Three views of a belief

From Gamma Poisson streams to Beta beliefs

Imagine two independent Poisson counters running in a laboratory: one recording support signals, the other recording opposing signals.

Their counts over a fixed interval follow Poisson distributions. For positive integer $\alpha$ and $\beta$, the waiting times to the $\alpha$-th support arrival and $\beta$-th opposing arrival follow Gamma distributions $G_{+}\sim \mathrm{Gamma}(\alpha ,\chi )$ and $G_{-}\sim \mathrm{Gamma}(\beta ,\chi )$ when both streams share rate $\chi$ (chi). Independent Gamma variables also make sense for any positive shape parameters.

Normalize $G_{+}$ by $G_{+}+G_{-}$, and a standard mathematical theorem gives our exact Beta distribution!

The integer-shape case gives this construction a concrete laboratory picture: a Beta variable emerges from the relative waiting times of two Poisson arrival streams.

Rapidity is half the log-odds

Adding nonnegative evidence to the two counts keeps $v_{\mathrm{E}}$ safely inside $(-1,1)$. But how can we compose changes in this bounded coordinate?

Special relativity uses rapidity: $\varphi =\mathrm{artanh}(v)$. When two collinear boosts combine, their rapidities add linearly: ${\varphi }_{\mathrm{total}}={\varphi }_1+{\varphi }_2$.

Substitute our epistemic velocity $v_{\mathrm{E}}=\frac{\alpha -\beta }{\alpha +\beta }$ into $\mathrm{artanh}$:

Mean-state epistemic rapidity ${\varphi }_{\mathrm{E}}$ is exactly half the log-odds (half the logit)! For two competing hypotheses, a Bayesian update multiplies their odds by a likelihood ratio. Evidence conditionally independent under each hypothesis supplies likelihood ratios that multiply, so their half-logarithms add like rapidities.

For these hypothesis-odds updates, Einstein’s relativistic velocity addition law is Bayes’ rule in disguise! Pooling Beta observations is a different operation: add their counts, retaining the shared prior once.

Turn the whole belief into rapidity

A mean is one point; a Beta belief is a whole distribution. Recall that $B$ is the random probability and $B_{\mathrm{E}}=2B-1$ is its centered version. Apply the rapidity transformation to every possible value and call the resulting random variable ${\mathrm{\Phi }}_{\mathrm{E}}$, capital phi.

Lowercase ${\varphi }_{\mathrm{E}}$ transforms the centered mean. Capital ${\mathrm{\Phi }}_{\mathrm{E}}$ retains the spread of the full belief on an unbounded axis. The inverse is $B_{\mathrm{E}}=\tanh {\mathrm{\Phi }}_{\mathrm{E}}$. Because this transformation bends the coordinate, averaging before transforming generally gives a different result from transforming before averaging: $\mathbb{E}[{\mathrm{\Phi }}_{\mathrm{E}}]$ generally differs from ${\varphi }_{\mathrm{E}}$.

The relationship that always holds is $\mathbb{E}[\tanh {\mathrm{\Phi }}_{\mathrm{E}}]=v_{\mathrm{E}}=\tanh {\varphi }_{\mathrm{E}}$. Keeping the two letters distinct lets us move between the simple cone point and the richer random belief.

The Lorentz factor and Bernoulli Fisher information

In special relativity, moving clocks run slow and moving rulers contract by the Lorentz factor $\gamma =1/\sqrt{1-v^2}$.

Calculate the epistemic Lorentz factor for our velocity $v_{\mathrm{E}}=2\mu -1$:

In statistics, Fisher information $I(\mu )=\frac{1}{\mu (1-\mu )}$ measures how much information a single sample carries about the parameter $\mu$.

The epistemic Lorentz factor is exactly half the square root of Bernoulli Fisher information: ${\gamma }_{\mathrm{E}}=\frac{1}{2}\sqrt{I(\mu )}$.

As you approach absolute certainty ($v_{\mathrm{E}}\to \pm 1$), the Lorentz factor blows up, reflecting both infinite relativistic dilation and infinite statistical information density.