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Any non-linear differentiable function, (,), of two variables, and , can be expanded as + +. If we take the variance on both sides and use the formula [11] for the variance of a linear combination of variables (+) = + + (,), then we obtain | | + | | +, where is the standard deviation of the function , is the standard deviation of , is the standard deviation of and = is the ...
Note that the values at 0 and 1 are given by the limit := + = (by L'Hôpital's rule); and that "binary" refers to two possible values for the variable, not the units of information. When p = 1 / 2 {\displaystyle p=1/2} , the binary entropy function attains its maximum value, 1 shannon (1 binary unit of information); this is the case of ...
The bounds often also enclose distributions that are not themselves possible. For instance, the set of probability distributions that could result from adding random values without the independence assumption from two (precise) distributions is generally a proper subset of all the distributions enclosed by the p-box computed for the sum. That ...
The entropy or the amount of information revealed by evaluating (X,Y) (that is, evaluating X and Y simultaneously) is equal to the information revealed by conducting two consecutive experiments: first evaluating the value of Y, then revealing the value of X given that you know the value of Y. This may be written as: [11]: 16
Very frequently, one is solving the inverse problem: given () =, one is solving for x, and thus the condition number of the (local) inverse must be used. [ 1 ] [ 2 ] The condition number is derived from the theory of propagation of uncertainty , and is formally defined as the value of the asymptotic worst-case relative change in output for a ...
where () = [= ()] and = is the Fisher information of Y relative to calculated with respect to the conditional density of Y given a specific value X = x. As a special case, if the two random variables are independent , the information yielded by the two random variables is the sum of the information from each random variable separately:
MI is the expected value of the pointwise mutual information (PMI). The quantity was defined and analyzed by Claude Shannon in his landmark paper "A Mathematical Theory of Communication", although he did not call it "mutual information". This term was coined later by Robert Fano. [2] Mutual Information is also known as information gain.
If r is fractional with an even divisor, ensure that x is not negative. "n" is the sample size. These expressions are based on "Method 1" data analysis, where the observed values of x are averaged before the transformation (i.e., in this case, raising to a power and multiplying by a constant) is applied.