Bounded density tilt
= Bounded density tilt
{title2=$p_t=p(1+tg),\ Pg=0$}
For bounded measurable $g$ with $Pg=0$, the formula $p_t=p(1+tg)$ defines a normalized positive <probability density function> for $|t|\|g\|_\infty<1$. A uniform <Taylor expansion> of the square root proves <differentiability in quadratic mean> with <score function> $g$: the squared remainder is $O(t^4Pg^4)$. Such paths realize all bounded centered directions of an unrestricted density model.