Definition of the bias of an estimator I'm quite confused about the definition of the bias of an estimator.

Suppose we have unknown distribution $P(x, \theta)$, and construct the
estimator $\hat{\theta}$ that maps the observed data sample to values
that we hope are close to $\theta$. Then the bias of an estimator is
defined by:
$$Bias(\hat{\theta}) = \mathbb{E}(\hat{\theta}) - \theta ,$$

How can we calculate the $Bias(\hat{\theta})$ if we know neither the parent distribution $P(x, \theta)$ nor the true value of parameter  $\theta$?
Is there a need for any assumptions?

upd: suppose I've proved, that estimator of the mean ($\hat{\mu}=\frac{1}{m}\sum_{i=1}^mx_i$) is unbiased for the Gaussian distribution, but is there any distribution for which the same estimator is biased? Or is it possible to prove such an estimator is unbiased for any distribution?
I guess such a question is caused by the lack of intuition about estimator and its bias. May be there is some convenient way to think about it?

Think of the bias as "if I repeat my experiment many times and apply my estimator each time, how much does the average estimation result differ from the true result?"
For example, the sample variance
$$
\hat{\sigma}_Y^2 = \frac{1}{N}\sum_{i=1}^N (Y_i-\bar{Y})^2 = \frac{N-1}{N} \sigma_Y^2
$$
is a biased estimator of the variance of a distribution, which means that on average over many repeated experiments it will under-estimate the true variance $\sigma_Y^2$. This does not mean that it will under-estimate it every single time.
To assess the biasedness or unbiasedness of the estimator, one will usually make some assumptions about $P(x, \theta)$. E.g., the derivation of the biasedness of the sample variance assumes independent and identically distributed (i.i.d.) measurements. The magnitude of the bias can in general depend on many factors such as the sample size $N$ and the true parameter vector $\theta$.
To put it another way, the bias of an estimator is something that is usually analyzed theoretically under specific assumptions. Therefore, one can assume full knowledge of the true parameters $\theta$ for the purpose of the derivation. In the above case of the sample variance, the bias is easy to correct (just multiply the resulting estimate by $\frac{N}{N-1}$), but in general this may not be possible, e.g., if the bias depends on the true parameter vector $\theta$.
As a sidenote, in many cases it may be desirable to use a biased estimator because, e.g., the mean squared error of a biased estimator may be lower than the mean squared error of an unbiased estimator. See, e.g., the bias-variance trade-off. One classical example is the James-Stein estimator, which is biased but has a lower MSE than ordinary least squares, which is unbiased.
