Foundations of Data Science

If the probability of a leaf being +1 is p, then the probability of i leaves being +1 and
d − i being -1 is ( d
i)
p i (1 − p) d−i
Thus, the probability of the root being +1 is proportional to
A =
d∑
i=1
( d
i)
p i (1 − p) d−i e (2i−d)β = e −dβ
and the probability of the root being –1 is proportional to
B =
d∑
i=1
= e −dβ d∑
d∑
( ) d (pe ) 2β i
(1 − p) d−i = e [ −dβ pe 2β + 1 − p ] d
i
i=1
( d
i)
p i (1 − p) d−i e −(2i−d)β
i=1
= e −dβ d∑
i=1
The probability of the root being +1 is
where
and
q =
( d
i)
p i [ (1 − p)e −2(i−d)β]
( d
i)
p i [ (1 − p)e 2β] d−i
= e −dβ [ p + (1 − p)e 2β] d
.
A = A+B
[pe 2β +1−p] d
[pe 2β +1−p] d +[p+(1−p)e 2β ] d
C = [ pe 2β + 1 − p ] d
= C D
D = [ pe 2β + 1 − p ] d
+
[
p + (1 − p) e
2β ] d
.
At high temperature, low β, the probability q of the root of the height one tree being
+1 in the limit as β goes to zero is
q =
p + 1 − p
[p + 1 − p] + [p + 1 − p] = 1 2
independent of p. At low temperature, high β,
q ≈
p d e 2βd
p d e 2βd + (1 − p) d e 2βd =
p d
p d + (1 − p) d = { 0 p = 0
1 p = 1 .
q goes from a low probability of +1 for p below 1/2 to high probability of +1 for p above
1/2.
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