User:IssaRice/Logical inductor construction: Difference between revisions

Revision as of 00:50, 25 June 2019

The following is used in the Fixed Point Lemma (5.1.1):

Writing the $n$ -strategy as

T_{n}=\sum _{j=1}^{k}\xi _{j}\phi _{j}-\sum _{j=1}^{k}\xi _{j}\phi _{j}^{*n}

we have

\mathbb {V} (T_{n}(\mathbb {P} _{\leq n-1},\mathbb {V} ))=\sum _{j=1}^{k}\xi _{j}(\mathbb {P} _{\leq n-1},\mathbb {V} )\cdot \mathbb {V} (\phi _{j})-\sum _{j=1}^{k}\xi _{j}(\mathbb {P} _{\leq n-1},\mathbb {V} )\cdot \phi _{j}^{*n}(\mathbb {P} _{\leq n-1},\mathbb {V} )

But $\phi _{j}^{*n}(\mathbb {P} _{\leq n-1},\mathbb {V} )=\mathbb {V} (\phi _{j})$ so the two sums cancel to obtain $0$ .

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+==Lemma 5.1.1 (Fixed Point Lemma)==
+The following is used in the Fixed Point Lemma (5.1.1):
+Writing the <math>n</math>-strategy as
+:<math>T_n = \sum_{j=1}^k \xi_j \phi_j - \sum_{j=1}^k \xi_j\phi_j^{*n}</math>
+we have
+:<math>\mathbb V(T_n(\mathbb P_{\leq n-1}, \mathbb V)) = \sum_{j=1}^k \xi_j(\mathbb P_{\leq n-1}, \mathbb V)\cdot \mathbb V(\phi_j) - \sum_{j=1}^k \xi_j(\mathbb P_{\leq n-1}, \mathbb V) \cdot \phi_j^{*n}(\mathbb P_{\leq n-1}, \mathbb V)</math>
+But <math>\phi_j^{*n}(\mathbb P_{\leq n-1}, \mathbb V) = \mathbb V(\phi_j)</math> so the two sums cancel to obtain <math>0</math>.
 ==See also==
 * https://machinelearning.subwiki.org/wiki/User:IssaRice/Logical_induction_notation