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Current Trends in Bayesian Methodology with Applications
book

Current Trends in Bayesian Methodology with Applications

by Satyanshu K. Upadhyay, Umesh Singh, Dipak K. Dey, Appaia Loganathan
May 2015
Intermediate to advanced content levelIntermediate to advanced
680 pages
22h 33m
English
Chapman and Hall/CRC
Content preview from Current Trends in Bayesian Methodology with Applications
Priors on Hypergraphical Models via Simplicial Complexes 401
Proposition 19.4.1. Every feasible graph in R
d
may be represented in the
form G(V, r, A) for a collection V of n points in the unit ball B
d
and for r =
1
n
.
Proof
Let V = {V
1
, . . . , V
n
} B
d
be a set of points and r > 0 a radius such that
G(V, r, Čech) is the empty graph. Then the balls V
i
+rB
d
are disjoint and their
union with d-dimensional volume
d
r
d
lies wholly within the ball (1 + r)B
d
of volume ω
d
(1 + r)
d
(where ω
d
= π
d/2
/Γ(1+d/2) is the volume of the unit
ball), so n < (1 +
1
r
)
d
.
Slightly stronger, the empty graph may not be attained as G(V, r, Čech)
for any r 1/[(n/p
d
)
1/d
1] where p
d
is ...
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Publisher Resources

ISBN: 9781482235128