I am not very sure if I am able to pose the question properly, but I am having a lot of problem trying to justify a joint distribution derived from the marginals.

I am given that:

(y_{t+1}| x_{t+1}, θ) ~ N (x_{t+1}, σ^{2})

(x_{t+1}| x_{t}, θ) ~ N (x_{t+1}, σ^{2})

(y_{t+1}| x_{t}, θ) ~ N (x_{t}, σ^{2}+ τ^{2})

(Basically θ represents the parameters sigma and tau)

Then how can we derive (x_{t+1}| y_{t+1}, x_{t}, θ)?

The answer is:

(x_{t+1}| y_{t+1}, x_{t}, θ) ~ N (µ, w^{2}),

µ = w^{2}(σ^{-2}y_{t+1}+ τ^{-2}x_{t}) and w^{-2}= σ^{-2}+ τ^{-2}

but I don’t know how to calculate it. Any help will be much appreciated.

## Bookmarks