Intuition behind the Posterior
We can estimate the posterior even BEFORE calculating it.
If you remember from a previous post, the posterior is a careful construction of three pieces: likelihood, prior, and denominator.
Turns out, once you don’t really need the denominator to visualize and at least get a “sense” of the posterior’s shape:
Why can we do this?
From the same post linked above, we see that the denominator is the only piece of the posterior sculpture that doesn’t depend on the parameter (θ).
In that case, we can treat the denominator as a sort of “factor” that we multiply to the numerator:
So, if we know the prior and likelihood, we can essentially start pushing away the fog that obscures the behavior of our posterior.
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WARNING: This does NOT mean the data is no longer of use or important. In fact, it is extremely important in our understanding of the shape of the posterior. Let the end of this post be a reminder, but let’s review it here too:
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What affects the posterior?
Answer: prior and data (likelihood)
Here is a table to prime your intuition before we use some visualizations:
Case 1: uniform prior
Case 2: strong prior
Case 3: amount of data increases (Remember: affect of prior decreases)
Main Takeaway: The posterior is proportional to the prior and likelihood. Depending on the strength of the prior and amount of data collected, the posterior’s shape will be affected in predictable ways.
Research done from: “The Posterior- The Goal of Bayesian Inference” A Student’s Guide to Bayesian Statistics, by Ben Lambert, SAGE, 2018.
