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: $$p_i={\frac{e^{- {\varepsilon}_i / k T}}{\sum_{j=1}^{M}{e^{- {\varepsilon}_j / k T}}}} $$

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

Of course, there is a close link between the Boltzmann distribution and the multinomial distribution/multinomial regression (of which the binomial distribution/logistic regression is a special case).

A quick google for "history of logistic regression" does seem to suggest a long history of logistic functions for growth processes and then in bio-assay analysis (probit/logit-models) that arose from other rationales than the Boltzmann distribution.

Actual logistic regression seems to originate in the 1960s in the work of Cox. I do not see much of a hint that the Boltzmann distribution was his motiviation rather than having a suitable transformation for extending the idea of regression models to the binary/binomial outcome case (i.e. what we now call "generalized linear models"). Soon thereafter the multinomial regression was described, too. Perhaps with the multinomial regression the link is more obvious, but evidently the idea of the Boltzman disribution was around for 100 years or so, before people hit upon the idea of logistic regression.

I guess it is not unusual that related ideas in different fields are invented independently.

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  • $\begingroup$ 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. . $\endgroup$ – Nick Cox Apr 11 '18 at 8:22

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