Can you use Bayes theorem to transform a likelihood function into a probability of the parameters, given the data? Example 1 of the wikipedia article on the Likelihood Function suggests that given the likelihood function $\mathcal{L}(\theta | X)$, we can find the probability of $\theta$ given $X$, $p(\theta|X)$.
Is this just a confusion of notation? That is, some of the literature likes to write $\mathcal{L}(\theta|X) = p(X|\theta)$ even though the likelihood is $\textit{not}$ a conditional probability distribution, and is perhaps more aptly notated as $p(X;\theta)$.
 A: 
Let ${\displaystyle p_{\text{H}}}$ be the probability that
  a certain coin lands heads up (H) when tossed. So, the probability
  of getting two heads in two tosses (HH) is ${\displaystyle
  p_{\text{H}}^{2}}$. If ${\displaystyle
  p_{\text{H}}=0.5}$, then the probability of seeing two heads is 0.25:
$${\displaystyle P({\text{HH}}\mid p_{\text{H}}=0.5)=0.25.}$$
With this, we can say that the likelihood of ${\displaystyle
  p_{\text{H}}=0.5}$, given the observation HH, is 0.25, that is
$${\displaystyle {\mathcal {L}}(p_{\text{H}}=0.5\mid
  {\text{HH}})=P({\text{HH}}\mid p_{\text{H}}=0.5)=0.25.}$$
This is not the same as saying that the probability that
  ${\displaystyle p_{\text{H}}=0.5}$, given the observation HH, is
  $0.25$. For that, we could apply Bayes' theorem, which implies that
  the posterior probability (density) is proportional to the likelihood
  times the prior probability. [Wikipedia Example 1 on Likelihood
  function]

This Wikipedia Example states exactly what it should: 
the likelihood (function) $$\mathcal{L}(\theta|x)$$ as a function of $\theta$ indexed by the realised observation $x$, takes an image value at a particular value of the parameter (like $\theta={\displaystyle p_{\text{H}}=0.5}$) that is the value of the sampling distribution (pmf or pdf) at the observed sample for that value of the parameter $$p(x|\theta).$$ 
The final paragraph is a proper warning that a likelihood value or function is in general not a probability value or density/mass function on the parameter. To turn the likelihood function into a density function, the parameter space needs to be endowed with a probability structure, including a prior distribution/measure, which turns the sampling probability density into a conditional probability density.
The last sentence could always be turned into something clearer, like

For producing a probability statement on a value of the parameter, one
  needs to consider this parameter as a random variable, which requires
  a probability measure on the parameter space, called a prior
  distribution. With this preliminary, one applies Bayes' theorem,
  defining the posterior probability (density) on the parameter as
  proportional to the likelihood times the prior probability.

A: Yes you can. The distribution function you get when you do that is called the posterior distribution function. You need to specify a marginal distribution for the parameter though, which is called the prior distribution. 
A: I think wiki uses a confusing notation and term. It uses likelihood to distinguish it from posterior probability.
If we write down the full Bayes'r rule, it should be as:
$P(p_H=0.5|HH) = \frac{P(HH|p_H=0.5)*P(p_H=0.5)}{p(HH)}$.
In Bayesian inference, $P(HH|p_H=0.5)$ is always referred to as likelihood where as $P(p_H=0.5)$ is referred to as prior and $P(p_H=0.5|HH)$ is referred as posterior probability. 
Given the formula above, obviously wiki cannot claim $P(p_H=0.5|HH) = P(HH|p_H=0.5)$ so I guess that is why it uses likelihood here.
