ax.plot(bins[:-1],counts, alpha = .5) line1, line2, line3 = ax.lines ax.legend((line2, line3), ('Likelihood of Theta for New Campaign' , 'Frequency of Theta Historically') , loc = 'upper left') ax.set_xlabel("Theta") ax.grid() ax.set_title("Evidence vs Historical Click Through Rates") plt.show() Clearly, the maximum likelihood method is giving us a value that is outside what we would normally see.Perhaps our analysts are right to be skeptical; as the campaign continues to run, its click-through rate could decrease.The denominator simply asks, "What is the total plausibility of the evidence?
Prior distributions reflect our beliefs before seeing any data, and posterior distributions reflect our beliefs after we have considered all the evidence.
Alternatively, this campaign could be truly outperforming all previous campaigns. Ideally, we would rely on other campaigns' history if we had no data from our new campaign.
And as we got more and more data, we would allow the new campaign data to speak for itself.
Because we are considering unordered draws of an event that can be either 0 or 1, we can infer the probability and pick the value that is most aligned with the data.
This is known as maximum likelihood, because we're evaluating how likely our data is under various assumptions and choosing the best assumption as true.
This post is an introduction to Bayesian probability and inference.