Bayesian inference for parameter estimation I am new in theBayesian inference domain, so sorry if my questions seem silly. However,   I realy need some help in understanding this concept, since reading the same information each time from references is useless.
As I understood, that with Bayesian inference one is able to find an "update" estimate of the parametr of  model  $F(\theta)=y$. Where $\theta$  represents the parameter and $y$ is the output.  It is stated that to use Bayesian inference one should have a prior distribution for  $\theta$  and "data" . What is meant by data? data for the output  after run with $\theta$ ? what if I have only a prior distribution of $\theta$  and a distribution of $y$ derived  by $F$ from  a sample of the prior distribution of $\theta$  ?  
 A: Let's look at a simple concrete example; it might help focus your questions.
The aim is to start with what you know/understand about the parameters $\mathbf{\theta}$ before you see these data (summarized as a prior) and then update that knowledge with the observed data.
Consider inference on a single parameter, $\mu$ in the model
$f_Y(y) = \frac{1}{\mu} e^{-y/\mu};\, y>0, \, \mu>0$
where $Y_1,Y_2,...Y_n$ are i.i.d draws from $f_Y$. We observe values $y_1,y_2,...y_n$ (in this case, these are our data).
In this case I'll take the easy shortcut of assuming a nice prior so that the calculations are easy. (This is simple convenience, it's not necessary to make such choices.)
Let our prior $p(\mu)={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}\mu^{-\alpha -1}e^{-{\frac {\beta }{\mu}}}$, where $\beta=10$ and $\alpha=2$. This prior has a mean of 10 but its variance isn't finite.
Let $\mathbf{y}=(y_1,y_2,...,y_n)$
\begin{eqnarray}
p(\mu|\mathbf{y}) &\propto& f(\mathbf{y}|\mu)\,\cdot\,p(\mu)\\
 &\propto& \,\prod_i \frac{1}{\mu} e^{-y_i/\mu}\:\cdot\: \mu^{-\alpha -1}e^{-{\frac {\beta }{\mu}}} \\
 &\propto& \,\mu^{-\alpha -1}e^{-{\frac {\beta }{\mu}}} \mu^{-n} e^{-\sum_iy_i/\mu}\\
 &\propto& \,\mu^{-(\alpha+n) -1}e^{-\frac{1}{\mu}({\beta }+\sum_iy_i)}
\end{eqnarray}
We can recognize this as an inverse gamma density with parameters $\alpha+n=n+2$ and ${\beta }+\sum_iy_i=\sum_iy_i+10$. So that's our posterior distribution for $\mu$. As $n$ grows large the posterior becomes more concentrated (and the peak moves slowly toward $\bar{y}$), and less skew.
Of course it's not necessarily just parameters you may be interested in making inferences about; you may be interested in prediction, for example.
In practice you can extract information about the posterior without it necessarily being a distribution you can identify. We then either need some way of evaluating the denominator in Bayes theorem that I pushed into the proportionality constant: $\int_\mu f(\mathbf{y}|\mu)\,p(\mu)\,d\mu$ - perhaps by numerical methods - or we may have some way of avoiding evaluation of that integral altogether.
