Editing Beta distribution (section)

====Bayes–Laplace prior probability (Beta(1,1))====
The beta distribution achieves maximum differential entropy for Beta(1,1): the [[Uniform density|uniform]] probability density, for which all values in the domain of the distribution have equal density.  This uniform distribution Beta(1,1) was suggested ("with a great deal of doubt") by [[Thomas Bayes]]<ref name="ThomasBayes"/> as the prior probability distribution to express ignorance about the correct prior distribution. This prior distribution was adopted (apparently, from his writings, with little sign of doubt<ref name=Laplace/>) by [[Pierre-Simon Laplace]], and hence it was also known as the "Bayes–Laplace rule" or the "Laplace rule" of "[[inverse probability]]" in publications of the first half of the 20th century. In the later part of the 19th century and early part of the 20th century, scientists realized that the assumption of uniform "equal" probability density depended on the actual functions (for example whether a linear or a logarithmic scale was most appropriate) and parametrizations used.  In particular, the behavior near the ends of distributions with finite support (for example near ''x'' = 0, for a distribution with initial support at ''x'' = 0) required particular attention. Keynes (<ref name=KeynesTreatise/> Ch.XXX, p.&nbsp;381) criticized the use of Bayes's uniform prior probability (Beta(1,1)) that all values between zero and one are equiprobable, as follows: "Thus experience, if it shows anything, shows that there is a very marked clustering of statistical ratios in the neighborhoods of zero and unity, of those for positive theories and for correlations between positive qualities in the neighborhood of zero, and of those for negative theories and for correlations between negative qualities in the neighborhood of unity. "