Visualization of Posterior, Likelihood and Prior

Question

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?

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

Answer 1

Yes, that’s correct.

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.