Test-time context optimization adapts frozen language models through textual feedback, but ordinary conditional generation provides no separate control over how strongly to reinforce the distributional changes induced by a context update. We introduce EMO, a training-free framework inspired by expectation--maximization that alternates question-specific context estimation (E-step) and distribution refinement (M-step). The E-step revises a system context using textual gradients from an external judge without access to gold answers. The M-step contrasts next-token distributions under the revised and preceding contexts at a shared prefix to construct a confidence-constrained logit correction. Feedback on the steered response drives the next context update, coupling adaptation across rounds with intervention within each generation. Theoretically, we connect parameter and context optimization through a shared target-KL objective and additive logit representations, and derive the steering vector from context-induced logit displacements. Under context-improvement and faithful-steering assumptions, we prove a one-round decrease in target KL divergence for sufficiently small steering strength. Under greedy decoding across eight models and six benchmarks, EMO improves the zero-shot baseline in 39 of 40 aggregated settings, averaging $+9.10$ percentage points. With the same judge, it adds $+1.53$ points on average over context-only optimization (TextGrad), with consistent MMLU-Pro gains across models. At matched target-model generation budgets, accuracy advantages over best-of-$N$ are strongest on Llama, while oracle coverage is higher in most comparisons despite fewer candidates. The coverage--selection gap motivates allocating test-time computation jointly to candidate generation and verification.

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