Does entropy $H(y)$ decrease as $H(x,y)$ decreases when $H(x)$ is fixed? Can't find any proof in Shannon's 1948 paper. Can you provide one or disproof?
Thank you.
P.S.
$H(x)$(or $H(y)$) is the entropy of messages produced by the discrete source $x$(or $y$).
$H(x,y)$ is the joint entropy.
They are all entities in information theory.

Both cases are possible.
 H(X)            [=========================]
 H(Y)   [===============]                         (original)
 H(X,Y) [==================================]


 H(X)            [=========================]  constant 
 H(Y)        [=======]                        less than original
 H(X,Y)      [=============================]  decreases


 H(X)            [=========================]  constant 
 H(Y)        [=======================]        greater than original
 H(X,Y)      [=============================]  decreases

The assertion would be true only if we state that the mutual information $I(X;Y)$ is kept constant.
