ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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A two-state hidden Markov filter and smoother

Example

Let $E_{0} = E_{1} = E_{2} = F_{1} = F_{2} = {0, 1}$ with counting measure. Take $μ_{0} = (1 / 2, 1 / 2)$ and transition matrix $M = (\begin{array}{cc} 3 / 4 & 1 / 4 \\ 1 / 4 & 3 / 4 \end{array}),$ where rows index the current state. At each time let $G_{t} (1 ∣ 0) = 1 / 4$ and $G_{t} (1 ∣ 1) = 3 / 4$ , with the complementary probabilities for observation $0$ . For observations $Y_{1} = 1, Y_{2} = 0$ , the exact filter at time $2$ is $(9 / 14, 5 / 14)$ on states $(0, 1)$ , while the smoothing law of $X_{1}$ given both observations is $(5 / 14, 9 / 14)$ . This finite model has positive evidence at both observed times.

Verification

Given: AC, the finite-horizon kernel model, and observations in the Example.

[A1] AC states that every family of nonempty sets has a choice function. The Axiom of Choice

[F1] Filtering and smoothing kernels are conditional laws given the stated observation histories. Filtering, predictive, and smoothing laws in a finite-horizon state-space model

[F2] For dominated observation kernels, the prediction and update are given by the exact recursion, with evidence computed by summing the likelihood against the predictive state law. Exact state-space filtering recursion under dominated observations

[F3] The finite-horizon model law is the ordered product of the initial, transition, and observation kernels. Finite-horizon state-space kernel law exists and is unique

[F4] Under AC the data-indexed conditional filtering and smoothing kernels exist for this finite standard-Borel model. Filtering and finite-horizon smoothing laws exist

[F5] The model supplies transition and observation probability kernels. General finite-horizon state-space models by kernels

[F6] The recursive filtering procedure uses the prediction/update formulas and predictive filler on zero or infinite evidence. Recursive Bayesian filtering under dominated observations

1.1A1givenF2F3F5F6algebra

Let $g_{t} (i, j) : = G_{t} (i, {j})$ for $i, j \in {0, 1}$ . Counting measure integrates by finite sums, so $\sum_{j} g_{t} (i, j) = G_{t} (i, {0, 1}) = 1$ and $G_{t} (i, C) = \sum_{j \in C} g_{t} (i, j)$ ; every function on these finite discrete spaces is product-measurable. Thus [F2] applies. Since $μ_{0}$ is uniform and each transition-matrix column sums to one, $X_{1}$ is uniform; hence $P (Y_{1} = 1) = \frac{1}{2} (\frac{1}{4} + \frac{3}{4}) = \frac{1}{2}$ , and Bayes' finite sum gives $P (X_{1} = 0 ∣ Y_{1} = 1) = \frac{(1 / 2) (1 / 4)}{1 / 2} = 1 / 4$ and $P (X_{1} = 1 ∣ Y_{1} = 1) = 3 / 4$ .

2.1A1givenF2F6step 1.1algebra

Predicting through $M$ gives $P (X_{2} = 0 ∣ Y_{1} = 1) = \frac{1}{4} \frac{3}{4} + \frac{3}{4} \frac{1}{4} = \frac{3}{8}$ and $P (X_{2} = 1 ∣ Y_{1} = 1) = \frac{1}{4} \frac{1}{4} + \frac{3}{4} \frac{3}{4} = \frac{5}{8}$ .

3.1A1givenF2F6step 2.1algebra

The evidence for $Y_{2} = 0$ is $(3 / 8) (3 / 4) + (5 / 8) (1 / 4) = 7 / 16$ ; thus $P (X_{2} = 0 ∣ Y_{1} = 1, Y_{2} = 0) = 9 / 14$ and $P (X_{2} = 1 ∣ Y_{1} = 1, Y_{2} = 0) = 5 / 14$ , which sum to one.

4.1A1givenF1F3F4step 1.1step 3.1algebra∎

By the ordered path law [F3], for $i = 0, 1$ the conditional probability of $Y_{2} = 0$ given $X_{1} = i, Y_{1} = 1$ is $\sum_{j = 0}^{1} M_{i j} G_{2} (j, {0})$ ; the factor $G_{1} (i, {1})$ cancels in this conditional probability because it depends only on the fixed state $i$ and event $Y_{1} = 1$ . These sums are $(3 / 4) (3 / 4) + (1 / 4) (1 / 4) = 5 / 8$ for $i = 0$ and $(1 / 4) (3 / 4) + (3 / 4) (1 / 4) = 3 / 8$ for $i = 1$ . Multiplying by the time-1 filter $(1 / 4, 3 / 4)$ gives conditional smoothing masses $(5 / 32, 9 / 32)$ ; their sum is $7 / 16$ , the conditional evidence in step 3.1, so the smoothing law of $X_{1}$ is $(5 / 14, 9 / 14)$ . The history event has positive probability by step 1.1 and step 3.1, so its conditional distribution is fixed by the event identities.

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Sources

Simo Särkkä and Lennart Svensson, Bayesian Filtering and Smoothing, 2nd ed., Chapter 6 (standard reference, not scraped)
Arnaud Doucet and Adam M. Johansen, A Tutorial on Particle Filtering and Smoothing: Fifteen Years Later, §2.1 (standard reference, not scraped)