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Smoothing a two-point data law

Example

For $z \in R^{d}$ , $δ_{z}$ denotes the probability concentrated at $z$ . Let $d \geq 1$ , $T > 0$ , $a, b \in R^{d}$ , $α \in [0, 1]$ , and $μ = α δ_{a} + (1 - α) δ_{b}$ . Let $g : [0, T] \to (0, \infty)$ be continuously differentiable with $0 < g_{\min} \leq g (t) \leq g_{\max} < \infty$ , set $v (t) = \int_{0}^{t} g (r)^{2} d r$ , and define $ϕ_{c} (z) = (2 π c)^{- d / 2} \exp (- ∣ z ∣^{2} / (2 c))$ for $c > 0$ . For every $0 < t \leq T$ , the smoothed law has density

$p_{t} (x) = α ϕ_{v (t)} (x - a) + (1 - α) ϕ_{v (t)} (x - b) .$

At time zero its law remains $μ$ . If $a = b$ , or if $α$ is $0$ or $1$ , the displayed density reduces to the single Gaussian centered at the atom carrying all mass.

Example references

Särkkä and Solin, Applied Stochastic Differential Equations, Chapter 5, §§5.1–5.2

Hunter, Notes on Partial Differential Equations, Chapter 5, §5.1.1

Facts & Assumptions

Given: $d, T, a, b, α$ , the finite probability measure $μ$ , and the scalar-noise smoothing convention.

[A1]

The full Axiom of Choice is inherited from the Gaussian smoothing interfaces. The Axiom of Choice

[F1]

For a compactly supported initial law, positive-time smoothing has density $p_{t} (x) = \int ϕ_{v (t)} (x - z) μ (d z)$ and initial law $μ$ . Gaussian-smoothed data law and transition density

Proof

Given: The example's parameters and the Gaussian smoothing density in [F1].

1.1A1F1givenalgebra

The measure $μ$ is a Borel probability supported in the closed ball of radius $\max (∣ a ∣, ∣ b ∣)$ , so [F1] applies. For $0 < t \leq T$ , substitute $μ = α δ_{a} + (1 - α) δ_{b}$ in the defining integral. By the defining property of point masses, integration of any nonnegative Borel function $h$ gives $\int h (z) μ (d z) = α h (a) + (1 - α) h (b)$ . Taking $h (z) = ϕ_{v (t)} (x - z)$ yields the displayed formula, using $ϕ_{c} (z) = ϕ_{c} (- z)$ . The initial law is $μ$ by [F1].

2.1F1step 1.1algebra∎

If $a = b$ , the two weights sum to one and both Gaussian terms have the same center. If $α = 1$ , only the first term remains; if $α = 0$ , only the second remains. In each case the stated single Gaussian is normalized and positive by [F1].

Depends on

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Sources

Simo Särkkä and Arno Solin, Applied Stochastic Differential Equations (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (standard reference, not scraped)