When a question changes the shape of uncertainty

You have a mysterious object, a detector that answers yes or no, and permission to turn the detector in any direction. Point it north. Ask ten questions. Receive five answers each way.

You have learned something about north. What happened to your uncertainty about east?

Our first experiment has a delicious answer: it grew. Learning squeezed the possibilities into a pancake, and pancakes are wide sideways.

Bring that pancake aboard. It will take us from Bayesian inference to probabilities with principal axes, then to a question menu that can discover new answers without spending its prior all over again. Along the way, Epistemic Physics gains a richer kind of belief: one that remembers the directions of its trials and the structure of its questions.

An unknown point and a movable question

Let the object’s unknown condition be a vector $\mathbf{v}=(v_x,v_y,v_z)$ inside the unit ball. A vector is an ordered list of components; its length $\Vert \mathbf{v}\Vert$ is at most one. Each component describes a possible directional bias.

Choose a unit vector $\mathbf{u}$ for the detector’s orientation, and give its positive answer the probability

The dot product $\mathbf{u}\cdot \mathbf{v}$ picks out the bias along the question. It lies between $-1$ and $1$, so the displayed probability lies between zero and one. The negative answer has the complementary probability.

Think of $\mathbf{v}$ as a plant’s hidden response tendencies and $\mathbf{u}$ as a chosen combination of light, temperature, and moisture. We have chosen a small model in which only their alignment controls the response. The point’s ball-shaped constraint couples those tendencies: a large bias in one direction leaves less room for the others.

A prior describes the possible conditions before collecting the current trials. Choose a uniform prior inside the three-dimensional ball: equal volumes receive equal probability. The detector’s answers then update this distribution into a posterior.

Point upward and write $s=v_z$. A horizontal slice at height $s$ is a disk with area $\pi (1-s^2)$. There are more possible points near the equator than near a pole. Thus the prior density of the vertical coordinate is

“Uniform inside the ball” has become a curved density when viewed through one coordinate. Geometry has already entered the statistics.

Five answers each way make a pancake

Let $Y$ count positive answers and $N$ negative answers, all collected along this same direction. Assume they are independent conditional on the fixed unknown point. Their likelihood, the probability of these answers as a function of that point, is proportional to

The proportionality symbol $\propto$ allows us to omit a factor that does not depend on $s$. Multiplying by the prior gives

Set $p=(1+s)/2$. This is exactly a Beta distribution for $p$, with shape parameters $Y+2$ and $N+2$. Those initial twos come from the chosen uniform-ball geometry.

Now set $Y=N=5$. The vertical density becomes proportional to $(1-s^2{)}^6$, favoring the equator. Its mean remains zero, and its variance becomes $1/15$. Variance is the average squared distance from the mean.

What happens sideways? Conditional on a particular height $s$, the remaining coordinates are still uniform inside a disk of radius $\sqrt{1-s^2}$. Either sideways coordinate has conditional variance $(1-s^2)/4$. Average over the updated heights:

The symbol $\mathbb{E}$ means an average under the stated distribution. The same sideways result holds for $v_y$.

The sideways variance rose by one sixth. Balanced vertical answers favor the broad middle of the ball, opening more room sideways.

Make a prediction: add several positive answers instead. They pull the posterior toward a pole. Will the sideways variance still grow?

There is a compact formula behind every batch in the laboratory. Define $T=Y+N+4$ and the posterior axial mean $q=(Y-N)/T$. Then

The subscripts $\parallel$ and $⟂$ mean parallel and perpendicular to the question; $j$ chooses either sideways coordinate. This posterior retains both the answer balance and the direction in which the answers were collected.

Average the sideways variance over every possible batch of ten answers, weighted by the prior’s predictions, and it returns to $1/5$. Particular experiences can widen one coordinate even though the average across experiences obeys the familiar variance bookkeeping.

A trial has a direction

A single total trial amount cannot describe this pancake. Ten repetitions along north and ten questions spread across three directions can leave very different unknowns.

The local sensitivity of one answer is captured by its Fisher information matrix. This matrix measures how strongly small changes in the unknown vector affect the answer distribution. For an interior point, where , differentiating our likelihood gives

The superscript $\mathsf{T}$ turns a column into a row. Their product $\mathbf{u}{\mathbf{u}}^{\mathsf{T}}$ is an outer product, a matrix whose action follows one direction. Its rank is one: it supplies one independent direction of local sensitivity.

This suggests an adventurous learning machine. Keep a posterior shape and a library of possible questions. Predict how each question would reshape that posterior, then choose a direction that helps the next decision. A robot could ask where its map is thinly supported. A tutor could choose an exercise that separates two plausible misconceptions. An experimentalist could turn a sensor toward the long axis of the remaining uncertainty.

At the origin, repeating one question $n$ times supplies information $n\mathbf{u}{\mathbf{u}}^{\mathsf{T}}$. Spreading the questions equally across three orthogonal axes instead supplies $(n/3)I$, where $I$ is the identity matrix. The same trial count now reaches every direction. Question diversity has acquired a precise geometric meaning.

Put an amount and a direction in the same object

An ordinary Beta probability can be assembled from two independent Gamma amounts with the same rate:

A Gamma distribution describes a positive random amount; its shape parameter controls its distribution. The notation $\sim$ means “has this distribution.” With the chosen rate one, the mean Gamma amount equals its shape. The resulting share $p$ has a Beta distribution with shapes $\alpha ,\beta$.

Now give the amounts directions. A Gaussian vector $\mathbf{z}$ has bell-shaped coordinates, and its outer product $\mathbf{z}{\mathbf{z}}^{\mathsf{T}}$ is a positive matrix. Summing such outer products produces a Wishart matrix, a matrix relative of the Gamma amount.

Take independent Wishart matrices $W_{+},W_{-}$ with a common scale, form their total $S$, and normalize:

The operation $S^{-1/2}$ rescales the principal axes of $S$ until the total becomes $I$. Applying that same transformation on both sides preserves positivity. This construction gives a matrix Beta distribution. In real dimension $d$, with Wishart degrees of freedom , its density is proportional to

Here $det$ is the determinant, the matrix’s volume-scaling factor; degrees of freedom count independent Gaussian vectors when they are integers. Díaz-García and Gutiérrez-Jáimez describe the matrix Beta construction and density.

The matrix order $Q⪰0$ means ${\mathbf{w}}^{\mathsf{T}}Q\mathbf{w}\ge 0$ for every vector $\mathbf{w}$. Consequently, every unit direction reads an ordinary probability ${\mathbf{w}}^{\mathsf{T}}Q\mathbf{w}$ between zero and one.

The smallest and largest possible readouts are the eigenvalues of $Q$. Their eigenvectors are its principal directions. Imagine a probability crystal: turn it, and another share appears.

The object and the question get different matrices

Let $\rho$ describe the state being examined. It is a positive matrix with trace one; the trace, written $\mathrm{tr}$, adds the diagonal entries. Let $Q$ describe a binary question, with $0⪯Q⪯I$. Their meeting produces

This is the state-and-effect pairing of quantum measurement theory. A menu of positive question matrices that sums to $I$ gives outcome probabilities summing to one. Such a menu is called a positive operator-valued measure, usually shortened to POVM. An operator here is a matrix acting on vectors. Heinosaari, Jivulescu, and Nechita study random versions of these menus.

Our opening ball fits directly into this language. Let $i$ be the imaginary unit, defined by $i^2=-1$. For any three-component vector $\mathbf{a}$, define

Then choose $\rho =M(\mathbf{v})/2$ and $Q=M(\mathbf{u})/2$. Direct multiplication gives

The matrix’s eigenvalues are $(1\pm \Vert \mathbf{a}\Vert )/2$ after division by two, so the ball constraint supplies the needed positivity. The earlier directional likelihood is now a matrix pairing.

This opens two different statistical adventures. We can learn an unknown state using calibrated questions, or learn an unknown question using calibrated states. Put priors on both and the instrument can learn about its world while the world teaches it about the instrument. A laboratory’s calibration procedure becomes part of its inference model.

The gold handle selects the state direction. The colored curves belong to the question menu. Changing the state preserves that menu; changing a raw directional amount rebuilds the menu through its shared total. These are two separate ways an ordinary probability can move.

Discover a finer answer without losing the old one

Suppose a detector originally reports “yes” or “no,” and later learns to distinguish two kinds of yes. Give every outcome a positive raw matrix $W_i$ and normalize them together:

Now split a raw outcome into two pieces, $W_i=W_{i,A}+W_{i,B}$. The shared total remains exactly the same. Therefore

The whole old answer is recoverable for every state. Refinement reveals distinctions inside an existing answer. Nechita explains the shared Wishart normalization behind random measurement menus.

Picture a field microscope that first recognizes “organism,” then “alga,” then several algal forms. Each refinement should preserve the older broad question. If the original tracelets include images or other identifying features, their lineage can also carry the distinction into the finer categories. The question tree grows while its trunk remains connected.

Give an expandable menu one prior budget

Can the prior itself survive that growth? Choose positive weights $a_j$ summing to one, unit directions ${\mathbf{u}}_j$, and independent Gamma amounts

Attach an outcome label to each contribution, then build each raw outcome matrix by summing the contributions carrying that label:

This is a finite marked Gamma construction: each random amount has a label and a direction attached. Choose directions spanning the state space so that the shared total is invertible. Normalize once, exactly as before.

The laboratory’s four contributions illustrate this construction with $a_j=1/4$. Coarse “yes” receives $1/2$ of the concentration budget. Splitting it gives $1/4$ to each child, while the total remains one. These prior shapes belong to the menu model; the earlier uniform-ball model has its own geometrically determined shapes.

There is even an exact recipe for subdividing one Gamma amount. If $a=a_A+a_B$, draw

independently, and set $G_A=BG$, $G_B=(1-B)G$. The children are independent Gamma amounts with shapes $a_A,a_B$, and their sum is the original $G$. Retain the parent’s direction for these children, and both the raw matrix and its prior budget refine consistently.

A continuum of labels and directions suggests a Gamma random measure: an additive random supply of amounts over regions, with additive shape budgets. Assign total base concentration one and carry the same shared normalization through every partition. Work on infinitely divisible matrix Gamma laws offers mathematics for taking that construction further. Pérez-Abreu and Stelzer develop such cone-valued distributions.

Now imagine an Epistemic Physics laboratory that invents better questions as it learns. Early observations build a rough instrument. Later tracelets divide its answers into finer distinctions. Its state posterior develops ridges and pancakes; its question prior grows branches; its lineage keeps the old observations connected to their newer interpretations. When evidence takes different roads builds an exact reconstruction from joined observations, then lets the questions themselves change the state.

An ambitious next experiment would let this laboratory choose both the direction of its next trial and whether to refine its answer menu. Give each action a cost, predict its effect on the next decision, and let the instrument grow where a new distinction becomes useful. Learning would then reshape what the observer believes and what the observer knows how to ask.