting rate is κ = 1 / memory time, and g is the fraction of a very slowly changing input that gets through. The circuit uses matched units with capacitance C = 1, input conductance κg = 0.594, and ground conductance κ(1 − g) = 0.031. Conductance is the reciprocal of resistance; the ground path supplies the pull toward neutral.

$$\kappa =0.625,n_0=1,{\tau }_{\ast }=1+\lambda /\kappa =20.20$$

The same equation also lives in a well-mixed thermal room, where heat capacity stores the past and thermal conductance connects the room to its surroundings. Electrical storage, thermal storage, and evidence storage invite the same questions about delay, damping, and responsiveness.

Kalman filters: let uncertainty choose the pull

A navigation system must combine two stories: where its motion model predicts it should be, and where a noisy sensor says it is.

Consider a scalar position estimate with predicted mean $m^{-}$ and predicted variance $s_{-}^2$. The minus superscript means “before this measurement.” A direct sensor returns $z$ with independent error variance $R$. The scalar Kalman correction is

$$K=\frac{s_{-}^2}{s_{-}^2+R},m^{+}=m^{-}+K(z-m^{-}),s_{+}^2=(1-K)s_{-}^2.$$

The difference $z-m^{-}$ is the innovation: the part of the reading the prediction did not anticipate. $K$ is the Kalman gain. A scattered prediction and a precise sensor produce a large gain. A precise prediction and a noisy sensor produce a small one.

This is the two-spring calculation wearing a navigation badge. The predicted estimate brings precision $1/s_{-}^2$; the sensor brings precision $1/R$. With Gaussian uncertainty, their product gives the full Gaussian posterior. The familiar prediction-and-correction equations are illustrated in Carnegie Mellon’s Kalman filtering notes.

Where does prediction variance come from? In the simplest random-walk model, we carry the previous mean forward and add a process variance $Q$:

$$m^{-}=m_{\mathrm{previous}}^{+},s_{-}^2=s_{\mathrm{previous},+}^2+Q.$$

Process noise represents uncertainty introduced as the state evolves. Repeated measurements tighten the estimate; uncertain motion opens it up again. That competition gives a tracking system its memory. The gain adapts to the variances propagated by the model; choosing or learning $Q$ and $R$ is part of designing the estimator.

Now return to the Beta ledger. We already found

$${\mu }^{+}=\mu +K(z-\mu ),K=\frac{1}{{\tau }_{\mathrm{E}}+1}.$$

The connection goes deeper than matching the typography. If $P$ is our uncertain success probability and $Z$ is the next binary result, then

$$\mathrm{Cov}(P,Z)=\mathrm{Var}(P),\mathrm{Var}(Z)=\mu (1-\mu ).$$

Covariance measures how two quantities vary together. Their ratio gives exactly our gain:

$$\frac{\mathrm{Cov}(P,Z)}{\mathrm{Var}(Z)}=\frac{\mathrm{Var}(P)}{\mu (1-\mu )}=\frac{1}{{\tau }_{\mathrm{E}}+1}.$$

We can place the quantities directly into the usual Kalman formula. Write $V=\mathrm{Var}(P)$ and think of the binary reading as $Z=P+\epsilon$. At a given $P$, its error has mean zero and variance $P(1-P)$. Averaging that sensor noise over our current belief gives

$$R=\mathbb{E}[P(1-P)]={\tau }_{\mathrm{E}}V,K=\frac{V}{V+R}=\frac{1}{{\tau }_{\mathrm{E}}+1}.$$

The binary sensor’s noise depends on $P$, but its error is uncorrelated with $P$. That is enough for this covariance calculation. The two mean updates now agree term by term, using the same prior and the same reading.

A surprise awaits in the variance. For a Beta belief, an unexpected outcome can widen the posterior. Averaging over the two possible outcomes restores the familiar contraction:

$${\mathbb{E}}_Z[\mathrm{Var}(P\mid Z)]=(1-K)\mathrm{Var}(P).$$

The mean correction is exact for each binary result. The variance contraction holds on average over the next result. The demo lets us watch both at once.

Belief updating ↔ Kalman filtering

One reading, the same mean update.

Reset

Give both rules the same starting belief and the same binary reading. The Beta rule adds a trial; the Kalman rule combines variances. Watch them produce the same gain and move the mean to exactly the same place.

Supporting trials

9

Opposing trials

0

Try a success: z = 1 Try a failure: z = 0

Beta belief update

$${\mu }^{+}=\frac{\alpha +z}{{\tau }_{\mathrm{E}}+1}$$ $$K=\frac{1}{{\tau }_{\mathrm{E}}+1}$$

Mean before

95.00%

Reading z

0

Gain K

9.09%

Mean after

86.36%

Kalman mean update

$${\mu }^{+}=\mu +K(z-\mu )$$ $$K=\frac{V}{V+R}$$

Mean before

95.00%

Reading z

0

Gain K

9.09%

Mean after

86.36%

Mean beforeMean afterNew reading

Both rules move the mean from 95.00% to 86.36%. The shared calculation is 0.9500 + 0.0909 × (0 − 0.9500) = 0.8636.

For the Kalman column, V = 0.004318 is the starting belief variance. R = 0.043182 is the binary sensor's average noise variance. Because R = τ_EV, its gain V/(V + R) is exactly 1/(τ_E + 1).

What happens to the variance?

The Kalman variance matches the Beta variance averaged over the next result. This particular failure widens the Beta curve.

Before

0.004318

Kalman / expected Beta

0.003926

Beta: this outcome

0.009814

See the Beta curve before and after

Before the reading

The density diverges at p = 1. The curve is sampled only inside (0, 1).

After this reading

The mean is μ; prior-inclusive trial amount is τ_E = 10. The prior contributes half a trial per channel. “Expected Beta” weights a success by 95.00% and a failure by 5.00%.

Try the initial nine-success streak, then deliver one failure. The gain is only $1/11$, yet the variance more than doubles. The ledger becomes harder to move with the next trial while its probability curve becomes wider. Malleability and variance each tell us something useful.

Give the filter a working memory

The circuit at the previous bench used a fixed response rate. A discrete memory can do something similar: move a fixed fraction of the distance toward every reading. A Kalman tracker uses the same form of correction and chooses that fraction from its current variances.

Suppose the sensor goes quiet while the world keeps moving. With , the tracker becomes less certain where the signal has wandered. When a reading finally returns, the enlarged prediction variance gives it more influence. Watch the teal band open during the gap, then tighten on the first returning reading.

Prediction opens the estimate; a reading pulls it together

Keep track when the sensor goes quiet.

Reset

Follow the same moving signal with a Kalman filter and a fixed-gain memory. Scrub through the readings. During a sensor outage, watch the Kalman band widen; then see how strongly the returning measurement pulls.

Continuous readingsSensor goes quietWorld makes a jump

Expected motion per step · SD

0.15

Sensor noise · SD

0.60

Fixed memory's gain

0.20

WorldReadingsKalmanFixed gain

Drag the gold time handle to walk forward or back through the readings.

Walk through the experiment

Step 46 / 80

RewindNext stepLast quiet stepFirst returning reading

Predicted mean / SD

-0.549 / 0.663

Updated mean / SD

-0.881 / 0.445

This reading

-1.153

Kalman gain

0.550

The reading pulls the prediction 55.0% of the way toward it. The fixed memory always moves 20% of the distance from its own estimate to the reading.

SD means standard deviation; the shaded band is the Kalman estimate ±2 model SDs. The world takes independent steps with SD 0.15, with an added jump in the jump scenario. The sensor's SD is known to the filter; the expected motion is your model choice. The same random sequence is reused as you adjust the settings. Both filters use only readings up to the selected step.

Try setting expected motion to zero. The filter now acts as though it is estimating one fixed quantity, so repeated readings progressively reduce its gain. It also stops widening its estimate during an outage. Switch to the sudden-jump example and see the price of that assumption.

An unexpected reading changes the mean through its innovation. With $Q$ and $R$ fixed, surprise alone does not instantly change the gain. The gain follows the variance model. A filter that learns when the world has changed needs an additional mechanism to revise that model.

Nonlinear sensors: a tangent or a scouting party?

Real sensors enjoy bending things. A camera projects a three-dimensional scene onto a plane. A rangefinder measures a square root of squared distances. A chemical detector saturates. A robot’s turning motion mixes angles and positions.

To isolate the difficulty, imagine a hidden rapidity $\rho$ and a noisy analog sensor:

$$z=h(\rho )+\epsilon ,h(\rho )=\tanh \rho .$$

The noise $\epsilon$ has zero mean and variance $R$. We choose a Gaussian starting distribution in rapidity, described by mean $m$ and variance $s^2$. Each candidate rapidity corresponds to a candidate probability $p=(1+\tanh \rho )/2$.

An extended Kalman filter, or EKF, replaces the curved sensor by its tangent near the current estimate. The slope is

$$H=h^{\prime }(m)=1-{\tanh }^{2}m.$$

It then uses

$$K_{\mathrm{EKF}}=\frac{s^{2}H}{H^2{s}^2+R},m^{+}=m+K_{\mathrm{EKF}}(z-h(m)).$$

An unscented Kalman filter, or UKF, sends a small, carefully weighted collection of representative points through the actual curve. These sigma points are a scouting party placed around the estimated state. Their transformed positions approximate the predicted reading, its variance, and its covariance with the state. Those quantities supply the correction gain. MathWorks documents both constructions and their prediction/correction steps.

The EKF asks how the sensor bends right here. The UKF samples how it behaves across the current spread. A numerical Bayesian calculation can go further and retain the whole posterior curve. Let’s give all three exactly the same problem.

A tangent, three scouts, and the whole curve

Send a belief around a bend.

Reset

A sensor measures a hidden rapidity ρ. Its response can be straight or saturating, like a detector running out of range. Predict which filter will stay closest to the gold curve after one noisy reading.

Example: straight agreement Example: broad belief Example: near saturation

Straight sensor: h(ρ) = ρ Saturating sensor: h(ρ) = tanh ρ

Starting rapidity mean

0.00

Starting standard deviation

1.00

Sensor reading z

0.75

Sensor noise standard deviation

0.15

What the sensor sees

Sensor responseEKF tangentUKF scoutsNew reading

What the filters believe

PriorEKFUKFBayes reference

Drag the teal center scout along the sensor curve, or move the sky-blue reading up and down.

Three ways to process this same reading

Method	Predicted reading	Updated ρ mean	Updated ρ spread
● EKF	0.000	0.733	0.148
● UKF	0.000	0.948	0.193
● Bayes reference	0.000	1.081	0.408

The EKF follows the tangent at ρ = 0.00. The UKF sends scouts to 0.00, -1.00, 1.00. The gold curve keeps the full posterior shape.

Spread here means standard deviation. The gold curve comes from numerical integration. The prior is Gaussian in rapidity; probability is p = (1 + tanh ρ)/2. Noise is added to the analog reading, so a reading can pass ±1. This lab isolates one correction step; a tracking filter repeats prediction and correction.

How the three scouts are weighted

These are deterministic sigma points, not observed trials. Their mean weights are 0, ½, ½; covariance weights are 2, ½, ½. This is the scalar scaled unscented transform with α = 1, β = 2, κ = 0. Both approximations retain a Gaussian in rapidity.

Start with the straight-agreement example: the rose tangent lies on the sensor curve, and all three posterior answers meet. Try the broad belief and watch the teal scouts reach around the bend. Then choose the saturation example. The tangent becomes nearly flat while a reading of zero asks the belief to move toward the middle. The gold curve can become asymmetric while the two filters keep bell-shaped summaries. Which summary follows its center better depends on the setting.

This is an especially useful lesson for Epistemic Physics. Moving the mean through a curved function generally differs from moving the whole distribution and then taking its mean:

$$\mathbb{E}[\tanh \rho ]\ne \tanh \mathbb{E}[\rho ].$$

A beautiful coordinate change can simplify the dynamics while changing the shape of uncertainty. Rapidity makes multiplicative changes in odds additive. We still get to choose how much of the resulting distribution our estimator carries.

There is another connection hiding in the slope. For $n$ independent binary measurements with probability $p=(1+\tanh \rho )/2$, the Fisher information, the expected curvature of the negative log likelihood, is

$${\mathcal{I}}_{\rho }=n(1-{\tanh }^2\rho ).$$

The sensor learns most near balance and least near saturation. The same geometric factor appears in the evidence cone:

$$\frac{m_{\mathrm{E}}^2}{{\tau }_{\mathrm{E}}}={\tau }_{\mathrm{E}}(1-v_{\mathrm{E}}^2).$$

This suggests an engineering question with teeth: should a sensing system change its operating point to stay where new measurements can still move its estimate? Our receptor-bank experiment explores that idea by spreading sensors across different tunings.

Aerospace: the estimate needs a reference frame

A spacecraft carries more than a position estimate. It has velocity, orientation, drifting instrument biases, and measurements expressed in different frames. NASA’s GEONS navigation software, for example, uses an extended Kalman filter architecture for onboard orbit determination. NASA describes its implementation and flight applications.

Orientation introduces a wonderful geometric snag. You cannot treat every rotation like an ordinary displacement on a flat sheet. Turn a book around two different axes in opposite orders and you can end in different orientations.

Rotations form a group: they have a composition rule, an identity rotation, and inverses. Smooth groups are called Lie groups. Robotics often uses $SO(3)$, the group of three-dimensional rotations, and $SE(3)$, which combines rotation with translation.

An invariant extended Kalman filter uses the symmetry of such a state space to define and propagate its estimation error. Barrau and Bonnabel show how this can give useful stability properties for a class of systems, including mobile robotics and inertial navigation. Their paper develops the construction.

Epistemic rapidity offers a small, unusually friendly workshop for this kind of thinking. A boost by $r$ acts by

$${\rho }^{+}=\rho +r.$$

Its composition law is simple addition. An uncertainty model can be built around errors in that coordinate, then projected back to probability when a reading is needed. In a spacecraft, the corresponding design question is which geometric error to propagate for orientation and motion.

Both invite us to design the estimator around the transformations its state actually undergoes. A coordinate system becomes part of the machinery.

Civil engineering and SLAM: let the network settle

Now roll our carriage into a larger structure. Several nodes are connected by springs. Each connection has a preferred displacement and a stiffness. The whole structure finds a compromise by balancing the forces at every free node.

A robot building a map faces much the same problem. SLAM means simultaneous localization and mapping: estimating where the robot is while also estimating the surroundings it uses to locate itself. Wheel motion supplies relative displacements. Cameras and rangefinders connect positions to landmarks. Revisiting a place adds a loop closure, a constraint linking parts of the route that were previously connected only through a long chain of uncertain motion. GTSAM’s tutorial introduces these factor graphs and mapping examples.

A factor is one contribution to the joint model, such as a measurement likelihood. In a simple one-dimensional Gaussian model, an edge from node $i$ to node $j$ contributes

$$E_{ij}=\frac{1}{2}{w}_{ij}(x_j-x_i-d_{ij}{)}^2.$$

Here $x_i$ and $x_j$ are estimated positions, $d_{ij}$ is the measured displacement, and $w_{ij}$ is measurement precision. As a spring, the same expression uses a preferred displacement and a stiffness.

Add all the edges, and we have both a spring network and a map estimator. Before trying the demo, make a prediction: if two steps say “four plus four” but another route says “six,” where should the disagreement go?

Mechanical, civil, electrical, and robotic viewpoints

A robot map held together by springs.

Reset

A robot records two steps of four units each. A second route measures the separation from A to C. When the routes disagree, the map settles like a network of springs competing for their preferred lengths.

A-to-C measurement

6.0 units

Loop precision / spring stiffness

1.00

First step precision / stiffness

1.00

Position assigned to anchor A

0.0

Try a weaker first step Remove the loop reading Mistakenly count it twice

Drag the sky-blue anchor A to shift the whole map. The whiskers under B and C show position ±1 standard deviation relative to that fixed anchor.

Two-route disagreement

2.00 units

Reported position spread at C

σ = 0.816

First / second step error

-0.667 / -0.667

Loop's fitted error

0.667

A stronger loop pulls A and C closer to its requested separation. Move the anchor to shift the entire map while keeping every fitted separation unchanged.

$$E=\frac{w_1}{2}(x_B-x_A-4{)}^2+\frac{1}{2}(x_C-x_B-4{)}^2+\frac{w}{2}(x_C-x_A-d{)}^2$$

The positions are x_A, x_B, x_C; d is the loop reading and w its effective weight. The first step has precision w₁ and noise standard deviation 1/√w₁; the second has precision one and standard deviation one. Minimizing E gives spring equilibrium and the Gaussian map estimate. A is fixed exactly; σ comes from the inverse of the remaining precision matrix.

The network distributes the mismatch according to the weights. A stiff connection gives up little; softer ones deform more. In the map, a precise observation pulls harder and less precise observations absorb more of the residual error.

With all three weights equal, the two four-unit steps each shorten by $2/3$, while the six-unit loop lengthens by $2/3$. Now choose the weaker-first-step example. Its precision drops to $1/4$, so it absorbs four times as much error as the second step: the first shortens by $4/3$, the second by $1/3$, and the loop lengthens by $1/3$. The springs and the measurements negotiate by exactly the same rule.

The matrix shared by three workshops

We can pack the network into a matrix $A$ that takes node values and returns their differences along the edges. Such a matrix is called an incidence matrix. Put the edge weights on the diagonal of a matrix $W$, and collect the preferred differences in a vector $d$. Then

$$E(x)=\frac{1}{2}(Ax-d{)}^{\mathsf{T}}W(Ax-d).$$

The superscript $\mathsf{T}$ means transpose. Setting the derivative to zero gives

$$Lx=A^{\mathsf{T}}Wd,L=A^{\mathsf{T}}WA.$$

In mechanics, $L$ is an assembled stiffness matrix. In the Gaussian map, it is a precision matrix. For an electrical resistor network, use conductances as weights, node voltages as $x$, and injected currents on the right-hand side: the same $L$ appears in Kirchhoff’s current law. This family of network constructions is developed in MIT’s computational science and engineering notes.

“Laplacian” is the name for this matrix of weighted differences. Each workshop assembles it from local connections and uses it to solve a global balance.

Civil engineering extends the assembly idea to trusses and finite elements, where each element contributes a small stiffness matrix to the structure. Estimation assembles local information in the same spirit. A network can carry directional resistance that a single scalar strength could never describe.

Why the map needs an anchor

Notice what happens when we add the same constant to every position. Every difference $x_j-x_i$ stays unchanged. Relative measurements cannot tell whether the entire map has moved.

Mechanics calls this a rigid translation. Circuits let us choose a ground potential. Mapping chooses an origin. Mathematically, there is a direction in which the unanchored network has zero stiffness and zero information.

Fixing an anchor removes that freedom. In the demo, moving the anchor slides the map while preserving all its internal compromises. A precise relative map can still have an arbitrary absolute origin.

A copied measurement is a counterfeit spring

Try counting the loop reading twice. The solver adds another spring in the same place. The map shifts farther toward that reading, and its reported uncertainty shrinks.

But the robot has learned nothing new.

This is where Epistemic Physics brings lineage into the workshop. A tracelet is an elementary evidence contribution; its lineage records where it came from. Two messages carrying the same observation still descend from one source event. A fleet of robots exchanging estimates needs that distinction just as much as a scientific community exchanging claims.

Distinct observations can also share errors: two sensors may use the same calibration, or two maps may inherit the same earlier measurements. When those correlations are unknown, methods such as covariance intersection combine estimates while accounting for that uncertainty about their dependence. Julier and Uhlmann explore this in SLAM.

Lineage and covariance therefore answer complementary engineering questions: Did this contribution arrive before? And how do its errors move with the others? Our one trial, many echoes field trip follows the first question in detail.

Chemical engineering: composition needs an inventory

At the chemical bench, put amounts $a$ of conjugate base and $b$ of acid into an ideal buffer. Two mixtures can have the same ratio $a/b$, and therefore the same pH, while containing wildly different total amounts.

That should sound familiar. A ratio tells us where the system sits. Its inventory helps tell us how far a new dose can push it.

The Henderson-Hasselbalch equation reads

$$\mathrm{pH}=\mathrm{p}{K}_a+{\log }_{10}\frac{a}{b}.$$

Here $\mathrm{p}{K}_a$ characterizes the acid’s dissociation equilibrium. Set $p=a/(a+b)$ and define its rapidity $\rho =\frac{1}{2}\ln (a/b)$. Then

$$\mathrm{pH}=\mathrm{p}{K}_a+\frac{2\rho }{\ln 10}.$$

pH is an affine rapidity coordinate: a scaled rapidity plus an offset. The concentration-ratio equation is given in IUPAC’s chemical terminology reference.

For a fixed-volume buffer, add a small amount $q$ of strong base. In our ideal model it converts acid into conjugate base: $a\mapsto a+q$, $b\mapsto b-q$. The total inventory $T=a+b$ stays fixed. Differentiating the pH gives

$$\frac{dq}{d\mathrm{pH}}=(\ln 10)\frac{ab}{a+b}.$$

This is an amount-based buffer capacity. Define the balanced amount $m=2\sqrt{ab}$, the same two-channel construction used for epistemic mass, and it becomes

$$\frac{dq}{d\mathrm{pH}}=\frac{\ln 10}{4}\frac{m^2}{T}.$$

For a fixed inventory, balance maximizes resistance to a pH change. Near either end, the remaining minority ingredient is easily exhausted. Try the same dose in a balanced mixture and an unbalanced one.

Acid, base, and resistance to change

Give a buffer a little shove.

Reset balanced buffer

Add the same small dose of acid or base. Does a balanced mixture move as far in pH as one that is already mostly acid or mostly base?

Total buffer amount

20 mmol

Starting pH minus pK_a

0.00

HA · acid 10.00 mmolA⁻ · conjugate base 10.00 mmol

Add 1 mmol acid Add 1 mmol base Undo additions

pH

7.000

Rapidity ρ

0.000

Epistemic mass

20.00 mmol

Buffer capacity

11.51 mmol/pH

One curve, two rulers: the pH values on the left and rapidity values on the right mark the same mixtures. At pH 7, rapidity is zero.

Capacity / maximum capacity(Mass / total amount)²

The two curves coincide: both equal 4p(1 − p), where p is the fraction in the base form. Here each is 1.000 of its maximum. Capacity reaches 11.51 mmol/pH and mass reaches 20 mmol at balance.

Start with equal amounts, then try a mixture away from balance. At a fixed total amount, both buffer capacity and epistemic mass reach their maximum at a 50:50 mixture.

Here a and b are the acid and base amounts. Added strong base converts HA to A⁻; added strong acid does the reverse. The model holds volume fixed, uses pK_a = 7, and neglects water's contribution. Capacity means the local amount of added base per unit pH: ln(10) ab/(a + b). The shared curve compares their shapes after dividing out their different units and maxima. Additions stop before either component falls below 1%.

This suggests a chemical engineer’s reading of an evidence state: carry the composition and the inventory. In process control, a ratio may look perfectly steady while the reservoir that supports it is quietly running dry.

A drawer for Gamma and the Beta extensions

The richer distributions become useful when we start asking different questions about the inventory. Suppose two independent uncertain positive amounts satisfy

$$G_{+}\sim \mathrm{Gamma}(\alpha ,\chi ),G_{-}\sim \mathrm{Gamma}(\beta ,\chi ),$$

where $\chi$ is their shared rate parameter. Their total is Gamma distributed, while their fraction $P=G_{+}/(G_{+}+G_{-})$ is Beta distributed. The total and the fraction are independent in this construction. Gamma carries uncertain amount; Beta carries uncertain composition.

The odds $O=P/(1-P)=G_{+}/G_{-}$ have a beta-prime distribution. This gives us an unbounded ratio coordinate for the same composition. SciPy documents the Beta-to-beta-prime transformation.

Now imagine a process that rescales every candidate odds value by a positive factor $\ell$. The transformed candidate fraction is

$$q=\frac{\ell p}{1+(\ell -1)p}.$$

This translates each candidate rapidity by $\frac{1}{2}\ln \ell$. Starting with a Beta distribution and transporting its whole curve this way produces a Libby-Novick distribution. It also appears when the two Gamma amounts have different rates. Its density contains an extra factor of the form $(1+zq{)}^{-c}$, with constants $z$ and $c$ determined by the transformation. The VaRES documentation gives the normalized family.

If fresh binary observations follow that transport, their likelihood multiplies the density by powers of $q$ and $1-q$. The result belongs to the Gauss-hypergeometric family, which retains that extra factor while allowing the endpoint powers to change. SciPy gives its density and normalization.

Here is a concrete place for those extensions to earn their shelf space: an odds-changing process followed by observations. More elaborate sequences of processes can need richer families again. We can choose the distribution to match the operation we actually want our device to perform.

What would we build next?

A robot could carry a map whose observations retain their lineage. When another robot shares a map fragment, it could distinguish new measurements from familiar measurements taking a longer route home. The spring-network demo is a little blueprint for how duplicated evidence can distort the result.

A chemical controller could track both an operating ratio and the amount available to resist disturbances. The controller might move its sensors away from saturation, keeping them in the region where their readings remain informative.

A civil monitoring system could represent uncertainty about a structure in the same sparse pattern used for the structure’s mechanical model. A new strain measurement would then stiffen particular directions in the estimate, while a poorly observed mode could remain loose.

A navigation system could assign different memory times to different kinds of evidence: a slowly changing instrument bias, rapidly changing motion, and a stable landmark each deserve a different relationship with the past. The circuit analogy makes those timescales tangible; the Kalman equations turn the choice into a prediction-and-correction calculation.

There is also a direct route into reliability engineering. A component’s pass/fail record already fits the Beta notebook on our first bench. Keeping its trial amount alongside its estimated success probability lets a controller distinguish an extensively tested component from an unfamiliar one with the same estimated performance. NIST’s reliability handbook describes this Beta/binomial pairing and Gamma models for failure processes.

The interesting design question is often where to spend the next trial. Inspect the uncertain joint? Turn the camera? Retune the receptor? Revisit the landmark? Once an action changes what a system can learn, sensing itself becomes part of the engineering problem.

We came into the workshop with two columns of evidence. We leave with springs that fuse measurements, circuits that remember, maps that negotiate, and buffers whose balance determines how much they can absorb. There are plenty of spare parts on the bench. Pick a system, give its belief something to push against, and see what moves.