Calibration under model misspecification is inherently ill-posed because calibration parameters and structural discrepancy are statistically confounded without additional assumptions. Bayesian formulations address this ambiguity through prior and covariance modeling choices, including multi-response observations and cross-source alignment. However, how these choices, together with output-dependent uncertainty, shape posterior geometry and practical parameter identifiability remains poorly understood. We present a latent-variable multi-response calibration model that combines empirical Bayes estimation of nuisance hyperparameters with a Fisher-information-based local Gaussian approximation of the calibration-parameter posterior. Closed-form expressions are derived for squared-exponential and Matérn covariance functions, revealing how multi-response correlations, latent alignment, and output-specific uncertainty influence local posterior geometry. Controlled numerical experiments show that additional responses provide complementary geometric constraints on calibration parameters, while noise modeling can improve uncertainty quantification by reducing overconfidence without necessarily improving parameter localization. A lithium-ion battery study further illustrates how the model distinguishes strongly from weakly constrained parameters. Together, these results establish expected-Fisher posterior geometry as a scalable diagnostic of local parameter identifiability in multi-response calibration.

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