A note on the capacity region of the multiple-access channel (Corresp.)

In the classification problem for chromosomes there are <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> chromosomes which must be classified into <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> populations <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A_(1}, ⋯ ,Aₖ</tex> having known probability distributions. It is further known that these <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> chromosomes have <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Nᵢ</tex> in class <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Aᵢ, i = 1,2, ⋯ ,k</tex> . This is a compound decision problem whose optimal solution gives a classification algorithm which is not currently useful in practice because of its long computation time. Two other classification methods are considered, and the results are compared. One is the method often used for the classification of the 46 human chromosomes, where the knowledge about the exact number of chromosome types is disregarded, and only the a priori probability that a chromosome originates from population <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Aᵢ</tex> is used. The other method permits only classifications with the correct number of objects in each class and selects from all the possible classifications that one which has the maximum likelihood function. This last method has some advantages for a small number of objects, particularly if the numbers of objects in the classes are equal.