Abstract
This chapter begins with a concise primer, or aide mémoire, of probability theory. The fundamentals are recapitulated, drawing on the material of Chap. 4. Moments of distributions are reviewed. The theory of runs (successions of similar events preceded and succeeded by different events), which is useful for analysing nucleic acid sequences and series of events, is reviewed, and it leads naturally onto the hypergeometric distribution. The chapter then moves on to likelihood as a valuable means of determining the degree of support for a proposition. A reliable method of assessing the support of any proposition basing its validity on data as the evidence for it is required for a great deal of work in bioinformatics. The chapter closes with a brief review of the method of maximum entropy, which is already used for image restoration but which has the potential for a great many more applications.
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Notes
- 1.
Called Merkmalraum (“label space”) in R. von Mises’ (1931) treatise Wahrscheinlichkeitsrechnung.
- 2.
Its protagonists include Laplace, Keynes, and Jeffreys.
- 3.
According to J.M. Keynes, probability is to be regarded as “the degree of our rational belief in a proposition”.
- 4.
Made during the 17th Guthrie Lecture to the Physical Society in London.
- 5.
Notation: in this chapter, \(P\{X\}\) denotes the probability of event X; \(N\{X\}\) is the number of simple events in (compound) event X. S denotes the certain event that contains all possible events. Sample space and events are primitive (undefined) notions (cf. line and point in geometry).
- 6.
The proof is given in Feller (1967), Chap. 4.
- 7.
Indeed, Reichenbach, Popper, and others have taken the view that conditional probability may and should be chosen as the basic concept of probability theory. We should in any case note that most of the results derived for unconditional probabilities are also valid for conditional probabilities.
- 8.
Stochastic independence is formally defined via the condition
which must hold if the two events A and H are stochastically (sometimes called statistically) independent.
- 9.
If and only if.
- 10.
Due to P.V. Sukhatme and V.G. Panse, quoted by Feller (1967), Chap. 6.
- 11.
\(\mathbf {X}\) may assume the values \(x_1,x_2,\ldots \) (i.e., the range of \(\mathbf {X}\)).
- 12.
The distribution function F(x) of \(\mathbf {X}\) is defined by
(i.e., a nondecreasing function tending to 1 as \(x\rightarrow \infty \)).
- 13.
Also denoted by angular brackets or a bar.
- 14.
Notice the mechanical analogies: centre of gravity as the mean of a mass and moment of inertia as its variance.
- 15.
Older literature uses the term “dispersion”.
- 16.
Strictly speaking, one should instead refer to propositions. A hypothesis is an asserted proposition, whereas at the beginning of an investigation it would be better to start with considered propositions, to avoid prematurely asserting what one wishes to find out. Unfortunately, the use of the term “hypothesis” seems to have become so well established that we may risk confusion if we avoid using the word.
- 17.
As Fisher and others have pointed out, it is not strictly correct to associate Bayes with the inverse probability method. Bayes’ doubts as to its validity led him to withhold publication of his work (it was published posthumously).
- 18.
- 19.
Implicitly, Platonic reality is meant here.
References
Feller W (1967) An introduction to probability theory and its applications, vol 1, 3rd edn. Wiley, New York
von Mises R (1931) Wahrscheinlichkeitsrechung. Deuticke, Leipzig
Mood AM (1940) The distribution theory of runs. Ann Math Stat 11:367–392
Planck M (1932) The concept of causality. Proc Phys Soc 44:529–539
Sommerhoff G (1950) Analytical biology. Oxford University press, London
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Ramsden, J. (2015). Probability and Likelihood. In: Bioinformatics. Computational Biology, vol 21. Springer, London. https://doi.org/10.1007/978-1-4471-6702-0_5
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DOI: https://doi.org/10.1007/978-1-4471-6702-0_5
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