Facts.

[MP2].

Facts: \[\begin{eqnarray} (1)& f_{X|Y}\left(x|y\right)=\frac{f_{XY}\left(x,y\right)}{f_{Y}\left(y\right)}\\ (2)& f_{X|YZ}\left(x|y,z\right)=\frac{f_{XY|Z}\left(x,y|z\right)}{f_{Y|Z}\left(y|z\right)}\\ (3)& f_{X_{n}|X_{1:n-1}}\left(x_{n}|x_{1:n-1}\right)=f_{X_{n}|X_{n-1}}\left(x_{n}|x_{n-1}\right).\\ (4)& f_{X|Y}\left(x|y\right)=\int f_{XZ|Y}\left(x,z|y\right)dz\\ (5)& f_{XZ|Y}\left(x,z|y\right)=f_{Z|Y}\left(z|y\right)f_{X|ZY}\left(x|y,z\right)\\ (6)& f_{Y_{n}|X_{0:N},Y_{1:n-1},Y_{n+1:N}}\left(y_{n}|x_{0:N},y_{1:n-1},y_{n+1:N}\right)=f_{Y_{n}|X_{n}}\left(y_{n}|x_{n}\right)\\ (7)& f_{X|YZ}\left(x|y,z\right)=\frac{f_{Y|XZ}\left(y|x,z\right)f_{X|Z}\left(x|z\right)}{f_{Y|Z}\left(y|z\right)} \end{eqnarray}\] Question 5.1.

[MP2].

5.1

\[\begin{eqnarray} f_{X_{0:N}}\left(x_{0:N}\right)&=&f_{X_{1:N}|X_{0}}\left(x_{1:N}|x_{0}\right)f_{X_{0}}\left(x_{0}\right) &by&(1)\\ &=&f_{X_{2:N}|X_{0:1}}\left(x_{2:N}|x_{0:1}\right)f_{X_{1}|X_{0}}\left(x_{1}|x_{0}\right)f_{X_{0}}\left(x_{0}\right) &by& (2) \\ &=&f_{X_{2:N}|X_{1}}\left(x_{2:N}|x_{1}\right)f_{X_{1}|X_{0}}\left(x_{1}|x_{0}\right)f_{X_{0}}\left(x_{0}\right) &by& (3)\\ &=&...& & \\ &=&f_{X_{0}}\left(x_{0}\right)\prod_{n=1}^{N}f_{X_{n}|X_{n-1}}\left(x_{n}|x_{n-1}\right)& by\ iteratively\ or\ formally\ by\ induction & \end{eqnarray}\] Question 5.2.

[MP4].

\[\begin{eqnarray} &&\int f_{X_{\mathrm{n}-1}|Y_{1:\mathrm{n}-1}}(x_{n-1}|y_{1:n-1}^{*})f_{X_{\mathrm{n}}|X_{\mathrm{n}-1}}(x_{n}|x_{n-1})dx_{n-1}&\\ &=&\int f_{X_{\mathrm{n}-1}|Y_{1:\mathrm{n}-1}}(x_{n-1}|y_{1:n-1}^{*})f_{X_{\mathrm{n}}|X_{\mathrm{n}-1}Y_{1:n-1}}(x_{n}|x_{n-1},y_{1:n-1}^{*})dx_{n-1}\quad &by\ (6) \\ &=&\int f_{X_{n}X_{\mathrm{n}-1}|Y_{1:\mathrm{n}-1}}(x_{n},x_{n-1}|y_{1:n-1}^{*})dx_{n-1}\quad&by\ (5)\\ &=&f_{X_{n}|Y_{1:\mathrm{n}-1}}(x_{n}|y_{1:n-1}^{*})\quad &by\ (4) \end{eqnarray}\]

Question 5.3.

[MP5].

\[\begin{eqnarray} f_{X_{\mathrm{n}}|Y_{1:\mathrm{n}}}(x_{n}|y_{1:n}^{*})&=&f_{X_{\mathrm{n}}|Y_{n}Y_{1:\mathrm{n-1}}}(x_{n}|y_{n}^{*}y_{1:n-1}^{*})&\\ &=&\frac{f_{Y_{\mathrm{n}}|X_{n}Y_{1:\mathrm{n-1}}}(y_{n}^{*}|x_{n},y_{1:n-1}^{*})f_{X_{\mathrm{n}}|Y_{1:\mathrm{n-1}}}(x_{n}|y_{1:n-1}^{*})}{f_{Y_{n}|Y_{1:\mathrm{n-1}}}(y_{n}^{*}|y_{1:n-1}^{*})}\quad &by\ (6)\\ &=&\frac{f_{Y_{\mathrm{n}}|X_{n}}(y_{n}^{*}|x_{n})f_{X_{\mathrm{n}}|Y_{1:\mathrm{n-1}}}(x_{n}|y_{1:n-1}^{*})}{f_{Y_{n}|Y_{1:\mathrm{n-1}}}(y_{n}^{*}|y_{1:n-1}^{*})}\quad &by\ (7) \end{eqnarray}\]

[MP6]. \[\begin{eqnarray} f_{Y_{\mathrm{n}}|Y_{1:\mathrm{n}-1}}(y_{n}^{*}|y_{1:n-1}^{*})&=&\int f_{Y_{\mathrm{n}}X_{n}|Y_{1:\mathrm{n}-1}}({\displaystyle y_{n}^{*},x_{n}|y_{1:n-1}^{*})}dx_{n}\quad &by\ (4)\\ &=&\int f_{X_{n}|Y_{1:\mathrm{n}-1}}({\displaystyle x_{n}|y_{1:n-1}^{*})}f_{Y_{\mathrm{n}}|X_{n}Y_{1:\mathrm{n}-1}}({\displaystyle y_{n}^{*}|x_{n},y_{1:n-1}^{*})}dx_{n}\quad &by\ (5)\\ &=&\int f_{X_{\mathrm{n}}|Y_{1:\mathrm{n}-1}}(x_{n}|y_{1:n-1}^{*})f_{Y_{\mathrm{n}}|X_{\mathrm{n}}}(y_{n}^{*}|x_{n})dx_{n}\quad &by\ (6) \end{eqnarray}\]

Question 5.4.

[MP8]. \[\begin{eqnarray} f_{Y_{\mathrm{n}:N}|X_{\mathrm{n}}}(y_{n:N}^{*}|x_{n})&=&f_{Y_{\mathrm{n}}|X_{\mathrm{n}}}(y_{n}^{*}|x_{n})f_{Y_{\mathrm{n}+1:N}|Y_{n}X_{\mathrm{n}}}(y_{n+1:N}^{*}|y_{n}^{*},x_{n})\quad &by\ (5)\\ &=&f_{Y_{\mathrm{n}}|X_{\mathrm{n}}}(y_{n}^{*}|x_{n})f_{Y_{\mathrm{n}+1:N}|X_{\mathrm{n}}}(y_{n+1:N}^{*}|x_{n})\quad &by\ (6) \end{eqnarray}\]

[MP9]. \[\begin{eqnarray} f_{Y_{\mathrm{n}+1:N}|X_{\mathrm{n}}}(y_{n+1:N}^{*}|x_{n})&=&\int f_{Y_{\mathrm{n}+1:N}X_{\mathrm{n}+1}|X_{n}}(y_{n+1:N}^{*},x_{n+1}|x_{n})dx_{n+1}\quad &by\ (4)\\ &=&\int f_{X_{\mathrm{n}+1}|X_{n}}(x_{n+1}|x_{n})f_{Y_{\mathrm{n}+1:N}|X_{\mathrm{n}+1}X_{n}}(y_{n+1:N}^{*}|x_{n+1},x_{n})dx_{n+1}\quad &by\ (5)\\ &=&\int f_{X_{\mathrm{n}+1}|X_{\mathrm{n}}}(x_{n+1}|x_{n})f_{Y_{\mathrm{n}+1:N}|X_{\mathrm{n}+1}}(y_{n+1:N}^{*}|x_{n+1})dx_{n+1}\quad &by\ (6) \end{eqnarray}\]

Question 5.5.

[MP10]. \[\begin{eqnarray} f_{X_{\mathrm{n}}|Y_{1:N}}(x_{n}|y_{1:N}^{*})&=&f_{X_{\mathrm{n}}|Y_{1:n-1}Y_{n:N}}(x_{n}|y_{1:n-1}^{*},y_{n:N}^{*})&\\ &=&{\displaystyle \frac{f_{X_{\mathrm{n}}|Y_{1:\mathrm{n}-\mathrm{l}}}(x_{n}|y_{1:n-1}^{*})f_{Y_{\mathrm{n}:N}|X_{\mathrm{n}}Y_{1:n-1}}(y_{n:N}^{*}|x_{n},y_{1:n-1}^{*})}{f_{Y_{\mathrm{n}:N}|Y_{1:\mathrm{n}-1}}(y_{n:N}^{*}|y_{1:n-1}^{*})}}\quad &by\ (7)\\ &=&{\displaystyle \frac{f_{X_{\mathrm{n}}|Y_{1:\mathrm{n}-\mathrm{l}}}(x_{n}|y_{1:n-1}^{*})f_{Y_{\mathrm{n}:N}|X_{\mathrm{n}}}(y_{n:N}^{*}|x_{n})}{f_{Y_{\mathrm{n}:N}|Y_{1:\mathrm{n}-1}}(y_{n:N}^{*}|y_{1:n-1}^{*})}}\quad &by\ (6) \end{eqnarray}\]