Notations to Use in Formulating of Maximum Likelihood Estimation
1) What notations to use for probability density function is it the one below: $\mathsf{P}_y(y_n|{\mathbf{u}_n})$ $\mathsf{P}_z(z_n|{\mathbf{u}_n})$ what goes in the subscript if I want to use $z$?


One way to write the density function is to subscripted upper case letters for the random variables. So $p_{Y_n}$ denotes the density of random variable $Y_n$ and $p_{Z_n}$ is the density of the random variable $Z_n$. Similarly, $p_\mathbf{Y}:=p_{Y_1,\ldots,Y_n}$ denotes the joint density of all the $Y_i$'s.


  2) Is $P(\mathbf{y}|\mathbf{\theta})$ correct or
  $P_y(\mathbf{y}|\mathbf{\theta})$ or
  $P_z(\mathbf{z}|\mathbf{\theta})$?


See my answer to (1) above. In some cases it is ok to drop the subscript if it is understood which random variable the density $p$ belongs to. However, if you have lots of different random variables it is good to explicitly include the subscripts.


  If I want to use expectation maximization, then would $\xi$ which
  denotes the complete data set consisting of the input signal and the
  received noisy signal be written as:
  $\xi = \{u_0,\ldots,u_{N-1},y_0,\ldots,y_{N-1} \}$ or
  $\xi = \{\hat u_0,\ldots,\hat u_{N-1},y_0,\ldots,y_{N-1} \}$


It is not clear what the hats mean.


  4) Would the posterior be written as 


You need to clarify the posterior of what conditioned on what. I assume you are interested in estimating the $u$'s so the posterior you are referring to is

\begin{eqnarray}
p_{\mathbf{U}|\mathbf{Y}}(\mathbf u|\mathbf y) &amp;\propto&amp; p_{\mathbf{Y}|\mathbf{U}}(\mathbf y|\mathbf u) p_\mathbf{U}(\mathbf u) \\
&amp;=&amp; p_{Y_1,\ldots,Y_N|U_1,\ldots,U_{N}}(y_1,\ldots, y_N|u_1,\ldots u_N) \; p_{U_1,\ldots,U_{N}}(u_1,\ldots u_N)\\
&amp;=&amp; \left(\prod_{i=1}^N p_{Y_i | U_{i},\ldots U_{1 \wedge (i-p+1)}} (y_i|u_{i},\ldots u_{1\wedge (i-p+1)}) \right) \; p_{U_1,\ldots,U_{N}}(u_1,\ldots u_N).
\end{eqnarray}

That $\wedge$ symbol denotes a min. operation and makes sure that the subscripts don't go below 1. If you have a probabilistic model for the $U_i$'s you can further factor out $p_{U_1,\ldots,U_{N}}$ into a product of conditional densities. For example, if it's a 1st order Markov model with a stationary initial distribution $p_{U_1}(\cdot)$ you can write, 
$$
p_{U_1,\ldots,U_{N}}(u_1,\ldots,u_N) = p_{U_1}(u_1)\;\prod_{i=2}^N p_{U_i|U_{i-1}}(u_i|u_{i-1}).
$$


  pdf, likelihood, log likelihood


Finally, I think, you may be interested in writing the "complete data" log-likelihood function i.e. the density of $\xi$ parameterized by $\theta$. This can be written as follows:

\begin{eqnarray}
p_{\xi|\theta}(\xi|\theta) &amp;=&amp; p_{U_1,\ldots ,U_N,Y_1,\ldots ,Y_N|\theta}(u_1,\ldots u_N, y_1,\ldots y_N | \theta) \\
&amp;=&amp; p_{Y_1,\ldots ,Y_N | U_1,\ldots ,U_N,\theta}(y_1,\ldots y_N | u_1,\ldots u_N,  \theta) \; p_{U_1,\ldots ,U_N|\theta}(u_1,\ldots u_N| \theta)
\end{eqnarray}

and simplify this further using your observation model for $Y_i$'s and Markov transition model for $U_i$'s.


It seems that your confusion stems from being unfamiliar with how random variable notation works. I would recommend reviewing a textbook on probability and random processes for electrical engineers.