![]() The relative importance of properties related to protein folding - like hydropathy and size - and function, including side-chain acidity, can also be estimated. By calculating the normalized mutual information, which measures the reduction in Shannon entropy, conveyed by single nucleotide messages, groupings that best leverage this aspect of fault tolerance in the code are identified. Each alphabet is taken as a separate system that partitions the 64 possible RNA codons, the microstates, into families, the macrostates. To evaluate these schemas objectively, a novel quantitative method is introduced based the inherent redundancy in the canonical genetic code. This fundamental insight is applied here for the first time to amino acid alphabets, which group the twenty common amino acids into families based on chemical and physical similarities. As with thermodynamic entropy, the Shannon entropy is only defined within a system that identifies at the outset the collections of possible messages, analogous to microstates, that will be considered indistinguishable macrostates. They argue that the name "uncertainty" would have been much more helpful since "Shannon entropy is simply and avowedly the 'measure of the uncertainty inherient in a pre-assigned probability scheme.The Shannon entropy measures the expected information value of messages. In the second place, and more important, nobody knows what entropy really is, so in a debate you will always have the advantage. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. You should call it entropy, for two reasons. Interestingly, Caianiello and Aizerman say the name entropy is thanks to von Neumann who said You can read more about this in Shannon's seminal paper A Theory of Mathematical Communication. H(x, y)=H(x)+H(y) only when x and y are independent events. The joint entropy of two events is less than or equal the sum of the individual entropies.Entropy H is maximized when the p_i values are equal.(Uncertainty vanishes only when we are certain about the outcomes.) Entropy H is 0 if and only if exactly one event has probability 1 and the rest have probability 0.Shannon observes that H has many other interesting properties: He named this measure of uncertainty entropy, because the form of H bears striking similarity to that of Gibbs Entropy in statistical thermodynamics. However, the independence property tells us that this relationship should hold: If the second flip is heads, x=1, if tails x=2. If the flip was tails, flip the coin again. Supposed we generate a random variable x by the following process: Flip a fair coin. Second, If each event is equally likely ( p_i=1/n), H should increase as a function of n: the more events there are, the more uncertain we are.įinally, entropy should be recursive with respect to independent events. A small change in a single probability should result in a similarly small change in the entropy (uncertainty). He thought that "it is reasonable" that H should have three properties:įirst, H should be a continuous function of each p_i. , p_n) describing the uncertainty of an arbitrary set of discrete events (i.e. In general, Shannon wanted to devise a function H(p_1, p_2. We might want to say the uncertainty in this case is 1. However, if the coin is fair and p=0.5, we maximize our uncertainty: it's a complete tossup whether the coin is heads or tails. Since there is no uncertainty, we might want to say the uncertainty is 0. Claude Shannon asked the questionĬan we find a measure of how much "choice" is involved in the selection of the event or of how uncertain we are of the outcome?įor example, supposed we have coin that lands on heads with probability p and tails with probability 1-p. Supposed we have a discrete set of possible events 1,\ldots, n that occur with probabilities (p_1, p_2.
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