A field of questions

Someone hands you a sealed jar of jelly beans. You estimate 250. Then they show you an identical empty jar and let you inspect one bean.

Your answer to “Are there more than 200?” changes. So does your answer to “Are there fewer than 300?” Your view of tomorrow’s weather probably stays put.

One observation has touched a whole neighborhood of questions. That neighborhood is the subject of this field trip. A question field assigns a belief state to each question, then asks how those states fit together, influence one another, and respond to being measured.

The physics analogy is inviting: a temperature field attaches a temperature to every location; our field attaches a distribution to every question. But questions have an extra talent. They can overlap, imply each other, or ask for different resolutions of the same unknown. A good map must show both logical connections and learned resemblance.

Two maps of the same territory

First, the logical map. “How many beans?” divides the possible worlds into exact counts. “More than 200?” groups those worlds into two larger regions. We call that grouping coarsening. Asking for a finer distinction is refinement.

Suppose we use three categories: fewer than 200, 200 through 300, and more than 300. A Dirichlet distribution is the multicategory relative of a Beta distribution. Give its three categories positive weights $(2,5,3)$. Merging the last two categories produces a Beta distribution with weights $(2,8)$. Total weight stays ten:

$$({\alpha }_1,{\alpha }_2,{\alpha }_3)⟶({\alpha }_1,{\alpha }_2+{\alpha }_3),\sum _i{\alpha }_i=\text{unchanged}.$$

This conservation comes from using one shared underlying distribution. Drawing a fresh prior for every question would generally break the correspondence. Likewise, counting the same jar inspection separately under three phrasings would create three ledger entries from one lineage. Independent trials, the small updates represented by tracelets, need to remain distinguishable from copies and downstream consequences.

We can extend a consistent family of such partitions using a Dirichlet process, a prior over probability distributions whose finite partitions have Dirichlet weights. That is one possible model for uncertain bean counts; continuity of the question alone does not select it. Ferguson’s original construction makes the shared base measure explicit.

The second map is semantic. It records which questions resemble each other. Two estimates of bean count are close in subject matter; bean count and sky color are distant. Such resemblance need not satisfy the mathematical rules of a distance, and it can depend on the learner. A gardener sees connections between soil acidity and leaf color that a novice misses.

Logical implication and semantic proximity do different jobs. “Both plants survive” implies “the first plant survives,” regardless of anyone’s expertise. Knowing that one plant survived might change expectations about a nearby plant, but the amount of that change needs a model of their relationship.

Can your answers inhabit one world?

Let $A$ and $B$ be events, and quote three probabilities:

$$a=P(A),b=P(B),c=P(A\cap B).$$

There are four possible worlds: neither event, only $A$, only $B$, and both. Their probabilities must be

$$(1-a-b+c,a-c,b-c,c).$$

Each entry must be nonnegative. Therefore all three quotes can belong to one joint distribution precisely when

$$max(0,a+b-1)\le c\le min(a,b).$$

Try $a=0.7$, $b=0.6$, $c=0.2$. The implied probability of neither is $-0.1$. Something has gone wrong before any observation arrives.

The mismatch has a concrete interpretation. Imagine a ticket paying one dollar if its event happens, with its quoted probability serving as its price. Assume a person accepts either side of a trade at that price. Sell them $A$ and $B$ tickets and buy their “both” ticket. You receive $1.10$ up front and owe a net payout of at most $1$. You gain at least ten cents, whatever happens, at the quoting forecaster’s expense. Dividing all three stakes by three bounds their total absolute size to one and leaves a guaranteed gain of $1/30$ dollars.

This is a Dutch book, a collection of accepted bets guaranteeing a loss to the incoherent forecaster. Its assumptions matter: common settlement events, two-sided prices, linear ticket payoffs, and no transaction costs. Probability reports collected in different contexts need not be offers to trade under those conditions.

There is an elegant geometric identity behind the example. Let $p$ be a vector of quoted prices, let $z(\omega )$ be the vector of ticket outcomes in world $\omega$, and let $C$ contain all coherent price vectors. Then

$$\underset{\text{largest guaranteed trading profit}}{\underbrace{\underset{\Vert s{\Vert }_1\le 1}{max}\underset{\omega }{min}s\cdot [z(\omega )-p]}}=\underset{\text{smallest maximum quote correction}}{\underbrace{\underset{r\in C}{min}\Vert p-r{\Vert }_{\mathrm{\infty }}}}.$$

Here $s$ lists signed ticket holdings; $\Vert s{\Vert }_1$ adds their absolute sizes. The infinity norm on the right takes the largest coordinate difference. To see why the two match, replace the worst world with a worst probability mixture of worlds, whose mean lies in the convex set $C$. Finite-dimensional minimax duality exchanges the choice of holdings and coherent mean; maximizing a dot product over the unit one-norm ball returns the infinity norm. These are standard tools of convex duality.

The connection to a perpetual-motion machine is a useful image of an exploitable inconsistency. The equation measures dollars per allowed stake, without introducing a physical energy or a rate of power production.

Before using the lab, predict which changes when sharing becomes broader: the total weight added, the number of questions affected, or both? Then make the “both” ticket more expensive than ticket $A$ and inspect every possible payoff.

Spread, sense, and audit

Send one observation through the garden

Reset field and quotes

Choose where a new trial enters. Its weight spreads across nearby beds. Can you shape the response with three trials? These are optional puzzles; keep exploring afterward.

Help the middleShare widelyRevise the middle

Try this: Raise bed 5 to 60% while beds 1 and 9 stay at or below 51%. 0 independent trials placed.

🌱Bed 150.0%🌱Bed 250.0%🌱Bed 350.0%🌱Bed 450.0%🌱Bed 550.0%🌱Bed 650.0%🌱Bed 750.0%🌱Bed 850.0%🌱Bed 950.0%

Narrow sharing Local sharing Broad sharing

Observe success Observe failure

Ask the garden instead: self-endorsement experiment

Self-endorsement weight 0: passive report0.2: weak endorsement0.8: strong endorsement

Ask with upward reporting bias

Site 5: probability mean

50.0%

Site weight α + β

4.00

Independent trials / asks

0 / 0

Choose a question site, then observe or ask.

Chosen model: each site starts at Beta(2, 2). A Gaussian-shaped kernel divides one trial's weight among related sites; these fractional updates share one lineage. This is a teaching rule for generalization, not an exact joint Bayesian posterior. Asking reports the current mean plus 15 percentage points, capped at 98%, then optionally treats that report as self-endorsement.

Another experiment: repair impossible probability quotes

Can these three quotes coexist?

A and B are two events. “Both” means A and B happen together. Edit the prices of tickets paying $1 when the event occurs.

A B Both

Coherent example Overpriced conjunction

Sell ⅓ A and ⅓ B; buy ⅓ both. Guaranteed buyer profit: $0.033 per unit absolute stake. Distance to the nearest coherent quote set: 3.33 percentage points.

Neither$0.367

A only$0.033

B only$0.033

Both$0.033

Assumptions: the same quotes apply to buying and selling, tickets settle on one shared outcome, total absolute stake is at most one, and there are no fees. The guarantee concerns all four possible worlds, regardless of their likelihood.

What carries an observation to its neighbors?

At each question $q$, assign positive Beta weights $\alpha (q)$ and $\beta (q)$. Write

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

The ordinary probability mean is $\mu$; the centered mean $v$ runs between $-1$ and $1$. Total weight $\tau$ includes the prior. At each site the familiar cone identity still holds:

$$m(q{)}^2=\tau (q{)}^2-[\alpha (q)-\beta (q){]}^2=4\alpha (q)\beta (q).$$

A collection of these identities does not specify which sites communicate. That requires a kernel, a rule describing how influence depends on the pair of questions.

The lab chooses Gaussian-shaped influence, strongest near the selected site. It normalizes the weights so that one observation adds exactly one unit across the whole field. This is a deliberately simple sharing rule. The sites retain correlated ancestry, and their separate Beta ledgers do not, by themselves, specify a coherent joint distribution over all questions.

A Gaussian process gives another, fully specified example. It models a real-valued latent function $f(q)$ using a mean $m(q)$ and covariance kernel $k(q,q^{\prime })$. For one observation $y$ at $q_{\ast }$ with Gaussian noise variance ${\sigma }^2$, conditioning gives

The first line is the footprint of the changed mean. The second is the reduction in uncertainty about the latent function. They share a kernel, but one depends on its value and the other on its square. These are Gaussian conditioning equations, not Beta update equations; using a latent function for probability requires an additional link or likelihood. See Gaussian Processes for Machine Learning, chapter 2.

Neural networks offer a related bridge. Under parameter gradient flow, the neural tangent kernel pairs the parameter gradients of two predictions. It determines how training error at one input changes another prediction. The kernel can evolve during learning; it becomes fixed in particular infinite-width limits. It is an instantaneous learning-response kernel, while a full time-dependent propagator describes the accumulated evolution. Jacot, Gabriel, and Hongler derive the distinction.

A human experiment could borrow the same design: measure a grid of responses, introduce targeted evidence at one question, then measure again. Repeating at several locations estimates a response matrix. Randomized untreated groups and separate remeasurement controls help distinguish the evidence’s effect from the questionnaire’s effect. Whether expertise narrows that footprint, or analogy creates distant connections, becomes a question for the resulting data.

Build a bridge between questions

Instead of choosing a ready-made Gaussian kernel, construct a small sharing network. Route one observation’s weight through two arms and split an arm when several destinations need a share. The goal is a specified footprint, with the original trial counted once across the whole map. This turns “which questions should move?” into a visible modeling decision.

One trial, several destinations

Build a bridge between questions

Reset bridge

Build a sharing rule for one observation. Each arm carries half its weight. Route the arms, or split one between neighboring destinations, to match the requested footprint.

One original trial↓ ½ through each arm ↓

Arm A · weight ½

Jar countPackingFill height

↓ Send the whole half

½ to Jar count

Arm B · weight ½

Jar countPackingFill height

↓ Send the whole half

½ to Jar count

Footprint 1 of 3 · Fill the outlined target bars

Jar count

1 routed / 0.5 wanted

Packing

0 routed / 0.5 wanted

Fill height

0 routed / 0 wanted

Send this observationNext footprint

Sky bars preview your routing. Gold outlines mark the requested weight at each question.

What this bridge assumes

The three names stand for related binary questions about a jar. We stipulate that this observation supports all three and design only how its unit weight is shared. Each destination starts at Beta(2, 2). Both arms share one original lineage; their fractions are not new independent trials. The kernel is chosen, and separate Beta marginals do not guarantee a coherent joint probability model.

A split arm sends quarters to its selected destination and the next one in the displayed cyclic order. Retrying reconstructs the experiment from a fresh baseline. Logical relationships or calibrated source reliability could impose additional constraints in an application.

Does an argument spread like heat or sound?

At one site, additive evidence obeys

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

The incoming bias $u$ is the direction of the next increment; $v=x/\tau$ summarizes the accumulated state. This equation says nothing yet about neighboring questions or elapsed clock time.

To produce spatial spreading, choose a coupling. For example, let $h(q,t)$ be a small perturbation of a belief field and postulate

$${\partial }_{t}h=D{\partial }_q^{2}h.$$

Here $t$ is elapsed time and controls local smoothing. This is diffusion: a sharp disturbance broadens like a drop of dye. Its continuum equation has arbitrarily small tails at every distance for any positive time. Individual traces can first rise and then fall as the disturbance passes; “diffusive” does not mean every site’s response decreases monotonically.

Now introduce a persistence variable, or equivalently a second time derivative:

This damped-wave model has characteristic speed $\sqrt{D/M}$. Its second initial condition, the initial rate of change, stores something the diffusion equation does not. Calling $M$ “commitment inertia” proposes a psychological interpretation that can be tested; it does not follow merely from accumulating many trials.

The equations suggest useful comparisons. Diffusion smooths a localized perturbation; a sufficiently underdamped wave can carry a front and overshoot. Apparent community opinion waves could also come from repeated external broadcasts or delayed communication. Observations of ringing need to be compared with those alternatives before assigning an inertia.

A map that learns, and attention that chooses a route

What if the question map itself changes? A novice groups many mushrooms together; an expert distinguishes small variations that once looked identical. We can model that learning by changing distances, similarities, or kernel parameters.

A geometric program would measure discrimination between neighboring questions and fit a metric, a local rule for converting small changes in coordinates into distances. The adventurous next step is to let accumulated evidence alter that metric and use the resulting shortest paths to model attention. This resembles curved-space physics. To turn the resemblance into an equation, we must specify how evidence changes the metric and how attention follows it. Evidence density alone supplies neither rule.

Attention also creates a resource-allocation problem. Suppose $a_i$ is the fraction of a fixed budget devoted to question $i$, with $\sum _i{a}_i=1$. Give each question a salience score $r_i$, and choose to reward both salience and diversity:

$$\underset{a_i\ge 0,\sum _i{a}_i=1}{max}[\sum _i{a}_i{r}_i-T\sum _i{a}_i\log a_i].$$

For , differentiation gives $a_i\propto e^{r_i/T}$. High $T$ spreads attention; low $T$ concentrates it. This is the same softmax form used for normalized attention weights in the original Transformer.

The correspondence is exact for the chosen entropy-regularized allocation. The total budget is a constraint we imposed. Its Lagrange multiplier is one shared shadow price, whereas the $r_i$ values are site-specific salience scores. Those are distinct quantities. A grand-canonical thermodynamic model would additionally allow exchange with a reservoir; softmax normalization alone does not provide that model.

How do you sense a belief?

Several instruments are possible, each exposing a different part of the system.

Actions are especially tempting: surely buying an umbrella reveals a rain belief? It also reveals how much getting wet matters, whether an umbrella was already available, and whether the purchase was a gift. Passive observation reduces one route of interference while leaving an inverse problem.

The proposed belief spectroscopy experiment makes that inverse problem explicit. Suppose a recorded response has the form

$$y(t)=\sum _{j=1}^J{w}_j{e}^{-{\lambda }_{j}t}+ϵ(t),$$

where $w_j$ are component amplitudes, ${\lambda }_j$ are decay rates, and $ϵ$ is measurement noise. If the rates are sufficiently distinct and the recording is long and precise enough, a fitted mixture can separate fast and slow responses. Nearby rates or short recordings can produce many nearly indistinguishable explanations.

Exponential decay is an assumption here. With constant incoming bias and accumulating weight, the additive Beta rule instead yields $v(\tau )-u\propto 1/\tau$. Exponential relaxation can arise under a different clock or a fixed-weight forgetting model. Identifying a decay component with a particular hidden belief also needs independent evidence. The productive experiment first calibrates which response law actually fits.

The question that nudges its own answer

Asking can disturb a belief in at least three ways: assembling an opinion that was previously unformed, encouraging commitment to the stated answer, and increasing attention to the subject. Experiments on question-induced behavior changes motivate measuring these effects rather than presuming their size.

We can isolate self-endorsement with a small model. Suppose the current centered mean is $v$, and the reported centered value is $v_s$. The report is assigned weight $w$, of which a fraction $s$ is taken back into the person’s ledger. Weighted addition gives exactly

$$v^{\prime }=\frac{\tau v+sw{v}_s}{\tau +sw},\mathrm{\Delta }v=\frac{sw(v_s-v)}{\tau +sw}.$$

If the report matches the current mean, this mechanism produces no displacement. If $s=0$, it produces none even for a noisy report. With the same report error and weight, a larger $\tau$ reduces the displacement: the belief has lower malleability under this update rule.

If $v_s=v+\epsilon$ with conditionally unbiased report noise of variance ${\sigma }_{\epsilon }^2$, then

$$\mathrm{Var}(\mathrm{\Delta }v\mid v,\tau )={\left(\frac{sw}{\tau +sw}\right)}^2{\sigma }_{\epsilon }^2.$$

Repeated asking changes $\tau$ in this model, so the step size generally shrinks. A constant-step random walk would be an additional approximation. The demo uses upward-biased reports so you can see the difference between “the instrument read a larger number” and “the ledger moved toward that number.”

A minimally disturbing instrument would first estimate which mechanism matters. Low-pressure reports might reduce commitment; questions about neighboring topics might reduce direct salience. Reconstructing the target from neighbors then depends on an accurately calibrated kernel. Anonymity could change reporting noise as well as self-endorsement, so its effect should be measured rather than assigned automatically.

There is no universal positive information-times-disturbance lower bound in these equations. Setting $s=0$ permits informative reports with zero self-endorsement disturbance. A nontrivial bound would require additional constraints on the channel, noise, and definition of disturbance.

Question order is another useful test. Quantum-probability models predict specific patterns such as the QQ equality. Ordinary Beta increments commute: adding two fixed packets in opposite orders gives the same endpoint. Order effects require state-dependent packets or additional state. A proposed rotation-based account must define its multidimensional geometry and response probabilities before it can be compared with the quantum prediction.

Where free energy enters the picture

A final bridge points toward the free energy principle. For a latent state $z$, observation $y$, model $p$, and approximating distribution $Q$, variational free energy is

$$F[Q]={\mathbb{E}}_Q[\log Q(z)-\log p(y,z)]=\mathrm{KL}(Q\Vert p(z\mid y))-\log p(y).$$

The Kullback-Leibler divergence $\mathrm{KL}$ measures the mismatch between two distributions and is nonnegative. Minimizing $F$ over $Q$ therefore approaches the posterior within the allowed family. Friston’s account connects this inference framework to perception and action.

Epistemic Physics gives a concrete family in which to investigate such a bridge. Yet precision, meaning inverse variance, depends on both weight and mean:

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

Thus $\tau$ alone is not the precision. To connect a field model to active inference, specify its likelihood, dynamics, action policy, and conditional-independence structure. Then ask whether its updates minimize a particular functional. The cone identity supplies useful coordinates for that work; a derivation of an entire theory of cognition remains an open project.

A jar of beans has led us to joint probability, learning kernels, waves, instruments, and inference. The unifying question is practical: when one answer changes, which other answers should move, by how much, and how can we tell without pushing them ourselves?

Further reading

• Ferguson: Dirichlet processes, for consistent beliefs across partitions.
• Boyd and Vandenberghe: Convex Optimization, for separating hyperplanes and duality.
• Rasmussen and Williams: Gaussian Processes for Machine Learning, for explicit learning kernels.
• Jacot, Gabriel, and Hongler: Neural Tangent Kernel, for the response geometry of gradient learning.
• Wang and colleagues: question-order experiments, for a competing framework with specific observable predictions.