By Magnus Ekdahl.

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Xj ) Pξ(j) (x1 , . . 1 and the induction assumption 1 2 log 2 x (1 − Pξj+1 |ξ(j) (y|x(j) ))Pξ(j) (x1 , . . ,xj = j(1 − Pξ (y)) 2 log 2 (j + 1)(1 − Pξ (y)) . 2 log 2 3. By 1, 2 and the induction axiom equation (37) holds. 3 1 − Pξ (y) Eξ(n) − log Pξ(n) (ξ (n) ) Eξ(n) − log Pˆξ(n) (ξ (n) ) n · 2 log 2 n · 2 log 2 . 10 and Eξ(n) − log Pˆξ(n) (ξ (n) ) . 5) for a Bayesian Network. As seen in the end of section 2 the SC will be minimized with Jeﬀreys’ prior. Jeﬀreys’ prior for a Bayesian Network is calculated in [50].

Xj ) log2 (Pξ(j) (x1 , . . ,xj Pξj+1 |ξ(j) (xj+1 |x1 , . . , xj ) Pξ(j) (x1 , . . 1 and the induction assumption 1 2 log 2 x (1 − Pξj+1 |ξ(j) (y|x(j) ))Pξ(j) (x1 , . . ,xj = j(1 − Pξ (y)) 2 log 2 (j + 1)(1 − Pξ (y)) . 2 log 2 3. By 1, 2 and the induction axiom equation (37) holds. 3 1 − Pξ (y) Eξ(n) − log Pξ(n) (ξ (n) ) Eξ(n) − log Pˆξ(n) (ξ (n) ) n · 2 log 2 n · 2 log 2 . 10 and Eξ(n) − log Pˆξ(n) (ξ (n) ) . 5) for a Bayesian Network. As seen in the end of section 2 the SC will be minimized with Jeﬀreys’ prior.

Find There are many ways to solve problem 1 so we turn to the procedure of which turns out to be best in this context. The earliest known algorithm for solving problem 1 was given by [7]. However, a provable time optimal algorithm remains to be found. 9). A history of the problem as well as an English translation of [7] can be found in [56]. 8). 7, equation (65)). When restricting the attention to complete directed graphs, which is the problem at hand, (where E contains (d − 1)2 elements, d = |V |), the February 13, 2006 (13:19) 44 problem can be solved linearly (O(|E|)) using [7] as well as [12] but in this case [7] has a lower actual running time.