Move to ask a better question

Imagine inspecting a sculpture in a dark room. You could turn up your flashlight. You could walk around the sculpture. Those actions help in different ways: one can make a signal easier to measure; the other can make rival interpretations predict different signals.

Nervous systems do both. Active sensing means acting to shape the sensory input you receive. Sometimes the action creates a signal, as when an animal emits a sound or electric field. Sometimes it changes the viewpoint, the sampling location or the timing.

In Epistemic Physics language, this takes us upstream of accumulating trials. Before asking how a tracelet changes a belief, ask which action makes the next tracelet worth obtaining.

A fish that moves to feel

Weakly electric fish produce an electric field and sense how their surroundings alter it. In a refuge-tracking experiment, Stamper, Roth, Cowan and Fortune varied illumination and water conductivity while fish followed a moving shelter. In darkness, fish added whole-body oscillations; those oscillations grew as electrosensory salience decreased. Tail bends appeared only in dark trials. The shelter remained inside sensing range, so simply reaching previously unsensed space could not explain the behavior. Tracking gain remained similar across conditions, while the extra oscillations increased tracking error. Read the original 2012 study.

There is a useful lesson here: an animal can change the stream arriving at its sensors by changing how it moves. The geometry of observation belongs inside the perception problem.

Let’s build a tiny world where we can separate two effects that real movements can combine.

Turn up the signal, or change the question?

Suppose an object lies at one of two possible positions, $\theta =-1/2$ or $\theta =+1/2$. Our sensor sits at position $x$. Choose the noiseless signal

where is a gain, a factor that scales signal amplitude. Positions and signal levels use dimensionless units. This quadratic is a deliberately simple teaching model: two possible worlds have intersecting signal curves.

At the center, both worlds predict exactly the same reading:

Increasing $g$ raises both equally. Their equality survives. But move right or left, and the difference becomes

The two operations line up like this:

Our demo holds the fresh measurement noise fixed when changing gain. Under that assumption, amplifying an existing difference helps. If the same gain amplified all noise by the same factor, that signal-to-noise benefit would disappear.

Try this sequence: double the gain at the center. Then move the sensor. Finally, keep your position fixed and enable the shared offset. Which sampling route becomes better, and why?

Open the active-sensing experiment in the playground.

The curves are candidate predictions, so neither is secretly designated as the observed truth. The bars show an expected separation between their observation distributions. They let us compare sensing plans before drawing noisy data.

A changing signal can answer a different question

At the center, the signal slopes, their rates of change with position, point in opposite directions:

A single center reading hides the difference. Two nearby positions expose it. If the sensor follows a known path $x(t)$ through time $t$, then the chain rule gives

Spatial shape becomes a temporal signal. Moving faster changes that temporal signal even along the same route. How a biological sensor responds then depends on its own filtering and time scales. The quadratic demo isolates the spatial part; the fish study motivates asking how actual movements reshape the full stream.

This is a particularly playful connection to evidence geometry: a learner can change the direction of the evidence it expects to receive by choosing a measurement context. Simply accumulating more readings in a blind spot might never resolve the ambiguity.

Count readings, then inspect their shared history

Let the $i$th measured value be

The term ${\epsilon }_i$ is fresh Gaussian noise, a bell-shaped random error with mean zero and standard deviation $\sigma$. Different readings have independent ${\epsilon }_i$. The term $b$ is a single Gaussian offset shared across the entire four-reading episode, with standard deviation ${\sigma }_b$. Think of a sensor whose zero setting drifts between episodes.

That shared offset is a simple source lineage. Four measurements have four fresh errors, but all inherit one calibration condition. Labeling each reading with a new identifier would not remove this dependence.

Let $d_i=f_A(x_i)-f_B(x_i)$ be the difference between the two candidate predictions at reading $i$. For $n$ readings, our model has the exact separation

Here $D$ is the Kullback-Leibler divergence from the observation distribution under A to that under B: the average natural logarithm of the likelihood ratio when A generates the data. Its units are nats, because it uses the natural logarithm. In this equal-covariance Gaussian model the separation is symmetric between A and B. It is an expected evidence measure, not a realized posterior probability or a trial count.

The first sum rewards differences between predictions. The second term discounts a difference that a common offset could explain. The derivation comes from the covariance matrix $\mathrm{\Sigma }={\sigma }^{2}I+{\sigma }_b^2\mathbf{1}{\mathbf{1}}^{\mathsf{T}}$ and the Gaussian identity $D=\frac{1}{2}{\mathbf{d}}^{\mathsf{T}}{\mathrm{\Sigma }}^{-1}\mathbf{d}$. Here $I$ is the identity matrix and $\mathbf{1}$ is a column of ones.

Now compare four readings at $x=a$ against two at $x=-a$ and two at $x=a$:

• With only fresh noise, both plans have the same sum of squared differences and therefore tie.
• With a shared offset, the one-sided plan loses separation. The two-sided plan has $\sum _i{d}_i=0$, so its differences survive the offset unchanged.
• At $a=0$, both plans return to the blind spot.

The movement benefit follows from the measurement model. The comparison matches reading count; traveling between positions could cost extra energy or time. A decision model can add those costs explicitly and ask how much separation each route buys.

When vision and touch vote together

A second way to improve sensing is to combine different cues. Suppose vision estimates an object’s height as $m_V$, and touch estimates it as $m_H$. Let their error variances be $s_V^2$ and $s_H^2$. A variance measures squared spread around the mean; a smaller value here means a more precise cue.

For unbiased, independent errors, the minimum-variance weighted estimate is

The quantities $r_V=1/s_V^2$ and $r_H=1/s_H^2$ are precisions. Rewrite the expression as a balance divided by a total:

Ernst and Banks tested visual and haptic estimates of object height while varying visual uncertainty. Their results were consistent with weighting the cues according to their relative precision under the tested conditions. Read their 2002 Nature paper.

For a numerical example, let vision say $10$ millimeters with standard deviation $1$ millimeter, and touch say $14$ millimeters with standard deviation $3$ millimeters. Their precisions have ratio $9:1$. The combined estimate is $10.4$ millimeters with variance $0.9$ square millimeters. The more precise cue has more influence.

Place this beside the evidence-cone calculation:

The common operation is weighted averaging. The weights have different meanings and units. Gaussian precision is inverse squared measurement units; Beta shape amounts are dimensionless. A cue estimate becomes a Beta trial contribution only after defining an observation model that supports that conversion.

If the cue errors share a common cause, their covariance changes the optimal weights and the combined variance. The same source-identity question from our moving sensor returns: which part of the apparent agreement is fresh information, and which part is shared?

Three expeditions to try next

Choose a path by predicted separation. Give a learner a budget of readings and let it compare possible next positions. An action can be selected because it separates its current leading explanations. Add a distance cost and see when small oscillations beat long excursions.

Learn the sensor’s reliability. Our cue variances were supplied. A richer learner could estimate them from experience, distinguishing a drifting calibration from independent noise. Then the weight assigned to a cue becomes a learned state with its own malleability.

Carry lineage through perception. A visual and tactile tracelet may both come from the same object encounter. Preserve that encounter’s identity while recording the separate modalities and errors. Ask which combinations deserve an increased trial amount, and which primarily clarify a shared hidden cause.

In all three expeditions, action and inference become a conversation. The learner forms a question, movement makes a new observation possible, and the resulting tracelet changes what the learner can ask next.