Bayesian updating normal distribution
Is Brown so good that she makes nearly every shot, or is she so bad that she misses nearly every shot?This prior says that Brown’s shooting rate is probably near the extremes, which may not necessarily reflect a reasonable belief for someone who is a college basketball player, but it has the benefit of having less influence on the posterior estimates than the uniform prior (since it is equal to 1 prior observation instead of 2).
I'm assuming that they're normally distributed because most things done by humans tend to be normally distributed.This would mean I could reasonably use the common Beta(1, 1) prior, which represents a uniform density over [0, 1].In other words, all possible values for Brown’s shooting percentage are given equal weight before taking data into account, because the only thing I know about her ability is that both outcomes are possible (Lee & Wagenmakers, 2005).If you had normal data you could use a normal prior and obtain a normal posterior.Conjugate priors are not required for doing bayesian updating, but they make the calculations a lot easier so they are nice to use if you can.I outlined the basic idea behind likelihoods and likelihood ratios.
Likelihoods are relatively straightforward to understand because they are based on tangible data.
The data are counts, so I’ll be using the binomial distribution as a data model (i.e., the likelihood. Her results were the following: Round 1: 13/25 Round 2: 12/25 Round 3: 14/25 Round 4: 19/25 Total: 58/100 The likelihood curve below encompasses the entirety of statistical evidence that our 3-point data provide (footnote 1).
The hypothesis with the most relative support is .58, and the curve is moderately narrow since there are quite a few data points.
Collect your data, and then the likelihood curve shows the relative support that your data lend to various simple hypotheses.
Likelihoods are a key component of Bayesian inference because they are the bridge that gets us from prior to posterior.
Another common prior is called Jeffreys’s prior, a Beta(1/2, 1/2) which forms a wide bowl shape.