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Minimax Lower Bounds
Minimax Lower Bounds
2 questions
Difficulty 7-9
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conceptual
Fano's method for minimax lower bounds works by packing
M
well-separated parameter values and showing no test can distinguish them. The bound is
P
(
error
)
≥
1
−
(
I
(
θ
;
X
n
)
+
lo
g
2
)
/
lo
g
M
. What is the role of the packing number
M
?
Hide and think first
A.
Larger
M
shrinks the separation
ϵ
between hypotheses, weakening the resulting lower bound
B.
Fano's method is restricted to
M
=
2
binary hypothesis testing and does not generalize to more hypotheses
C.
Larger
M
tightens the bound because the information requirement
I
(
θ
;
X
n
)
≥
lo
g
M
grows
D.
M
only shows up in a constant prefactor and does not affect the minimax rate itself
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