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Evolution in a nutshell an altrnative outline on evoution and some consequences concerning valuations by Gregor Kjellström
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4 Disorder, informaion and biological diversity In this chapter we will get familiar with the concept of information and we will see that mean information is equivalent to disorder and biological diversity. Further it will be shown that an increase in disorder also means an increase in the variance of a normal distribution. We will also introduce a definition of efficiency based on the concept of information. Because reproduction may be seen as sending a DNA-message from parent to offspring it is motivated to introduce the concept of information, originally defined by Claude E. Shannon, 1948, (see for instance Åslund). In this pamphlet the theory is applied to phenotypes rather than DNA-messages, but first a brief recapitulation of the theory. Information is an abstract concept that depends solely on probabilities but which ignores the mental meaning of the messages. Let us consider a system of r mutually exclusive events, each of which may occur with a probability = 1/r. For example, when tossing a coin there are two possible outcomes which may be denoted by ‘0’ and ‘1’ and they both appear with a probability = 0.5. And in a sequence of two tosses there will be four possible outcomes (messages) ‘00’, ‘01’, ‘10’ and ‘11’ each appearing with a probability = 0.25. Intuitively it is felt that the information obtained from the outcome of an experiment should not depend on whether we regard an experiment as one pair of tosses or as a sequence of two tosses. This means that the information of a pair should equal the total information from the separate tosses. It is also reasonable to assume that when the number of possible equally probable outcomes increases, more experiments have to be done before a certain interesting outcome appears. Consequently more information will be gained at this event. When only one alternative is available the outcome is given beforehand and no information will be gained. Considering all possible partitions of different messages written in different alphabets, the negative logarithm of the probability is evidently a very good candidate for being a measure of information. We have for instance -log(1/2) -log(1/2) = -log(1/4), -log(1/2) < -log(1/3), and log(1) = 0. Now, suppose that we have some statistical search process that gains information by iterative enclosing of interesting sub-volumes in smaller and smaller volumes in a parameter space. For instance, if the volume V1 may be decreased to V2 after some search, then we may say that the information gain is I1, because we may now focus on that smaller volume in what follows. Then, if V2 is further reduced to V3, we have gained I2 and so on. Let us further assume that the search is carried out with the aid of a uniform distribution over each Vj. If I should still be an additive measure, so that the total information gain in a sequence of r reductions is I = I1 + I2 + ... + Ir. then the only function satisfying this postulate for any arbitrary set of reductions is Ij = log( Vj/Vj+1) so that I = log( V1/V2 ) + log( V2/V3 ) + ... + log( Vr-1/Vr ) = log( V1/Vr ). We can easily pass on to probabilities: pj = Vj+1/Vj and Ij = -log(pj). in agreement with earlier findings. Definition 4.1. Thus we stick to the earlier definition of information as the negative logarithm of a probability. But reality is more complicated, and we can not always assume that different outcomes are equally probable. In a sequence of events it may also happen that a preceding event changes the probability of an event to follow. For instance, in English the sequence ‘we’ may probably be followed by ‘are’ but hardly by ‘am’. In order to examine this more complicated situation let us consider the mean value of information. |
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