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Model specification

Now that we have the data, we need to specify the model. Remember that this is done by specifying the likelihood and the prior using probability distributions. For the likelihood, we will use the binomial distribution with and , and for the prior, a beta distribution with the parameters . A beta distribution with such parameters is equivalent to a uniform distribution in the interval [0, 1]. We can write the model using mathematical notation:

This statistical model has an almost one-to-one translation to PyMC3:

with pm.Model() as our_first_model:
    θ = pm.Beta('θ', alpha=1., beta=1.)
    y = pm.Bernoulli('y', p=θ, observed=data)
    trace = pm.sample(1000, random_seed=123)

The first line of the code creates a container for our model. Everything inside the with-block will be automatically added to our_first_model. You can think of this as syntactic sugar to ease model specification as we do not need to manually assign variables to the model. The second line specifies the prior. As you can see, the syntax follows the mathematical notation closely.

Please note that we use the name θ twice, first as a Python variable and then as the first argument of the Beta function; using the same name is a good practice to avoid confusion. The θ variable is a random variable; it is not a number, but an object representing a probability distribution from which we can compute random numbers and probability densities.

The third line specifies the likelihood. The syntax is almost the same as for the prior, except that we pass the data using the observed argument. This is the way in which we tell PyMC3 that we want to condition for the unknown on the knows (data). The observed values can be passed as a Python list, a tuple, a NumPy array, or a pandas DataFrame.

Now, we are finished with the model's specification! Pretty neat, right?