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Machine Learning: a Probabilistic Perspective

Exercise 23.3 - Optimal proposal for particle filtering with linear-Gaussian measurement model

Answers

The model is specified by:

p ( 𝐳 t | 𝐳 t − 1 ) = 𝒩 ( 𝐳 t | f ( 𝐳 t − 1 ) , 𝐐 t − 1 ) ,

p ( 𝐲 t | 𝐳 t ) = 𝒩 ( 𝐲 t | 𝐇 t 𝐳 t , 𝐑 t ) .

For p ( 𝐳 t | 𝐳 t − 1 , 𝐲 t ) , we have:

p ( 𝐳 t | 𝐳 t − 1 , 𝐲 t ) = p ( 𝐳 t , 𝐳 t − 1 , 𝐲 t ) p ( 𝐳 t − 1 , 𝐲 t ) ∝ p ( 𝐳 t | 𝐳 t − 1 ) ⋅ p ( 𝐲 t | 𝐳 t ) ∝ exp ⁡ { − 1 2 ( 𝐳 t − f ( 𝐳 t − 1 ) ) T 𝐐 t − 1 − 1 ( 𝐳 t − f ( 𝐳 t − 1 ) ) } ⋅ exp ⁡ { − 1 2 ( 𝐲 t − 𝐇 t 𝐳 t ) T 𝐑 t − 1 ( 𝐲 t − 𝐇 t 𝐳 t ) } ,

from which we can readily read that the posterior covariance and mean for 𝐳 t are:

Σ = ( 𝐐 t − 1 + 𝐇 t T 𝐑 t − 1 𝐇 t ) − 1 ,

μ = Σ ( 𝐐 t − 1 f ( 𝐳 t − 1 ) + 𝐇 t T 𝐑 t − 1 𝐲 t ) .

On the other hand:

p ( 𝐲 t | 𝐳 t − 1 ) = ∫ p ( 𝐲 t , 𝐳 t | 𝐳 t − 1 ) d 𝐳 t = ∫ p ( 𝐲 t | 𝐳 t ) ⋅ p ( 𝐳 t | 𝐳 t − 1 ) d 𝐳 t ∝ exp ⁡ { − 1 2 𝐲 t T 𝐑 t − 1 𝐲 t } ⋅ exp ⁡ { 1 2 ( f T 𝐐 t − 1 + 𝐲 t T 𝐑 t − 1 𝐇 t ) Σ ( 𝐐 t − 1 f + 𝐇 t T 𝐑 t − 1 𝐲 t ) } ,

where Σ is the posterior covariance for 𝐳 t . Therefore the posterior corariance and mean for 𝐲 t given 𝐳 t − 1 are:

Σ ′ = ( 𝐑 t − 1 − 𝐑 t − 1 𝐇 t Σ 𝐇 t T 𝐑 t − 1 ) − 1 ,

and finally:

μ ′ = Σ ′ 𝐑 t − 1 𝐇 t Σ 𝐐 t − 1 f ( 𝐳 t − 1 ) .

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2021-03-24 13:42
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