4 · MaxEnt — the softmax is not an assumption

Where does the exponential form come from? It can be derived, twice over.

As a variational principle. Ask: which distribution maximises expected payoff plus a bonus for keeping options open, $$\sigma^* = \arg\max_\sigma \;\mathbb{E}_\sigma[U] + \tfrac{1}{\lambda}H(\sigma),$$ with H the Shannon entropy? The answer is exactly the logit response. Slide the 1/λ dial in the Lab and watch the optimum morph from uniform (entropy dominates) to argmax (payoff dominates). Physicists will recognise free energy F = U − TS with temperature T = 1/λ; the Gibbs variational principle, verbatim, with payoff as negative energy.

As a price of information. Rational inattention (Matějka–McKay 2015) derives the same logit from first principles: an agent who must pay for information about payoffs, at a per-bit price, optimally ends up choosing with exactly these probabilities — and λ is the inverse price of information. "Noise" is then not error but economised attention.

Both derivations matter practically. The variational form gives the log-partition function ψ = log Σ e^{λU}, whose gradient is the choice distribution and whose Hessian is the choice covariance C — the exact identities (verified to 10⁻¹²) on which every instrument in this project is built: the susceptibility is λC, and everything else follows from there.