a = e^{- (t_{2} - t_{1}) / τ}, b = e^{- (t_{3} - t_{2}) / τ} .

The Markov factorization and the conditional normal laws from part b give the fully univariate product

p (y ∣ t, μ, τ) = ϕ (y_{1}; μ, 1) ϕ (y_{2}; μ + a (y_{1} - μ), 1 - a^{2}) ϕ (y_{3}; μ + b (y_{2} - μ), 1 - b^{2}),

where

ϕ (\cdot; m, v)

denotes the

N (m, v)

μ

. Differentiating its log-likelihood gives

μ = \frac{y _{1} + \frac{y _{2} - a y _{1}}{1 + a} + \frac{y _{3} - b y _{2}}{1 + b}}{1 + \frac{1 - a}{1 + a} + \frac{1 - b}{1 + b}} .

Every innovation in the numerator has expectation equal to its coefficient in the denominator times

μ

. Therefore

E μ = μ

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