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Take independent and identically distributed draws from the uniform three-outcome reference above. An observed sample mean at least $1/2$ is unusual because the source expected value is zero. For two draws, the only successful ordered pairs are $(0,1),(1,0),(1,1)$, giving probability $3/9=1/3$.

For long samples, the relevant empirical distributions have mean at least $1/2$. The method of types assigns a profile $q$ a leading probability cost $e^{-nD(q\Vert p)}$. The least costly admissible profile is the information projection on three outcomes: in this example the minimizing mean is on the boundary $1/2$, and the leading exponent is approximately $0.197378$ nats per draw. The exponential scale is not an exact finite-sample probability formula.^[ref]

The attraction of this connection is that a constrained optimization problem predicts the most probable way an unusual aggregate observation can occur. It links relative entropy, expected value constraints and rare empirical distributions without identifying a rare event with just one exact attainable type class.

Under the finite-alphabet, positive-reference, closed-convex set assumptions above, every feasible law $r\in \mathcal{C}$ satisfiesThus a feasible law's excess relative entropy above the optimum is at least its relative entropy from the optimizer. The information projection supplies a quantitative comparison with every other feasible law, not merely the location of a minimum.^[ref]

For affine expected value constraints, the result becomes a Pythagorean identity for relative entropy. In relative entropy minimization under a mean constraint, the logarithm of $q_i^{\ast }/p_i$ is an affine function of the outcome $a_i$. Any feasible $r$ has the same normalization and mean as $q^{\ast }$, making the residual in the three-point identity for relative entropy zero.^[ref]