# Was logistic regression based on Boltzmann distribution from statistical mechanics?

Ever since seeing the logistic distribution for the first time many years ago, I always thought of it as an application of the Boltzmann distribution. Whoever developed it may had seen the Boltzmann distribution and applied it to economics or whatever the first application it was.

Is this true in historical context? Was it developed this way?

Here's the logistic equation as a log-linear model: $$\Pr(Y_i=c) = \frac{e^{\boldsymbol\beta_c \cdot \mathbf{X}_i}}{\sum_h e^{\boldsymbol\beta_h \cdot \mathbf{X}_i}}$$

Here's the Boltzmann distribution: $$p_i={\frac{e^{- {\varepsilon}_i / k T}}{\sum_{j=1}^{M}{e^{- {\varepsilon}_j / k T}}}}$$

• Historically, no. But as Richard Feynman was fond of pointing out, the same problems have the same solutions. – Nick Cox Apr 11 '18 at 8:26

• Joseph Berkson's 1944 paper is self-aware about history and parallel work by E.B. Wilson. Application of the logistic function to bio-assay. Journal of the American Statistical Association 39: 357-365. Sir David Cox's most important papers on this topic were in the 1950s. Logit, meaning $\log[p / (1 - p)]$ before it was named, was used as a transformation for $0 < p < 1$ at least in the 1930s. . – Nick Cox Apr 11 '18 at 8:22