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# Function `std.numeric.jensenShannonDivergence`

Computes the Jensen-Shannon divergence between `a` and `b`, which is the sum ```(ai * log(2 * ai / (ai + bi)) + bi * log(2 * bi / (ai + bi))) / 2```. The base of logarithm is 2. The ranges are assumed to contain elements in `[0, 1]`. Usually the ranges are normalized probability distributions, but this is not required or checked by `jensenShannonDivergence`. If the inputs are normalized, the result is bounded within `[0, 1]`. The three-parameter version stops evaluations as soon as the intermediate result is greater than or equal to `limit`.

``` CommonType!(ElementType!Range1,ElementType!Range2) jensenShannonDivergence(Range1, Range2) (   Range1 a,   Range2 b ) if (isInputRange!Range1 && isInputRange!Range2 && is(CommonType!(ElementType!Range1, ElementType!Range2))); CommonType!(ElementType!Range1,ElementType!Range2) jensenShannonDivergence(Range1, Range2, F) (   Range1 a,   Range2 b,   F limit ) if (isInputRange!Range1 && isInputRange!Range2 && is(typeof(CommonType!(ElementType!Range1, ElementType!Range2).init >= F.init) : bool)); ```

## Example

``````import std.math : approxEqual;

double[] p = [ 0.0, 0, 0, 1 ];
writeln(jensenShannonDivergence(p, p)); // 0
double[] p1 = [ 0.25, 0.25, 0.25, 0.25 ];
writeln(jensenShannonDivergence(p1, p1)); // 0
assert(approxEqual(jensenShannonDivergence(p1, p), 0.548795));
double[] p2 = [ 0.2, 0.2, 0.2, 0.4 ];
assert(approxEqual(jensenShannonDivergence(p1, p2), 0.0186218));
assert(approxEqual(jensenShannonDivergence(p2, p1), 0.0186218));
assert(approxEqual(jensenShannonDivergence(p2, p1, 0.005), 0.00602366));
``````

## Authors

Andrei Alexandrescu, Don Clugston, Robert Jacques, Ilya Yaroshenko