Stein's Identity
A core mathematical identity in the Stein method that constructs an expectation expression independent of the normalizing constant, making inference on unnormalized distributions possible.
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The Stein identity states that for a function φ that decays sufficiently fast on the boundary, when the probability density p(x) is sufficiently smooth on R^d, E_p[∇_x log p(x)·φ(x) + ∇_x·φ(x)] = 0.
The Stein identity constructs an expectation expression that does not depend on the normalization constant; even if the normalization constant Z is unknown, the score function can still be computed, which makes Stein methods suitable for scenarios requiring inference on unnormalized distributions (such as Bayesian posteriors and energy models).
All Facts (2)
The Stein identity states that for a function φ that decays sufficiently fast on the boundary, when the probability density p(x) is sufficiently smooth on R^d, E_p[∇_x log p(x)·φ(x) + ∇_x·φ(x)] = 0.
50%UnverifiedThe Stein identity constructs an expectation expression that does not depend on the normalization constant; even if the normalization constant Z is unknown, the score function can still be computed, which makes Stein methods suitable for scenarios requiring inference on unnormalized distributions (such as Bayesian posteriors and energy models).
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