lecture_slides-Chapter 2.3

# V p q x2x the entropy function is continuous at p if

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Unformatted text preview: ; 0 such that H (q )| < ✏ V ( p, q ) < . Deﬁnition 2.23 Let p and q be two probability distributions on a common alphabet X . The variational distance between p and q is deﬁned as X | p( x ) q ( x ) | . V (p, q ) = x2X The entropy function is continuous at p if ✓ ◆ lim H (p0 ) = H lim p0 = H (p), 0 0 p !p p !p or equivalently, for any ✏ > 0, there exists | H ( p) for all q 2 PX satisfying Thursday, 26 December, 13 > 0 such that H (q )| < ✏ V ( p, q ) < . An Example Thursday, 26 December, 13 An Example • • • • Let X = {1, 2, · · · , }, a countably inﬁnite alphabet. Let X = {1, 2, · · · , }, a countably inﬁnite alphabet. Let PX = {1, 0, 0, · · · }, and let Let PX = {1, 0, 0, · · · }, and let 9 8 > > ⇢ > > &l...
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