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If s is not included in D, let sq , sq+1 be two consecutive components of s such that sq D and sq+1 D Let E be the smallest set of Hd that includes {sq , sq+1 } Note that D E (because sk D E), and therefore, hd (D) hd (E) If E is obtained as the union of two disjoint sets E , E of Hd such that sk E and sk+1 E , we have D E Consequently, hd (E) = min{d(u, v)|u E , v E } d(sk , sk+1 ), which implies e(x, y) = hd (D) hd (E) d(sk , sk+1 ) ampd (s) Therefore, we conclude that e(x, y) ampd (s) for every s S(x, y) We show now that there is a sequence w S(x, y) such that e(x, y) ampd (w), which implies the equality e(x, y) = ampd (w).

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To this end, we prove that for every D k Hd there exists w S(x, y) such that ampd (w) hd (D) The argument is by induction on k For k = 1, the statement obviously holds Suppose that it holds for 1, , k 1 and let D k The set D belongs to k 1 or D is obtained by fusing the blocks B, C of k 1 In the rst case, the statement holds by inductive hypothesis The second case has several subcases: (i) If {x, y} B, then by inductive hypothesis, there exists a sequence u S(x, y) such that ampd (u) hd (B) hd (D) = e(x, y) (ii) The case {x, y} C is similar to the rst case (iii) If x B and y C, there exist u, v D such that d(u, v) = hd (D).

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We can think about our model as the theory that drives what we can observe. For a given parameter vector , we expect things to behave in a certain way. We use the data to give us additional information about , but we can never know . We know that our theory (model) is incorrect, so the question then becomes:

+

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By the inductive hypothesis, there is a sequence u S(x, u) such that ampd (u) hd (B) and there is a sequence v S(v, y) such that ampd (v) hd (C) This allows us to consider the sequence w obtained by concatenating the sequences u, (u, v), v; clearly, we have w S(x, y) and ampd (w) = max{ampd (u), d(u, v), ampd (v)} hd (D) To complete the argument we need to show that if e is an other ultrametric such that e(x, y) e (x, y) d(x, y), then e(x, y) = e (x, y) for every x, y S By the previous argument there exists a sequence s = (s0 , , sn ) S(x, y) such that ampd (s) = e(x, y) Since e (x, y) d(x, y) for every x, y S, it follows that e (x, y) ampd (s) = e(x, y).

(closure(D))

Thus, e(x, y) = e (x, y) for every x, y S, which means that e = e This concludes our argument..

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