On some confusion regarding the autoregressive model and the definition of a statistical model Citing Wikipedia the stationary AR(1) model (without constant trend parameter) is defined as 
$$
\begin{aligned}
y_{t} &=  + \beta y_{t-1} + \epsilon_{t}, \\
\epsilon_{t} &\stackrel{iid}{\sim} N(0,1).
\end{aligned}
$$
where  $|\beta|< 1$.
But what is the meaning of the word model? 
During my studies of statistics the definition of model has been that a model is a set of (usually parametrized) probability densities, e.g. supposing I have a set of observations $\{ X_1, \dots, X_n \}$ then I can choose to model them like IID Bernoulli random variables. That is I am choosing as the model the set of probability densities given by
$$\{ p^k (1-p)^{1-k} | p \in [0,1]     \}$$
where $k \in \{0,1 \}$. Once I have assumed the model I can estimate the parameter $p$ .
But what model (set of probability densities) does the difference equation $$y_{t} =  \beta y_{t-1} + \epsilon_{t}$$
define?     
At first I thought that the model was implicitly given by the distribution of the solution of the difference equation that is $y_t = \sum_{i=0}^{\infty} \beta^i \epsilon_{t-i}$. So the model would be
$$\left \{ f(y| \beta) = \frac{\sqrt{1-\beta}}{\sqrt{2 \pi }}  \exp \left( \frac{-y^2 (1-\beta)}{2 } \right)   \Bigg|  \beta \in ]-1,1[ \right \}$$ 
but I am unsure, maybe the model is given as a conditional distribution of some sort. What would the model be?
 A: Hi: I wouldn't worry too much about the term "model". There are time series "models" such as the AR(1) model you referred to. Then there are "models" for random variables where we say, for example, that an RV has a bernoulli distribution. So, the model for the RV is that it's bernoulli. But this is just semantics. They ( the term model in time series and the term model when used for RV's ) have no relation to each other. In fact, I would tend not to use the term "model" for RV's but you are correct that it is used.
As far as your other question, the AR(1) can be written in conditional likelihood form but what you wrote is not correct. The AR(1) model says that $y_t$ conditional on $y_{t-1}$ is normal with mean $\beta y_{t-1}$ and variance $\sigma^2$. So, if you have an AR(1) time series, you can write the full conditional likelihood out as a product of normal random variables or you can write the full conditional log likelihood out as a sum. But I'm not clear how you obtained what you wrote. 
My point is that you're understanding is mostly correct, except how you wrote the conditional likelihood for $y_{t}$. 
Note that I use the term conditional because each $y_{t}$ depends on $y_{t-1}$
so we use the term conditional. But don't worry about that either. It's really a likelihood just like any other likelihood.
