# Visualization of Posterior, Likelihood and Prior

I am confused by the visualizations of the likelihood, prior and posterior distribution that I usually see when the Bayes' theorem is explained. An example is the image below:

The x-axis shows the parameter $$\theta$$ and the y-axis represents the density. The definition of the distributions is the following:

Likelihood = P(Data | $$\theta$$)
Prior = P($$\theta$$)
Posterior = P($$\theta$$ | Data)

Given these definitions, I understand the Prior and Posterior plot (since we are visualizing the distribution of the parameters), but the plot of the likelihood distribution shown above is trickier. I understand the plot of the likelihood distribution where the x-axis shows the data. In the case shown above, if we assume that: (1) the likelihood = 0.8 when $$\theta$$ = 0.5, (2) the likelihood is Normally distributed and (3) the data is i.i.d, is it correct to say the following?

$$\prod_i^N \mathcal{N}(x_i \ \ | \ \ \theta=0.5, \sigma) = 0.8$$

Remember that the likelihood function varies the parameters, not the data. (See What is the difference between "likelihood" and "probability"?) The likelihood is a function of $$\theta$$ that assumes $$\text{Data}$$ is fixed, i.e.
$$L(\theta \mid \text{Data}) = p(\text{Data} \mid \theta)\text{.}$$
What you have is now three different functions of $$\theta$$: the prior, the likelihood, and the posterior. They’re able to be plotted on the same axes.
In the example you give, 0.8 is the value of the likelihood function at $$\theta=0.5$$. In other words, the likelihood of $$\theta=0.5$$ is 0.8.