Who did first perform maximum likelihood estimation? I am very interested in the historical development of statistical theories. Here is the research I've done: I've tried to read two old papers of Fisher. I think the first theory paper on MLE should be Fisher 1912.
Also, Thurstone had some models to construct a probability mass function which is "equivalent" to a likelihood function, but from two old papers of him, I cannot confirm that he mentioned the maximum likelihood estimation. He implemented his idea but I don't know if he had done something similar to an MLE estimation of the parameterized function.
So my question is, what is the paper on implementing MLE in practice, especially the discrete data set?  Who invented the probability mass function/likelihood function? Because Fisher invented MLE, do people usually attribute the invention of likelihood function to him?
 A: The most relevant reference imho is Steve Stigler's "Epic history of maximum likelihood" (2007)

"There were early intelligent comments related to this problem [of
seeking the most probable distribution for the observation] already
in the 1750s by Thomases Simpson and Bayes and by Johann Heinrich
Lambert in 1760, but the first serious assault related to our topic
was by Joseph Louis Lagrange in 1769." S. Stigler (2007)


"By introducing restrictions in the form of the curve only after
deriving the estimates of probabilities, Lagrange’s analysis had the
curious consequence of always arriving at method of moment estimates,
even though starting with maximum likelihood!" S. Stigler (2007)

He also points out at Daniel Bernoulli (1769) and Carl Friedrich Gauß (1809), albeit the later started using Bayesian arguments to see the MLE as a posterior mode.

"...a long memoir by Karl Pearson and Louis Napoléon George Filon,
published in the Transactions of the Royal Society of London in 1898
has a place in history, more for what in the end it seemed to suggest,
rather than for what it accomplished." S. Stiegler (2007)


"...the method of maximum likelihood was proposed independently by
Lambert and Daniel Bernoulli, but with no practical effect because the
maximum likelihood equation for the error distribution considered was
intractable." A. Hald (1999)


"It is an astounding fact that Edgeworth’s papers were unknown to Fisher when he wrote his paper on maximum likelihood estimation in 1912." A. Hald (1999)

A. Hald (1999) also mentions Encke (1832) and Hagen (1837) as maximising $p(\mathbf x|\theta)$ in $\theta$ to find the "most probable" sample. He further cites Chauvenet (1863) and Merriman (1884) before Edgeworth (1908).

"Edgeworth (1908) anticipated a good part of the (Fisher) 1922 version, but nobody noticed until a decade or so after Fisher had redone it." J. Aldrich (1997)


"...the [maximum likelihood] criterion appears at the head of the derivation of least squares in Chauvenet (1891, p.481), Bennett (1908, p.15) and Brunt (1917,p.77)"  J. Aldrich (1997)


" Pearson (1896, p.265) states that the "best" value of r is found by
choosing the value for which "the observed result is the most
probable." J. Aldrich (1997)

Looking at Thurstone's bibliography, it does not appear a relevant paper predates 1912.
