Learning travels and cools

Imagine tracking a flock of birds through fog. You could draw an arrow toward the flock’s center, or sketch the cloud occupied by all the birds. The arrow tells you where the flock is heading. The cloud tells you how scattered it is. Watching only one of them leaves half the story invisible.

A belief has a similar double life. Its mean probability can travel toward support or opposition while its distribution changes width. In the evidence cone, we describe the first story using direction and velocity. In statistical mechanics, we can describe the second using an energy landscape and temperature.

Can both descriptions belong to the same belief? They can, and there is an exact equation connecting them. The surprise is that cooling does not guarantee a narrower cloud after every new observation. Sometimes the ground beneath the cloud moves too.

One state, two maps

Let an unknown probability $P$ have a Beta distribution with positive parameters $\alpha$ and $\beta$. These are weights that include the prior. For a Bernoulli experiment, a supporting independent trial adds one to $\alpha$, and an opposing trial adds one to $\beta$. A lineage records which trials belong to this history; repeating somebody else’s observation does not create a fresh independent trial.

Define the total weight $\tau$, imbalance $x$, mean probability $\mu$, and centered mean $v$:

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

Our first map places the belief at $(\tau ,x)$ inside the cone . Its mass-like coordinate is $m=2\sqrt{\alpha \beta }$, giving the exact identity

$${\tau }^2-x^2=m^2.$$

Here $v$ describes the ray from the origin to the current state. The direction of an incoming stream is a different quantity, $u=dx/d\tau$. Differentiating $v=x/\tau$ gives

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

An incoming stream changes the mean when its direction differs from the existing balance. The same discrepancy produces a smaller change in a belief carrying more weight: greater resistance to change, or less malleability.

Our second map follows the entire distribution over possible probabilities. Its natural coordinate will be log odds,

$$L=\log \frac{P}{1-P},P=\frac{1}{1+e^{-L}}.$$

Log odds stretch the bounded interval into the whole real line. Even odds sit at zero; odds of three to one sit at $\log 3$; reversed odds sit at $-\log 3$. The random rapidity coordinate is $L/2$. Keeping the whole random variable matters: transforming the mean and averaging the transformed variable generally give different answers.

A Gibbs ensemble hiding inside a Beta belief

A Gibbs ensemble assigns greater weight to states with lower energy. With temperature measured in suitable dimensionless units, its density has the form $e^{-E/T}/Z$, where $Z$ normalizes the total probability.

The Beta density, including its normalization, is

$$p_P(p)=\frac{p^{\alpha -1}(1-p{)}^{\beta -1}}{\mathrm{B}(\alpha ,\beta )}.$$

Here $\mathrm{B}$ is the beta function. See NIST’s Beta distribution reference for the density and its standard moments.

Changing coordinates requires the Jacobian $dp/d\ell =p(1-p)$. That factor measures how much probability-coordinate space fits inside a little interval of log odds. Multiplying by it gives

This is an exact normalized Gibbs representation with energy $f_{\mu }$, inverse temperature $\tau$, and temperature $T=1/\tau$. It uses density relative to $d\ell$. Choosing a different coordinate measure changes the carrier factor, so specifying that measure is part of the statement.

The energy has a familiar meaning. If a candidate probability is $q=1/(1+e^{-\ell })$, then

$$f_{\mu }(\ell )=-\mu \log q-(1-\mu )\log (1-q).$$

This is the average surprisal, or cross-entropy, of predictions $q$ against a Bernoulli reference with success probability $\mu$. Surprisal is the negative logarithm of the probability assigned to an outcome: a poorly predicted outcome costs more. Our energy landscape is a prediction-loss landscape.

Its slope and curvature are simple:

The landscape therefore has one valley, at

$${\ell }_{\ast }=\log \frac{\mu }{1-\mu }=\log \frac{\alpha }{\beta }.$$

This is the mode of the log-odds density. Its inverse transform equals the probability mean $\mu$. The probability-density mode, when , is instead $(\alpha -1)/(\tau -2)$. A mode depends on the coordinate measure; the location of the tallest patch of a density can move when the horizontal ruler stretches.

Try moving the valley and changing the temperature

Predict what one opposing trial will do to a belief with weights $(20,1)$. Will its temperature fall? Will its mean move? Will its probability variance shrink?

Choose the cloud shape you expect, then add the opposing trial. The next two experiments change both weights together, holding the mean fixed. Open the density controls afterward to explore freely.

Three thermal experiments

Will cooling widen the cloud?

Reset experiments

Choose the cloud you expect, then carry out the experiment. Temperature and probability spread have different jobs.

Experiment 1 of 3 · A surprising trial

The belief is near the edge. Predict what happens to its probability spread.

Starting mean

95.2%

Starting weights

20 : 1

A narrower cloud A wider cloud

Add one opposing trial

Explore the actual density and weight controls

Rose dashed: experiment's starting density. Teal: current density over log odds. Both share the same vertical scale; the view is clipped to log odds −6 to 6.

Current mean 95.2%; weights (20.0, 1.0).

+1 support+1 oppositionDouble bothHalve both

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

Rescaling includes the prior and is hypothetical, rather than additional independent trials. The preview silhouettes indicate only wider or narrower probability spread; the plotted curves are normalized log-odds densities. Exploring these controls leaves the recorded mission verdict unchanged.

The rose curve remembers the preceding state, while teal shows the current one. The curves use log odds; the variance readout uses centered probability. These two measures of spread need not move together, because log odds stretch the probability axis unevenly. A new trial can move the valley and change the temperature together. Doubling both weights preserves their ratio, so the valley stays put while the Gibbs distribution concentrates around it.

That second operation is a hypothetical change of weight. It includes the prior and does not imply that the same data were observed independently twice. In log-odds coordinates, scaling $(\alpha ,\beta )$ by gives a density proportional to $p_L(\ell {)}^c$. In probability coordinates, the corresponding carrier factors differ: simply raising the probability density to $c$ would produce parameters $1+c(\alpha -1)$ and $1+c(\beta -1)$.

This distinction matters whenever “tempering” is used to mean changing a posterior’s temperature. Specify which density or likelihood is being tempered and which prior remains fixed.

The two factors in the cloud’s width

Let $B=2P-1$ be the centered random belief. Its mean is $v$. Directly from the Beta variance,

$$\boxed{\mathrm{Var}(B)=\underset{\text{directional factor}}{\underbrace{(1-v^2)}}\underset{\text{thermal factor}}{\underbrace{\frac{1}{\tau +1}}}.}$$

The directional factor is largest at equal support and opposition and smaller near either edge. The thermal factor shrinks as total weight grows. The variance of $P$ is one quarter of this centered variance.

These factors reveal why “learning is travel” and “learning is cooling” can coexist:

For $(\alpha ,\beta )=(20,1)$, $\mu =20/21$ and

$$\mathrm{Var}(B)=\frac{80}{{21}^2\cdot 22}\approx 0.00825.$$

One opposing trial produces $(20,2)$, with mean $10/11$, and

$$\mathrm{Var}(B)=\frac{160}{{22}^2\cdot 23}\approx 0.01437.$$

The temperature fell from $1/21$ to $1/22$, yet the variance increased. Moving toward the middle opened up the directional factor more than the extra weight compressed the thermal factor.

A scientist who receives a surprising measurement may revise a near-certain-looking estimate into a broader distribution. This is perfectly compatible with accumulating another trial. Nor is equal balance a unique psychological state: many opposing trials can represent sustained conflict, while a very small total weight near the same mean can represent little information.

How one count acquires several jobs

With a symmetric prior of total weight $n_0$ and $n$ new unit trials, $\tau =n_0+n$. Each fresh trial advances the evidence clock by one unit. Its cone increment is $(d\tau ,dx)=(1,+1)$ or $(1,-1)$, with zero incremental Lorentz interval. The same trial adds an exponent to the likelihood and increases the Gibbs inverse temperature parameter by one.

These are different roles of the same bookkeeping count:

Calling $d\tau$ energy-like is a modeling interpretation of the cone increment. A conversion to seconds, joules, or kelvin needs additional physical rules and units. The prior also means that the clock’s coordinate starts above zero before new observations arrive. At finer resolution, tracelets describe modeled pieces of a lineage; the relationship between those pieces and independent experimental trials must be stated.

The geometric bridge can be written another way. For a single Bernoulli observation, Fisher information about its probability $\mu$ is $I(\mu )=1/[\mu (1-\mu )]$. Fisher information measures local statistical distinguishability. The cone’s Lorentz factor is

$${\gamma }^2=\frac{1}{1-v^2}=\frac{I(\mu )}{4}.$$

Here the Fisher quantity belongs to the one-parameter Bernoulli model. The full two-parameter Beta family’s Fisher metric is a different object. The identity relates two useful quantities without identifying their whole geometries.

Cooling toward a Gaussian cloud

Hold fixed while increasing $\tau$. Near the valley floor, a Taylor expansion gives

$$f_{\mu }(\ell )\approx f_{\mu }({\ell }_{\ast })+\frac{1}{2}\mu (1-\mu )(\ell -{\ell }_{\ast }{)}^2.$$

Substitute this into the Gibbs density and recognize a Gaussian:

$$L\approx \mathcal{N}({\ell }_{\ast },\frac{1}{\tau \mu (1-\mu )}).$$

The log-odds width is approximately $1/\sqrt{\tau \mu (1-\mu )}$. Quadrupling the weight halves that width when the mean is held fixed. This is Laplace’s method: approximate a smooth valley by a parabola, then integrate the resulting bell-shaped cloud.

For repeated independent Bernoulli data generated at a fixed interior probability, with a suitable positive prior, the related Bernstein-von Mises result describes a posterior becoming approximately Gaussian around a data-dependent estimate that approaches the true parameter. The centering remains random across repeated experiments. Boundary cases require different limits; Bochkina and Green’s research on nonregular models shows how non-Gaussian components can arise.

The small parameter $1/\tau$ invites a semiclassical analogy. In quantum theory, decreasing Planck’s constant suppresses some fluctuations around classical behavior; here increasing evidence suppresses posterior spread along a fixed interior ray. We can explore a particle-like limit for a concentrated belief while retaining the actual derivation: a small-noise Gaussian approximation. A narrow posterior alone supplies neither a Hamiltonian nor a deterministic law of motion.

A warm belief in a changing environment

A learner that continually discounts old weights need never reach zero temperature. Consider a simple recursion for its centered mean,

where independent incoming outcomes $Y_k\in \{-1,+1\}$ have mean $u$. It arises when all weights are discounted by $1-\lambda$ before one fresh unit trial arrives, with total weight held at its stationary value $\tau =1/\lambda$.

Across many independent histories, the stationary variance $V$ of the running mean satisfies

$$V=(1-\lambda {)}^{2}V+{\lambda }^2(1-u^2),V=\frac{\lambda }{2-\lambda }(1-u^2).$$

For slow forgetting, $V\approx (1-u^2)\lambda /2$. Meanwhile the posterior variance reported inside a typical history is approximately $(1-u^2)\lambda$ at leading order. The factor of two compares two distinct fluctuations: variability of estimates across histories and uncertainty represented within a history.

This is a useful place to explore fluctuation-dissipation ideas, which connect random fluctuations with a system’s response to disturbances. The factor here follows from the chosen averaging recursion. It is not by itself an equipartition theorem, and changing the memory rule can change the comparison. Temperature language helps organize the experiment while the recursion supplies its actual dynamics.

Pilot a changing world

Take this recursion into a world with changing seasons. On each leg, choose whether to keep a long memory or let new trials turn the forecast quickly. The forecasting bill is charged before each outcome arrives. Compare your route with fixed-memory routes, then replay the same weather. When the gain changes between legs, the fixed-gain stationary formula above no longer describes the whole journey.

The memory pilot

Pilot a forecast through three seasons

Restart voyage

Choose how strongly the next eight trials can turn your forecast. A light touch smooths noise; a strong turn follows a changed world faster.

Leg 1 of 9 · 0/72 trials traveled

Sky: your forecast. Dashed: the season's true rate, revealed as you travel.

Forecast of plus

50.0%

Forecasting bill

0.0 bits

Hold courseLong memory · gain 0.04Trim the sailMiddle memory · gain 0.15Turn quicklyShort memory · gain 0.4

The first season begins with a forecast of 50%. Choose a memory response to launch.

Show fixed-memory comparison routes

Why memory helps and hurts

$$v_{k+1}=(1-{\lambda }_k)v_k+{\lambda }_k{Y}_k,Y_k\in \{-1,+1\}$$

The chart shows probability (1+v)/2. Gains are constant within each eight-trial leg, and may change between legs. The stationary-variance formula in the article applies only to a fixed gain. All routes start at 50% and see the same seeded, conditionally independent sequence with plus rates 80%, 20%, then 65%. Dashed lines show each season's rate after entering it. A forecast is scored before seeing its outcome using −log₂ of the probability assigned to that outcome. Lower total bits means a better prediction record for this one route; a single replay does not establish an optimal policy.

The imaginary-time trail

There is a deeper trail worth following. In quantum theory, a Hamiltonian $H$, the operator generating time evolution, appears in both

$$\underset{\text{real-time evolution}}{\underbrace{e^{-iHt/\hslash }}}\text{and}\underset{\text{thermal weighting}}{\underbrace{e^{-bH}}},$$

where $b$ is physical inverse temperature in energy units. Setting $t=-i\hslash b$ connects the expressions through imaginary time, a procedure called Wick rotation. David Tong’s path-integral lectures explain this connection and the conditions behind it.

Our Gibbs factor $e^{-\tau f_{\mu }}$ gives a promising thermal starting point. To extend it into a dynamical Wick bridge, we would need to choose an evolution generator, define time-dependent observables and their correlations, and show that an analytic continuation produces the proposed real-time process. A Lorentzian cone and a positive Fisher metric do not supply those ingredients by themselves. In particular, $f_{\mu }$ changes during ordinary learning, so even the candidate generator needs a rule for a moving landscape.

An especially inviting target is an epistemic counterpart of the Unruh effect. In quantum field theory, uniform acceleration through the vacuum produces a thermal detector response. Its derivation uses a field state, correlation functions along the accelerated trajectory, and the Kubo-Martin-Schwinger, or KMS, equilibrium condition. KMS relates differently ordered correlations by an imaginary-time shift; see Crispino, Higuchi, and Matsas.

An epistemic experiment could begin by specifying a fluctuating field over the cone and a detector coupled to it along a hyperbolic lineage. Then ask whether the sampled correlations satisfy KMS, and whether the detector’s transition-rate ratio defines a temperature proportional to acceleration. This makes the speculation testable as mathematics: the proposed field and coupling must produce the thermal relation, rather than having it inserted into the detector’s rule.

More thermodynamics from the same valley

At fixed $\mu$, our partition function already gives exact thermodynamic identities:

$$-\frac{\partial \log Z}{\partial \tau }=\mathbb{E}[f_{\mu }(L)],\frac{{\partial }^2\log Z}{\partial {\tau }^2}=\mathrm{Var}(f_{\mu }(L)).$$

The first derivative gives mean energy; the second gives its fluctuations. Define free energy $F=-T\log Z$ and heat capacity $C=d\mathbb{E}[f_{\mu }]/dT$. Then

$$C=\frac{\mathrm{Var}(f_{\mu }(L))}{T^2}\ge 0.$$

These identities follow by differentiating the normalized integral, holding the landscape fixed. Tong’s statistical mechanics notes develop the general partition-function method.

For one proper Beta belief, $Z=\mathrm{B}(\tau \mu ,\tau (1-\mu ))$ is smooth throughout , . There is no finite interior singularity signaling a first-order phase transition. Collective transitions become a richer question once many beliefs interact and an appropriate large-system limit is specified. Then each little prediction valley can become part of a population landscape.

For now, one lineage already carries plenty to explore: a point advancing through a cone, a cloud changing shape, a prediction valley being revised, and a temperature set by accumulated weight. The next surprising trial can affect all four at once.

Further reading

• NIST: Beta distribution, for the probability model and its moments.
• Tong: statistical mechanics, for ensembles, partition functions, and thermodynamic derivatives.
• Tong: path integrals, for the imaginary-time connection.
• Bochkina and Green: nonregular Bernstein-von Mises limits, for what changes at parameter boundaries.
• Crispino, Higuchi, and Matsas: the Unruh effect, for the field-theory target behind the proposed accelerated-detector experiment.