I placed to have a prior at p is equals to 0.5, a prior probability of 52% and

equally divided the remaining probability among the other models.

This equal distribution implies that the benefit of the treatment is symmetric,

that the treatment is equally likely to be better or

worse than the standard treatment.

And the 52% prior at peak equals 0.5 implies that we believe

that there's a 52% chance that there is no difference between the treatments.

One natural question that you might have at this point is how

did you come up with those priors?

We will discuss prior specification in detail later in the course, so for

now let's stick with the chosen priors and work through

the mechanics of calculating the posterior probabilities and making a decision.

Now we're ready to calculate the probability of observed data,

given each of the models that we're considering.

This probability is called the likelihood.

In this example, this is simply the probability of the data, given the model.

Which can be written as the probability that k is equal to 4,

given that n is equal to 20, and the various values of p we decided to

consider as plausible models, 10% through 90%.

As we did in the previous video,

we can express the probability of a given number of successes

in a given number of independent trials with a binomial distribution.

We consider a sequence of probabilities of success from 10% to 90%,

increasing by 10%, we assign a 52% prior probability to p equals 0.5,

and 6% probabilities to all other models.

We won't actually use these prior probabilities in the calculation of

the likelihood, but they will become relevant for

the calculation of the posterior in the next slide.

Finally, we can calculate the likelihood as a binomial with four successes and

20 trials, when p is equal to the variety of values we're considering.

The results are summarized in this table.

The header row lists the models that we're considering, and in the next row,

the priors we discussed earlier are shown.

The last row of the table lists

the likelihood calculated using the binomial distribution.

The number of successes and the number of trials are the same for

each of these likelihoods, four and 20 respectively.

However, each likelihood listed uses a different probability of success

based on which model is based on.