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A Gaussian pCN kernel leaves a normal posterior invariant

Example

Assume the Axiom of Choice. Let $H = R$ , give the unknown parameter $u$ the prior law $μ_{0} = N (0, 1)$ , and observe one datum $Y = 0$ from the model $Y ∣ u \sim N (u, 1)$ . The Gaussian conjugate update gives posterior $μ = N (0, 1 / 2)$ . Use the potential $Φ (u) = u^{2} / 2$ in the pCN algorithm. For $β \in [0, 1]$ , write $ρ = \sqrt{1 - β^{2}}$ , propose $v = ρ u + β ξ$ with $ξ \sim N (0, 1)$ independent of the current state, and accept with probability $α (u, v) = \min {1, \exp ((u^{2} - v^{2}) / 2)} .$ The resulting pCN probability kernel leaves $N (0, 1 / 2)$ invariant for every $β \in [0, 1]$ .

Example references

A normal prior and one normal observation with known variance give a normal posterior by the precision update.

The proposal combines the current state with an independent Gaussian reference draw and accepts by the potential ratio.

A proper pCN posterior target is invariant under its accept-reject kernel.

Facts & Assumptions

Given: The Axiom of Choice, the one-observation normal model and pCN rule in the Example.

[F1]

For prior $N (μ_{0}, τ_{0}^{2})$ and one observation from $N (u, σ^{2})$ , the posterior mean is $(τ_{0}^{- 2} + σ^{- 2})^{- 1} (μ_{0} / τ_{0}^{2} + y / σ^{2})$ and the posterior variance is $(τ_{0}^{- 2} + σ^{- 2})^{- 1}$ ; evidence is finite and positive. Known-variance Normal-Normal conjugate update

[F2]

The law $N (m, 1)$ has density $(2 π)^{- 1 / 2} \exp (- (y - m)^{2} / 2)$ with respect to Lebesgue measure. Standard normal and normal laws A positive-variance normal law has its location-scale density

[F3]

The pCN rule proposes $v = ρ u + β ξ$ and accepts with probability $\min {1, \exp (Φ (u) - Φ (v))}$ for finite potential values. Prior-preserving Crank–Nicolson sampler

[F4]

The pCN accept-reject probability kernel leaves its normalized posterior target invariant. The pCN accept-reject kernel preserves the posterior

Proof

Given: The Axiom of Choice, prior $N (0, 1)$ , observation variance $1$ , datum $y = 0$ , and potential $Φ (u) = u^{2} / 2$ .

1.1F1givenalgebra

Apply [F1] with $μ_{0} = 0$ , $τ_{0}^{2} = 1$ , $σ^{2} = 1$ , $n = 1$ , and $y = 0$ . The posterior variance is $(1 + 1)^{- 1} = 1 / 2$ , and its mean is $(1 + 1)^{- 1} (0 + 0) = 0$ . The evidence is finite and positive, so the posterior is the proper law $N (0, 1 / 2)$ .

2.1F2step 1.1givenalgebra

By [F2], the likelihood density at $y = 0$ is $(2 π)^{- 1 / 2} e^{- u^{2} / 2}$ . Its constant factor cancels when the likelihood is normalized against the prior, so the potential $Φ (u) = u^{2} / 2$ gives exactly the posterior from step 1.1. It is finite on $D = H = R$ , and the pCN target assumptions are satisfied.

3.1F3step 2.1givenalgebra

For the proposal in [F3], $Φ (u) - Φ (v) = \frac{u^{2} - v^{2}}{2},$ so the stated acceptance probability is precisely the algorithm's pCN acceptance rule for this target.

4.1F4step 1.1step 3.1given∎

Step 1.1 identifies the normalized target, and step 3.1 identifies the kernel as the pCN accept-reject kernel for that target. Applying [F4] proves that $N (0, 1 / 2)$ is invariant for every $β \in [0, 1]$ .

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Sources

MIT 18.05, Spring 2022, Class 15: Conjugate priors: Beta and normal (standard reference, not scraped)
S. L. Cotter, G. O. Roberts, A. M. Stuart, and D. White, MCMC Methods for Functions: Modifying Old Algorithms to Make Them Faster, Statistical Science 28(3) (2013) (standard reference, not scraped)