A compass for changing minds

Two travelers point their compasses toward the same mountain. One has wandered these hills for years; the other picked a direction five minutes ago. Their needles agree. Their response to the next landmark might be very different.

A probability report has the same ambiguity. “I give it 95%” tells us where someone points. It leaves open how readily they will change direction, which sources they accept, and what landscape of feedback surrounds them.

Let us build an expedition kit: an instrument that estimates a belief’s depth, a topographic map whose valleys emerge from selective exposure, and a speculative lens for detecting how nearby disputes distort interpretation. Each instrument needs its own measurement rule. That is where the physics becomes useful: it asks us to turn a picture into something we could actually measure.

Weighing a belief with a nudge

Our traveler carries a proper Beta distribution, $B\sim \mathrm{Beta}(\alpha ,\beta )$, with positive weights $\alpha$ and $\beta$. These include the prior. Define

$$\tau =\alpha +\beta ,v=\frac{\alpha -\beta }{\alpha +\beta },\mu =\frac{1+v}{2}.$$

Here $\mu$ is the mean probability, $v$ its centered version, and $\tau$ the total prior-inclusive weight. The geometric mass $m=2\sqrt{\alpha \beta }$ satisfies $m^2={\tau }^2-(\tau v{)}^2$. Our instrument will recover $\tau$, from which $m$ follows if $v$ is known. At a fixed mean, larger $\tau$ means less malleability under a fixed additive packet.

Imagine a controlled learning task with a coin of unknown bias. Show the traveler $w$ fresh heads, each accepted as an independent Bernoulli trial. Then $\alpha$ increases by $w$, while $\beta$ stays fixed. More generally, give a packet of weight $w$ with centered bias $d$: supporting-only has $d=+1$, opposing-only has $d=-1$. Its update is

$$v_1=\frac{\tau v_0+wd}{\tau +w},\mathrm{\Delta }v=\frac{w(d-v_0)}{\tau +w}.$$

Turn this around:

$$\boxed{\hat{\tau }=w\frac{d-v_1}{v_1-v_0}.}$$

A known nudge reveals the hidden depth through its recoil. This inversion is exact for the stated additive Beta model whenever the displacement is nonzero. A known likelihood ratio for a general proposition does not, by itself, specify an additive Beta packet. Calibrating what the traveler actually accepts is part of constructing the instrument.

For $v_0=0.9$, $\tau =40$, and one opposing trial, $v_1=35/41\approx 0.85366$. The mean probability moves from $95\mathrm{\%}$ to approximately $92.68\mathrm{\%}$. A supporting trial produces $v_1=37/41\approx 0.90244$, moving the mean to approximately $95.12\mathrm{\%}$. The opposing signal is much easier to see.

In fact, at the same starting state and packet weight,

$$\frac{|\mathrm{\Delta }{v}_{\mathrm{o}\mathrm{p}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{e}}|}{|\mathrm{\Delta }{v}_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}}|}=\frac{1+v_0}{1-v_0}=e^{2\varphi },\varphi =\mathrm{artanh}(v_0),$$

for a positive lean. The contrast is exactly the odds, $19$ at $v_0=0.9$. The rapidity coordinate $\varphi$, which appeared in the evidence cone, now predicts an instrument’s directional sensitivity.

Before trying the lab, predict which will help more: increasing the packet weight, or reversing its direction. Then switch to the terrain experiment and find a field strong enough to remove the left valley.

Bayesian navigation

Probe a belief and map its terrain

Reset instruments

Pick the stronger nudge, then send it. The recoil reveals how much trial weight holds the belief in place.

Recoil Terrain

Current centered mean v = 0.90. Each cartridge supplies 1 fresh, calibrated trial.

Which cartridge will move this belief farther? Make a prediction, then compare the two recoil arrows.

Opposition moves it fartherSupport moves it fartherSame distance

Change the starting notebook or inspect precise predictions

Start at τ = 4Start at τ = 40Start at τ = 200

Packet weight 1 trial5 trials20 trials

Opposing packet

Predicted displacement: -0.04634

Depth error SD: 1.24

Send opposition

Supporting packet

Predicted displacement: 0.00244

Depth error SD: 23.48

Send support

Recovered depth, probe log, and error budget

Current mean probability

95.00%

Current depth τ

40

Centered-report noise used in error budget

σ = 0.001

Error SD uses first-order propagation of two independent noisy reports, even though the displayed reports are exact. It becomes unreliable when noise is comparable to recoil. Every packet has known accepted weight; unknown trust would make absolute depth unidentifiable.

Satellites, noisy reports, and a missing ruler

A GPS receiver reconstructs its position from calibrated signals; its clock error is an additional unknown. NOAA’s GPS tutorial explains why ordinary positioning uses four satellite measurements to recover three position coordinates and a clock correction. Belief probing borrows that engineering question: which hidden quantities can these observations actually distinguish?

First account for noisy reports. Suppose the two centered reports have independent errors, each with standard deviation $\sigma$. Differentiating the recoil inversion gives the first-order estimate

$$\mathrm{sd}(\hat{\tau })\approx \frac{\sigma (\tau +w)\sqrt{{\tau }^2+(\tau +w{)}^2}}{w|d-v_0|}.$$

This uses the usual propagation of measurement uncertainty. When $\tau \gg w$, it becomes approximately $\sqrt{2}\sigma {\tau }^2/(w|d-v_0|)$. The approximation needs report noise small compared with recoil; dividing by a noisy displacement near zero can produce wild estimates. If the noise is specified on ordinary probability reports, use twice that standard deviation for centered reports.

This suggests two distinct design budgets. With few reporting opportunities, stronger probes enlarge the measurable displacement. With a fixed total packet weight and inexpensive reports, recording intermediate states supplies more observations to fit. Correlated reporting errors, rounding, fatigue, and learning from the elicitation itself can change that comparison. Fit the complete trajectory rather than relying on a single fragile ratio.

A more surprising obstruction appears when the accepted packet weight is unknown. Let be a common trust multiplier, and let $A_k,D_k$ denote cumulative supporting and opposing nominal weights. Every report then has the form

$$v_k=\frac{\tau v_0+c(A_k-D_k)}{\tau +c(A_k+D_k)}.$$

Multiplying both $\tau$ and $c$ by the same positive constant changes none of these reports. A deep belief accepting large packets looks identical to a shallow belief accepting proportionally small packets. Ten more probes cannot supply the missing ruler. The observations identify a ratio such as $\tau /c$ until trust or depth is anchored independently.

Counting equations is therefore only a necessary first check. The stronger question is identifiability: can two different parameter settings predict exactly the same data? A practical expedition begins with a task whose accepted trial weights can be calibrated, checks source lineage so repeated testimony is not counted as independent trials, and then estimates depth with uncertainty. In ordinary conversation, “depth” remains conditional on the update model.

A known reference learner exposed to the same framing might help remove shared reporting drift. That cancellation requires a model in which the drift acts comparably on both reports. Conversely, a panel of calibrated learners could help reconstruct an unknown broadcast from their responses. Locating its source would additionally require an exposure network and timing information; displacement alone carries no geographical address.

Open the detective cases below. Choose probes that separate candidate notebooks, then tackle a case where every possible probe leaves the same ambiguity. The difference between a weak measurement and a missing ruler becomes something you can test.

Find the hidden weight

The calibrated-probe detective

Restart cases

Case 1: A positive lean. Choose a probe cartridge, read its recoil, then name the compatible notebook. Every test uses a fresh copy at centered mean v = 0.90.

Lightτ = 8accepted weight × 1Still possibleMediumτ = 40accepted weight × 1Still possibleHeavyτ = 160accepted weight × 1Still possible

Fire a cartridge. Which direction produces a recoil large enough to distinguish the candidates?

Send 1 opposing trialSend 1 supporting trial

Reports are rounded to 0.01. Tiny displacements can disappear.

These candidates cannot be distinguished

Stronger probes, case selection, and the instrument rule

Send 5 opposing trialsSend 5 supporting trials

v after = (τ × v before + c × weight × direction)/(τ + c × weight), where c is the accepted-weight multiplier. This is a finite candidate experiment with deterministic rounding, not random reporting noise. Each test resets an identical calibration copy; the readings do not update one shared target. In the final case only τ/c is identified.

Case 1Case 2Case 3

A landscape made by the current

Now let the environment respond to the traveler. A recommender serves more of what they already like. Their friends share related material. Their current lean affects the bias of their next evidence packet.

Write $u(v)$ for that incoming centered bias. Differentiating $v=x/\tau$ gives

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

Incoming direction $u=dx/d\tau$ is a tangent along the journey; present belief $v=x/\tau$ is the direction from the origin. They need not agree. Using the normalized evidence clock $s=\log (\tau /{\tau }_0)$ removes the changing speed factor:

$$\frac{dv}{ds}=u(v)-v=-{\mathrm{\Phi }}^{\prime }(v),\mathrm{\Phi }(v)=-{\int }_0^v[u(z)-z]dz.$$

We have a real topographic map. Its elevation is the potential $\mathrm{\Phi }$. The traveler moves downhill because

$$\frac{d\mathrm{\Phi }}{ds}=-[{\mathrm{\Phi }}^{\prime }(v){]}^2\le 0.$$

Depth affects how rapidly this happens per added trial; it does not independently choose the valleys. With a physical clock $t$ and accepted evidence rate $r=d\tau /dt$, the speed becomes $r[u(v)-v]/\tau$. A hardness layer showing $\tau$ and a rainfall layer showing $r$ would distinguish a resistant belief from an environment supplying little new evidence.

What field should we draw? Start with a source whose positive probability is $p=(1+\tanh h)/2$. The number $h$ is the source rapidity, half its log-odds. Suppose positive and negative evidence are reweighted by $e^{Jv}$ and $e^{-Jv}$. After normalizing, their bias is exactly

$$u(v)=\frac{p{e}^{Jv}-(1-p)e^{-Jv}}{p{e}^{Jv}+(1-p)e^{-Jv}}=\tanh (Jv+h).$$

The exponential reweighting is an assumed selection rule; the hyperbolic tangent follows algebraically. Positive $J$ makes the stream favor the current lean. Interpreting $J$ as social coupling adds another modeling choice about how a community turns its members’ opinions into exposures. In either reading, the field gives a concrete feedback experiment.

Where the second valley comes from

Compare our fixed-point equation with a magnet:

The shared fixed-point equation is the familiar mean-field Ising equation, derived in David Tong’s statistical field theory notes. Here we can calculate the terrain directly from our chosen learning rule:

Subtracting the constant makes $\mathrm{\Phi }(0)=0$. At $J=0$, the continuous limit is $\mathrm{\Phi }(v)=v^2/2-v\tanh h$. This particular potential generates our particular dynamics; sharing a magnet’s stationary equation does not determine all of its thermodynamics or noise.

At zero field, its small-$v$ expansion is

$$\mathrm{\Phi }(v)=\frac{1-J}{2}{v}^2+\frac{J^3}{12}{v}^4+O(v^6).$$

Below $J=1$, the center is a valley. Above $J=1$, its curvature changes sign: the center becomes a ridge, with a valley on each side. A feedback strength has turned one destination into two.

Tilting the landscape with favors the right. A traveler in the left valley first moves within that valley as its location changes. Eventually the valley and its neighboring ridge merge and vanish. This disappearance is called a fold. Solve the stationary condition and zero-curvature condition together:

The negative-$v$ fold occurs at positive field. At $J=1.5$ its field is approximately $0.20755$. Slowly raising and lowering the field can carry the traveler along different surviving branches, producing hysteresis: the present destination depends on which valley the journey occupied before.

This is a candidate account of abrupt changes after gradual pressure. It also predicts movement before the tipping point and slow response near the fold. Finite-speed field changes can delay the observed jump beyond the static threshold.

If we add weak stationary noise to a fixed landscape, escape can occur before the valley disappears. In the high-barrier regime, Kramers theory gives an escape rate with an exponential dependence on barrier height divided by noise strength. A fixed escape rate produces an approximately exponential waiting-time distribution. Broad or heavy-tailed waiting times would need further structure, such as a mixture of barriers or changing environments. Those alternatives suggest different longitudinal measurements.

Rivers that refuse to settle

A one-dimensional continuous flow has a terrain potential obtained by integration. With three or more mutually exclusive choices, a Dirichlet belief lives on a multidimensional simplex, and the exposure field can circulate.

Imagine a local two-coordinate patch with displacement $\mathbf{z}=(z_1,z_2)$ and flow

$$\mathbf{F}(\mathbf{z})=-a\mathbf{z}+\omega (-z_2,z_1).$$

The first part descends the bowl $a|\mathbf{z}{|}^2/2$. The second circles its center. For $a=0$, a traveler keeps orbiting; for , the orbit spirals inward. A complete probability model must keep the motion inside the simplex, but this small patch already shows why elevation alone cannot describe every current.

Measuring motion around closed loops could expose this circulation. A gradient field has zero loop integral; the rotating term generally does not. Separating fields globally also requires boundary conditions and a choice of coordinates or metric. Panel reports and exposure records could begin locally: compare the current lean with the next accepted packet’s bias, then estimate the flow.

Ordinary additive Dirichlet updates still commute. Adding the same two fixed packets in opposite orders produces the same final parameter vector. Order dependence arises if earlier states change which later packets arrive or how they are weighted. Wigner rotation offers another adventurous route, but requires an additional multidimensional boost model and frame information. More categories alone do not produce it.

Can a hot topic act like a lens?

Consider a neutral sentence placed beside a contentious subject. Readers on opposite sides may hear different implications. Could we map that distortion as an astronomer maps deflection around a lens?

Here is a concrete hypothesis. Let $b$ measure semantic distance from the disputed topic, and let $M$ summarize a specified feature of its evidence population. Propose an interpretation shear

with fitted strength $A$ and a short-distance cutoff $b_0$. This is a trial response law inspired by lensing, not a gravity equation derived from the Beta identity. Other falloffs may fit better.

Construct matched messages at several semantic distances, randomize their placement, and measure how framing changes updates after accounting for baseline lean and calibrated depth. Compare the proposed falloff against a constant framing effect, topic familiarity, and source trust. An anomalous region would identify unexplained interpretive distortion worth investigating.

Choosing $M$ matters. Pooling two sharply opposed groups can produce a large $2\sqrt{\alpha \beta }$ even when neither group’s individual members carry both sides. Population disagreement, within-person ambivalence, and total accepted evidence require separate measurements. Calling all three “conflict” would hide the very geography the instrument is meant to reveal. Likewise, unexplained shear cannot by itself establish latent polarization: that inference needs alternative explanations to be tested.

More instruments for the expedition

The same habit of pairing a mechanism with a measurable quantity opens a larger workshop.

The lock-in idea is especially practical. In electronics, a lock-in amplifier multiplies a measured signal by a known reference and averages, isolating the response aligned with that reference. For elicitation, distribute balanced intervention signs $z_i\in \{-1,+1\}$ independently of unrelated drift, and consider $y_i=a{z}_i+b_i$. Then the average of $z_i{y}_i$ targets $a$ when the nuisance term satisfies $\mathbb{E}[z_i{b}_i]=0$. Random assignment can support this separation; carryover and nonlinear responses require a richer design. It is a specific measurement connection to experimental design.

The most promising first expedition is modest: use resolvable coin tasks to calibrate recoil, estimate an exposure field from repeated observations, and test whether adding feedback actually creates the predicted two-valley response. Along the way, the map becomes more interesting. A stable position might reflect deep accumulated trials, a quiet stream, or a valley maintained by the environment. Different causes call for different instruments.

Further reading

• The GPS position and clock problem, for recovering hidden state from calibrated observations.
• NIST’s law of propagation of uncertainty, for the recoil error budget.
• Tong: From Spins to Fields, for the mean-field magnet comparison.
• Hänggi, Talkner, and Borkovec: Reaction-rate theory, for noise-driven escape from a valley.
• Stanford Research Systems: Lock-in amplifier basics, for measuring a small response against a known reference.