Exercises 651a linear dynamical system governed by (13.75) and (13.76), with latent variables
{z 1 ,...,zN}in whichCbecomes the identity matrix and where the transition prob-
abilityA= 0 because the observations are independent. Let the parametersm 0
andV 0 of the initial state be denoted byμ 0 andσ^20 , respectively, and suppose that
Σbecomesσ^2. Write down the corresponding Kalman filter equations starting from
the general results (13.89) and (13.90), together with (13.94) and (13.95). Show that
these are equivalent to the results (2.141) and (2.142) obtained directly by consider-
ing independent data.13.26 ( ) Consider a special case of the linear dynamical system of Section 13.3 that is
equivalent to probabilistic PCA, so that the transition matrixA= 0 , the covariance
Γ=I, and the noise covarianceΣ=σ^2 I. By making use of the matrix inversion
identity (C.7) show that, if the emission density matrixCis denotedW, then the
posterior distribution over the hidden states defined by (13.89) and (13.90) reduces
to the result (12.42) for probabilistic PCA.
13.27 ( ) www Consider a linear dynamical system of the form discussed in Sec-
tion 13.3 in which the amplitude of the observation noise goes to zero, so thatΣ= 0.
Show that the posterior distribution forznhas meanxnand zero variance. This
accords with our intuition that if there is no noise, we should just use the current
observationxnto estimate the state variableznand ignore all previous observations.
13.28 ( ) Consider a special case of the linear dynamical system of Section 13.3 in
which the state variableznis constrained to be equal to the previous state variable,
which corresponds toA=IandΓ= 0. For simplicity, assume also thatV 0 →∞
so that the initial conditions forzare unimportant, and the predictions are determined
purely by the data. Use proof by induction to show that the posterior mean for state
znis determined by the average ofx 1 ,...,xn. This corresponds to the intuitive
result that if the state variable is constant, our best estimate is obtained by averaging
the observations.
13.29 ( ) Starting from the backwards recursion equation (13.99), derive the RTS
smoothing equations (13.100) and (13.101) for the Gaussian linear dynamical sys-
tem.
13.30 ( ) Starting from the result (13.65) for the pairwise posterior marginal in a state
space model, derive the specific form (13.103) for the case of the Gaussian linear
dynamical system.
13.31 ( ) Starting from the result (13.103) and by substituting forα̂(zn)using (13.84),
verify the result (13.104) for the covariance betweenznandzn− 1.
13.32 ( ) www Verify the results (13.110) and (13.111) for the M-step equations for
μ 0 andV 0 in the linear dynamical system.
13.33 ( ) Verify the results (13.113) and (13.114) for the M-step equations forAandΓ
in the linear dynamical system.