As a prior, we will use a beta distribution, which is a very common distribution in Bayesian statistics and looks as follows:

If we look carefully, we will see that the beta distribution looks similar to the binomial except for the first term, the one with all those . The first term is a normalizing constant that ensures the distribution integrates to 1, and  is the Greek uppercase gamma letter and represents what is known as gamma function. We can see from the preceding formula that the beta distribution has two parameters,  and . Using the following code, we will explore our third distribution so far:

params = [0.5, 1, 2, 3]
x = np.linspace(0, 1, 100)
f, ax = plt.subplots(len(params), len(params), sharex=True, 
                     sharey=True,
                     figsize=(8, 7), constrained_layout=True)
for i in range(4):
    for j in range(4):
        a = params[i]
        b = params[j]
        y = stats.beta(a, b).pdf(x)
        ax[i,j].plot(x, y)
        ax[i,j].plot(0, 0, label="α = {:2.1f}
β = {:2.1f}".format(a, 
                     b), alpha=0)
        ax[i,j].legend()
ax[1,0].set_yticks([])
ax[1,0].set_xticks([0, 0.5, 1])
f.text(0.5, 0.05, 'θ', ha='center')
f.text(0.07, 0.5, 'p(θ)', va='center', rotation=0)

I really like the beta distribution and all the shapes we can get from it, but why are we using it for our model? There are many reasons to use a beta distribution for this and other problems. One of them is that the beta distribution is restricted to be between 0 and 1, in the same way our  parameter is. In general, we use the beta distribution when we want to model proportions of a binomial variable. Another reason is for its versatility. As we can see in the preceding figure, the distribution adopts several shapes (all restricted to the [0, 1] interval), including a uniform distribution, Gaussian-like distributions, and U-like distributions. As a third reason, the beta distribution is the conjugate prior of the binomial distribution (which we are using as the likelihood). A conjugate prior of a likelihood is a prior that, when used in combination with a given likelihood, returns a posterior with the same functional form as the prior. Untwisting the tongue, every time we use a beta distribution as the prior and a binomial distribution as the likelihood, we will get a beta as the posterior distribution. There are other pairs of conjugate priors; for example, the Normal distribution is the conjugate prior of itself. For many years, Bayesian analysis was restricted to the use of conjugate priors. Conjugacy ensures mathematical tractability of the posterior, which is important given that a common problem in Bayesian statistics is ending up with a posterior we cannot solve analytically. This was a deal breaker before the development of suitable computational methods to solve probabilistic methods. From Chapter 2, Programming Probabilistically onwards, we will learn how to use modern computational methods to solve Bayesian problems, whether we choose conjugate priors or not.