Homework 4

Theoretical part

You can either write your answers to theoretical questions on paper or edit them in the file hw4/paper.tex. Please show all the mathematical derivations that you perform.

1. The following equality for the posterior model covariance was given in lecture notes without a proof:
   

   $$\widehat{\mathbf{C}_m} = \left(\mathbf{F}^T\,\mathbf{C}_n^... ...f{F}^T + \mathbf{C}_n\right)^{-1}\,\mathbf{F}\,\mathbf{C}_m\;.$$ (1)

   
   
   Prove it.

2. If the model shaping operator $\mathbf{S}_m$ admits a symmetric splitting $\mathbf{S}_m=\mathbf{H}_m \mathbf{H}_m^T$ with square and invertible $\mathbf{H}_m$, the model shaping equation can be rewritten in a symmetric form
   

   $$\left[\mathbf{I} + \mathbf{S}_m (\mathbf{B F - I})\right]^... ...}) \mathbf{H}_m\right]^{-1} \mathbf{H}_m^T \mathbf{B d}\;.$$ (2)

   
   
   1. Prove equation (2).
   2. Assuming a symmetric splitting for the data shaping operator $\mathbf{S}_d=\mathbf{H}_d^T \mathbf{H}_d$, find a symmetric form of the data shaping equation
      

      $$\mathbf{B} \left[\mathbf{I} + \mathbf{S}_d (\mathbf{F B - I})\right]^{-1} \mathbf{S}_d \mathbf{d} = \hfill \$$ (3)

      
      

Homework 4