Section 2.5.1: Conditional Compression (Paragraph starting with "At a first glance")
Numerical analysis is used to identify the value of \(t_1\) that maximises the prediction for outcomes \(x_t=0\) and \(x_t=1\).
We can use constrained optimisation (with lagrange multipliers) instead. For outcome \(x_t=1\), we have
maximise: \( \frac{(t_1+1)^{(t_1+1)}(t-1-t_1)^{(t-1-t_1)}}{(t_1+1)^{(t_1+1)}(t-1-t_1)^{(t-1-t_1)}+t_1^{t_1}(t-t_1)^{(t-t_1)}} \) subject to: \( 0 \leq t_1 \leq t-1 \)
The objective function attains its maximum value at at \(t_1=t-1\). Likewise, we can do it for outcome \(x_t=0\).