Appendix 2. The two ply brute force Likelihood Function

Opponent is the opponent at one ply, and Player is the player at one ply. Some definitions:

$$\begin{eqnarray*}P_{i_1 i_2 \ldots i_d} & = & \hbox{position resulting from $P$... .... and also makes early checkmates better than late ones);} } \\ \end{eqnarray*}$$

For $h\ge1$, define

$$v_h(P_{i_1 i_2 \ldots i_h}) = \left\{ \begin{array}{l} 0, ~~~... ...~~~ \hbox{ when evaluating the first ply.} \end{array}\right.$$

For values that are h-ply distant from P the function $v_h(\_)$ assigns values. The ordinary evaluation function, where the value is calculated from the $\lambda$ values and the attribute values is given by $v(\_)$. The summations are over all moves i₂ that can be made in position P_i1.

The move from position P is chosen probabilistically according to

$$\mbox{Pr(move $i^*(P)$ )} = \frac{\exp\left(\sigma_1(P) v_1(P... ...ight)} {\sum_{i=1}^{m} \exp\left(\sigma_1(P) v_1(P_i)\right)}.$$

This is the likelihood for a single position P. For a set of positions, $\mathcal{P}$, the likelihood is

$$\prod_{P \in \mathcal{P}} \mbox{Pr(move $i^*(P)$ )} .$$