= Information projection on three outcomes
{title2=$q^*=(1,t,t^2)/(1+t+t^2),\quad t=(1+\sqrt{13})/2$}
Let the possible numerical outcomes be $-1,0,1$, and give the reference law $p$ <uniform distribution on a finite set>: $p=(1/3,1/3,1/3)$. We want a law $q$ with <expected value> $1/2$. Write the three <probabilities> in outcome order as $(q_-,q_0,q_+)$. The <mean> constraint is $q_+-q_-=1/2$, while normalization requires $q_-+q_0+q_+=1$.
These conditions leave a whole line of possible <probability distributions>:
$$
q=(x,1/2-2x,1/2+x),\qquad 0\leq x\leq1/4.
$$
For example, $(0,1/2,1/2)$ and $(1/4,0,3/4)$ both have the required <mean>. The constraint alone does not choose between them. <Information projection> selects the point on this line with the smallest <relative entropy> from the reference.
<Exponential tilting> gives weights proportional to $e^{-\theta},1,e^\theta$. Set $t=e^\theta>0$ and multiply these weights by $t$ to obtain $q^*=(1,t,t^2)/(1+t+t^2)$. Enforcing the <mean> gives
$$
\frac{t^2-1}{1+t+t^2}=\frac12
\quad\Longleftrightarrow\quad t^2-t-3=0.
$$
The positive root is $t=(1+\sqrt{13})/2$, hence
$$
\boxed{q^*\approx(0.116204,0.267592,0.616204),\qquad
D(q^*\Vert p)\approx0.197378\text{ nats}.}
$$
The most favored outcome becomes $1$, but all three remain possible. This is an exact constructed example with a stipulated reference law and constraint, not a fitted distribution from measurements.
\Image[/information-projection-probability-simplex.png]
{title=A mean constraint selects the least-divergent law in a probability simplex}
{description=Each point represents a <probability distribution> on $\{-1,0,1\}$. The color is <relative entropy> from the uniform reference, in <nats>. The green segment fixes <expected value> $1/2$, and its marked point is the exact <information projection on three outcomes>. The diagram is calculated from the stated model.}
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