This homework gives you some experience at algebraic manipulation of POMP models by deriving the prediction, filtering and smoothing formulas in Chapter 9 of the notes. Please read the revised version of Chapter 9, with many typos corrected, posted online.
The calculations are all applications of basic identities for joint, conditional and marginal probability density functions. You may follow the hints in Section 9.2.5 of the notes.
Please say explicitly when you use the Markov property. Also, write explicitly when and how you use basic identities such as those provided in Section 9.2.5.
Please turn in your homework on paper, in class.
Question 5.1. Derive the identity [MP2].
Question 5.2. Derive the prediction formula, [MP4].
Question 5.3. Derive the filtering formulas [MP5] and [MP6].
Question 5.4. Derive the backward recursion formulas [MP8] and [MP9].
Question 5.5. Derive the smoothing formula [MP10].