Decision-Sufficient Posterior Approximation

The paper studies how a posterior approximation must preserve a specific downstream decision problem. It introduces a regret geometry defined by the target posterior and loss, and uses the forward‑Kullback‑Leibler divergence from a baseline approximation to quantify the information needed for action changes. By analyzing the generalized eigenproblem of the regret Hessian and the baseline information metric, the authors derive a tolerance‑dependent effective dimension and show how restricted approximation families can either miss crucial decision directions or incur excess Fisher information costs.

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