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This says that the likelihood function of the current observation y t is distributed as a multivariate
normal distribution with mean Ft T θ t and variance-covariance V t . We have already
outlined these terms in the list above.
Finally we have the posterior of θ t :
θ t |D t ∼ N (m t , C t ) (13.10)
This says that the posterior view of the current state θ t , given our current knowledge at time
t is distributed as a multivariate normal distribution with mean m t and variance-covariance C t .
The Kalman Filter is what links all of these terms together for t = 1, . . . , n. We we will not
derive where these values actually come from. Instead they will simply be states. Thankfully we
can use library implementations in Python to carry out the "heavy lifting" calculations for us:
a t = G t m t−1 (13.11)
R t = G t C t−1 G T t + W t (13.12)
e t = y t − f t (13.13)
m t = a t + A t e t (13.14)
f t = F T t a t (13.15)
Q t = F T t R t F t + V t (13.16)
A t = R t F t Q −1
t (13.17)
C t = R t − A t Q t A T t (13.18)
Clearly that is a lot of notation. As I said above we need not worry about the excessive
verboseness of the Kalman Filter since we can simply use libraries in Python to calculate the
algorithm for us.
How does it all fit together? Well, f t is the predicated value of the observation at time t,
where we make this prediction at time t − 1. Since e t = y t − f t , we can see easily that e t is the
error associated with the forecast–the difference between f and y.
Importantly the posterior mean is a weighting of the prior mean and the forecast error, since
m t = a t + A t e t = G t m t−1 + A t e t , where G t and A t are our weighting matrices.
Now that we have an algorithmic procedure for updating our views on the observations and
states we can use it to make predictions as well as smooth the data.
13.2.2 Prediction
The Bayesian approach to the Kalman Filter leads naturally to a mechanism for prediction.
Since we have our posterior estimate for the state θ t we can predict the next day’s values by
considering the mean value of the observation.
Let us take the expected value of the observation tomorrow given our knowledge of the data
today: