$($iv$)$ Reflect on the two recursive relationships we have studied:

Forward $($HW$4),{}_t=P(Y_{0\text{:}t},x_t)=P(Y_t|x_t){\displaystyle \underset{x}{\overset{?}{}}}P(x_t|x_{t-1}=x)P(Y_{0\text{:}t-1},x_{t-1}=x)$

Backward $($HW$5),{}_t=P(Y_{t+1\text{:}T}|x_t)={\displaystyle \underset{x}{\overset{?}{}}}P(Y_{t+1}|x_{t+1}=x)P(x_{t+1}=x|x_t)P(Y_{t+2\text{:}T}|x_{t+1}=x)$

Note ${}_t,{}_t$ are vectors with length equal to the number of states of $x.$ In this case, there are two

states. Then one can think of $$ and $$ as matrices with shape $TN$ for $N=2$ states. We have

${}_{t,i}=P(Y_{0\text{:}t},x_t=x_i)$ and ${}_{t,i}=P(Y_{t+1\text{:}T}|x_t=x_i)$ where $x_i$ is the $i^{th}$ state of $x.$

Find an expression of $P(Y_{0\text{:}T},x_t)$ in terms of the forward and backward probabilities.

$($v$)$ Find an expression for $P(x_t=x_i|Y_{0\text{:}T})$ using the forward and backward matrices.

$($vi$)$ Using the updated path.csv dataset from Brightspace $($see Homework $5$ code$),$ plot the noisy path

$(Y_{0\text{:}T}),$ the clean path $(x_{0\text{:}T}),$ and the most likely states using $P(x_t|Y_{1\text{:}t}),P(x_t|Y_{t+1\text{:}T}),$ and $P(x_t|Y_{0\text{:}T})$

for tin$\{0,$dots,$T\}.$ Plot the clean and noisy path over time so the x $-$axis is the position and the y $-$axis

is time. The provided helper code will be very useful for this problem.