How is this connection between Beta and Binomial possible?

Question

I'm learning stats, and I've reasoned about this for many days and I'm losing sleep over it.

Say I have a binomial distribution B(k; n, p) simulating a hypothesis p value for some events:

B(14; 41, 0.5)

In R:

dbinom(14, 41, 0.5) #=> 0.016

So in English, this could be said as "the probability of exactly 14 successes in a series of 41 fifty-fifty events is about 1.6%. OK, makes sense (a poor hypothesis).

Now, lets try iterating over different p values using R:

p <- seq(0, 1, by = 0.01)
qplot(
  x = p,
  y = dbinom(14, 41, p)
)

This is clearly starting to form a shape of the probabilities for different probability values:

I understand this is not the "proper" way to do a beta distribution because of the fact that probability is continuous, and this graph needs to be integrated, it doesn't add up to 1, etc.

With that said... it seems the above chart IS perfectly "accurate", because each single value by itself is accurate.

Question 1: how IS the graph above "different" than a proper beta distribution in terms of what they describe in English?

• Incorrect beta distribution is saying (I think...) "For these 100 exact values, chart the probability (Y) that each probability (X) is correct"
• Correct beta distribution is saying (I think...) "For all possible values, chart the probability density (Y) that each probability (X) is correct"

Question 2: Are the charts related somehow? Like can they convert between them? It's clear from the incorrect chart I CAN compute a probability of a probability in a non-continuous way, yet in the second chart it appears there's no way I can do this unless I say something like p > 0.49 & p < 0.51). How can both charts be "correct" (to my knowledge) yet in the continous one the probability of choosing that item is 0, and in the non-continous chart I can compute it just fine?

qplot(
  x = p,
  y = dbeta(p, 14, 41)
)

Answer 1

The function you have plotted is the kernel of a beta density function (i.e., it is a positive multiple of the beta density). Since you have really just plotted the binomial likelihood function for a particular observed outcome, from a Bayesian perspective your plotted function is proportionate to the posterior density that emerges from using a uniform prior for $p$. In Bayesian modelling terms, what you have illustrated here is that:

$$\underbrace{\text{Beta}(p | 15, 28)}_\text{Posterior} \ \propto \ \underbrace{\text{Bin}(14|41, p)}_\text{Likelihood} \times \underbrace{\text{U}(p|0,1)}_\text{Prior}.$$

(Note also that your last plot just appears to be a repitition of the first plot. Your code for the beta distribution uses the wrong parameters; see the parameterisation of the beta distribution to see why the parameters should be as shown here.) To plot the posterior density in this problem we can use the following R code:

#Plot posterior density
library(ggplot2)
XX <- seq(0, 1, by = 0.001)
DD <- dbeta(XX, shape1 = 15, shape2 = 28)
qplot(XX, DD, geom = 'line') + xlab('p') + ylab('Posterior Density')