The discussion revolves around whether the determinant can be considered a linear operation, particularly in the context of expectation operators. Participants explore the relationships between determinants and expectations, including inequalities involving logarithms of determinants.
Participants generally disagree on whether the determinant can be treated as a linear operation in the context of expectations, with multiple competing views presented throughout the discussion.
Participants rely on specific examples and properties of determinants and expectations, but the discussion does not resolve the underlying mathematical complexities or assumptions involved.
Nope. Suppose H=((1 0), (0 1)) with 50% probability and ((-1 0), (0 -1)) with 50% probability. Both of those have determinant 1, so E[log det H] = 0. But E[H] = ((0 0), (0, 0)), with determinant 0, so log det E[H] = -infinity.S_David said:Ok, now we have this inequality:
[tex]\mathbb{E}\left[\log(X)\right]\leq\log\left(\mathbb{E}[X]\right)[/tex]
Can we say in the same manner that:
[tex]\mathbb{E}\left[\log\left(\text{det}\left\{H\right\}\right)\right]\leq\log\left(\text{det}\left\{\mathbb{E}\left[H\right]\right\}\right)[/tex]
pmsrw3 said:Nope. Suppose H=((1 0), (0 1)) with 50% probability and ((-1 0), (0 -1)) with 50% probability. Both of those have determinant 1, so E[log det H] = 0. But E[H] = ((0 0), (0, 0)), with determinant 0, so log det E[H] = -infinity.