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Download Bayesian analysis of binary prediction tree models for by Pittman J., Huang E., Nevis J. PDF

By Pittman J., Huang E., Nevis J.

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13) The Objective Method . : Finite ; .. : Infinite : ;. : . 45 .. : Infinite ; :. : . L V" V" If the root is a leaf If the root is not a leaf Fig. 9. If the root of Goo has finite degree, we are sure to have 2Nfoo + Noooo 2: 2 because no edge of the MSF of Goo can connect two finite components of the MSF and because in the three remaining cases one always has either Nfoo 2:1 or N 0000 2:2. The key step turns out to be a very simple involution invariance argument that shows that Nfoo and Noof have the same expectation.

First we take a fixed 0 < x < oo, and we condition on the event that there exist an edge a the root that has length x. We call this edge (r, v) and note that as in Figure 10, the edge determines two subtrees of the PWIT that one could label T(r, v) and T(v, r). As we have noted before, a Poisson process conditioned to have a point at x is again a Poisson process when this point is deleted, so by the definition of the PWIT we see that T(u, v) and T(v, u) are conditionally independent copies of the original PWIT.

9. If the root of Goo has finite degree, we are sure to have 2Nfoo + Noooo 2: 2 because no edge of the MSF of Goo can connect two finite components of the MSF and because in the three remaining cases one always has either Nfoo 2:1 or N 0000 2:2. The key step turns out to be a very simple involution invariance argument that shows that Nfoo and Noof have the same expectation. 13) gives us E[N] = E[NJoo] + E[NooJ] + E[Noooo] = 2E[NJoo + E[Noooo] 2:: 2. Finally, to establish the equality of E[NJoo] and E[NooJL we first recall the measure [l that one uses to define involution invariance.

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