Let us assume that I have a continuous probability distribution with density function $x\to p(x)$ that is $1$ dimensional, monomodal and approximately symmetric.
We furthermore have a ternary tree of decisions. Every node $T_{i}$ of the ternary tree represents a situation where we need to decide $c_i$ :
$\begin{array}{rll} -1. &\text{Sub a real number }& x_{i+1} = x_i - r_i\\ 0. &\text{Nothing changes}\\ 1. &\text{Add a real number} &x_{i+1} = x_i + r_i \end{array}$
How can I decide the sequence (or vector if $N$ finite) ${\bf r} = [r_0,r_1,\cdots,r_N]^T$ so that the expected error $$ \left\|x_{true} - \left(x_0+\sum_{i}^N c_i\cdot r_i\right)\right\|$$ is minimized?
Own work: I have guessed that a reasonable starting point $x_0$ is one of the common mean values, for example arithmetic mean $$x_0 = E[p] = \int_{-\infty}^\infty \xi p(\xi)d\xi$$
or median
$$x_0\,\, s.t.\, \int_{-\infty}^{x_0} p(\xi)d\xi= \frac 1 2$$ For an approximate symmetric distribution these should be the same.