Monte Carlo Approximation

Okay, so we are going to start with the Monte Carlo approximation to the PPD. We learned the steps of Monte Carlo approximation last time in terms of the Beta-Binomial. Now, let’s see if we can apply these to the Poisson-Gamma.

And now…we need to pause.

Considerations with the Poisson-Gamma

When we got to this point with the Beta-Binomial, the next step was to determine the proportion of our samples \(y_{j}^{*}\) that were equal to 0 and equal to 1. In other words

\[P(Y^* =1 |y_1,y_2,...y_n) \approx \frac{1}{s} \sum_{j=1}^{s} y^{*}_{j}\]

\[P(Y^* =0 |y_1,y_2,...y_n) \approx 1- \frac{1}{s} \sum_{j=1}^{s} y^{*}_{j}\]

With these two probabilities, we could approximate the PPD with a Bernoulli distribution with probability of success \(\frac{1}{s} \sum_{j=1}^{s} y^{*}_{j}\).

\[P(Y^* |Y_i =y_1,Y_2 = y_2,...Y_n = y_n) \sim Bernoulli \left(\frac{1}{s} \sum_{j=1}^{s} y^{*}_{j} \right)\]

However, in our current data example we have a Poisson random variable \(Y^{*}\) rather than Bernoulli random variable \(Y^{*}\). This means that instead of just two unique outcome values \(y^{*}\), the \(Y^{*}\) can be 0, 1, 2, 3, and so on. Accordingly, our PPD is a not defined by just two probabilities.

This means that we want to look at (1) what different values for \(y^{*}\) we sampled and (2) how often we drew each of those values. This will help us determine the characteristics of the PPD.

This is very different from what we had before! Remember that the “D” in PPD stands for “distribution”. What we are looking at is an approximation of the pmf for the PPD \(Y^{*}|Y_1 = y_1, \dots, Y_n = y_n\).

What can we do with this approximation? One thing we can do is use this to determine which values of \(y^{*}\) are likely to occur. We are trying to predict \(Y^{*}\) for a new puppy, and we are now looking at a distribution that tells what values of \(Y^{*}\) we are likely to see for that puppy.

We can also use the PPD approximation how likely it is that \(Y^{*}\) would equal a specific value of \(y^{*}\).

This allows us to evaluate how likely it is that this new puppy would be adopted in a certain number of days. This means that with the Monte Carlo approximation of the PPD, we can (1) determine a posterior probability that \(Y^{*}\) is equal to a specific number of days and (2) determine which values of \(Y^{*}\) have the highest posterior probability.

Two Posteriors - Considerations

At this point in our process, you have probably noticed that we have two posterior distributions we are working with. The first is the posterior distribution for \(\theta\), \[\theta||Y_1 = y_1, \dots, Y_n = y_n\]

and a posterior predictive distribution for \(Y^{*}\),

\[Y^{*}|Y_1 = y_1, \dots, Y_n = y_n\]

Let’s make sure we are clear on the difference between these two posterior distributions.

Turning in your assignment

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