Complete Data Likelihood function intuition I am struggling with the intuition of the following model:
$y_{t} = (1-s_{t})x_{t}'\beta_{0} + s_{t}x_{t}'\beta_{1}+\varepsilon_{t}$   where $\varepsilon_{t} \sim N(0,\sigma{2}$
where $s_{t}$ is an unobserved process which can take the value 0 or 1. It is described by $Pr[S_{t} = 1] = \lambda$ and $Pr[S_{t} = 0] =1- \lambda$.
I understand how we get to the likelihood function:
$p(y|\beta_{0},\beta_{1},\sigma^{2},\lambda) = \prod\limits_{t=1}^{T}(1-\lambda)\phi(y_{t};x_{t}'\beta_{0},\sigma^{2})+\lambda\phi(y_{t};x_{t}'\beta_{0},\sigma^{2})$.
But then I cannot understand why we want the Complete Data Likelihood Function and what the intuition of raising to $s_{t}$ is such that:
$p(y,s|\beta_{0},\beta_{1},\sigma^{2},\lambda) = \prod\limits_{t=1}^{T}[(1-\lambda)\phi(y_{t};x_{t}'\beta_{0},\sigma^{2})]^{1-s_{t}}[\lambda\phi(y_{t};x_{t}'\beta_{0},\sigma^{2})]^{s_{t}}$
I would appreciate some intuition of why we want the complete data likelihood and why in such shape.
 A: It depends on whether $s$ is observable or not. 
If $s$ is not observable (also called latent variable) then the only likelihood you can calculate is the one of $y$. Naturally :
$p(y_t=a)=P(s_t=0)p(x'_t\beta_0+\epsilon=a)+P(s_t=1)p(x'_t\beta_1+\epsilon=a)=(1-\lambda)p(x'_t\beta_0+\epsilon=a)+\lambda p(x'_t\beta_1+\epsilon=a)$
But if $s$ is observable, you want to calculate the likelihood of your observation, that is $(y,s)$. This is somehow your output. It would be forgetting information to make yourself blind of something you actually see. Then simply :
$p(y_t=a,s_t=b)=$


*

*if $b=0$ :  $p(x'_t\beta_0+\epsilon=a,s_t=0)=(1-\lambda)p(x'_t\beta_0+\epsilon=a)$

*if $b=1$ : $p(x'_t\beta_1+\epsilon=a,s_t=1)=\lambda p(x'_t\beta_1+\epsilon=a)$


The final formula is just using powers as a trick to write the two cases as a single formula.
Notes :


*

*I made conditioning on the parameters implicit in my formulae. 

*you seemed to assume $s_t$ and $x'_t$ are independent so I did it too. 

*I use denotation $p(y_t=a)$ instead of $p(y_t)$ for clarity.
