Welcome

Likelinesses

"Likeliness" is a term I introduced into my thinking because I did not like the generally-used term "best-estimate of (frequentist) probability": it's too wordy and is based on too many assumptions.

It is easy to confuse frequentist probabilities and likelinesses, especially as the latter can be used as a predictor of the former; it can at times be very difficult to tell whether one is dealing with specifically a frequentist probability or a likeliness. That confusion can give rise to some very serious errors, not only in practical calculations but also in the theory, so it is worthwhile sorting out the differences betwen the two.

The main points are:-

• (Frequentist) Probabilities on a finite domain are a special case of likelinesses: we can identify probabilities as being likelinesses over a singleton underlying set. Put the other way round, likelinesses are a generalisation of probabilities.
• Likelinesses can be calculated without going to the limit. This means they don't need a large sample; indeed -apart from a few simple cases which can be handled theoretically rather than numerically- they struggle with sample sizes greater than about 100. In many ways, likelinesses are best thought of as a theory of small samples, stepping in when sample sizes are so small that Relative Frequencies cannot be thought of as a reliable estimate of frequentst probabilities.
• There is nothing on this site which seeks to redefine the academic concept of "probability". Instead, we use Likelinesses to define a practical concept of "probability", to sit alongside the academic version in the same way that frequentist probabilities sit alongside the academic version.

This site summarises the main results. A more detailed discussion, including some proofs not given here, can be found in the pdf: Fundamentals of Likelinesses

Distributions

A side-issue to work on Likelinesses is their actual calculation. This is best-done by numerical means, and so requires the development of algorithms to generate different types of distribution, such as

• Ranked Distributions
• Sawtooth distributions
• Plateau distributions
• Multimodal distributions
• U-shaped distributions

The approach used on this site is to use those algorithms not only as a means of calculating values of distributions but also as the formal definition of the distributions.

It is distributions that I am currently mostly interested in, for their own sake rather than because of their use when calculating likelinesses.