Remark. A penalty coefficient is not a hard KL budget [ftip-008V]
Remark. A penalty coefficient is not a hard KL budget [ftip-008V]
For fixed \(\beta \), the exponential tilt solves the penalized problem
\[ \max _{q\in \Delta (\mathcal Y)} \left \{\mathbb E_q[r] -\beta D_{\mathrm {KL}}(q\Vert p)\right \} \]under the conventions of Definition [ftip-008I]. This is not the same specification as choosing a number \(\kappa \) and solving
\[ \max _q\mathbb E_q[r] \quad \text {subject to}\quad D_{\mathrm {KL}}(q\Vert p)\leq \kappa . \]A Lagrange multiplier can relate the two problems when the relevant duality and activity conditions hold. The coefficient \(\beta \) alone does not declare a hard budget, and the identities Theorem [ftip-008O]--Theorem [ftip-008Q] do not supply those conditions.