# Intuition behind posterior predictive distribution

I've recently encountered the "posterior predictive distribution" $$p(\bar{x}|X)=E_\theta[p(\bar{x}|\theta)]=\int_\theta p(\bar{x}|\theta)\hspace{0.5mm}p(\theta|X)d\theta$$ where $$\bar{x}$$ is a new point, $$\theta$$ is the (vector or 1-D) parameters of the distribution and $$X$$ is the already observed sample.

However I'm not sure I understand where this formula comes from. I know that we don't know the true value of $$\theta$$. I think that since we don't know $$\theta$$, we say "why settle for one estimate of it, and not scan all it's possible values?" Is that correct? Even if it is, I'm not certain that I understand the expected value part.

Sorry if my thoughts are a bit jumbled. Any insight would be appreciated, thanks.

Let $$X$$ denotes the observations and $$\theta \in \Theta$$ the parameter. In a Bayesian approach, both are considered random quantities. The first step of modeling is to define a statistical model, i.e. the distribution of $$X$$ given $$\theta$$, which can be written as $$X \mid \theta \sim p(\cdot \mid \theta)$$. This is mainly done by expliciting a likelihood function.
Thus our statistical model describe the conditional distribution of $$X$$ given $$\theta$$.
From a Bayesian perspective, we also define a prior distribution for $$\theta$$ on $$\Theta$$: $$\theta \sim \pi(\theta)$$.

### The prior predictive distribution

Before observing any data, what we have is simply the chosen model, $$p(x \mid \theta)$$, and the prior distribution of $$\theta$$, $$\pi(\theta)$$. One can then ask to see what is the marginal distribution of $$X$$, that is, the distribution of $$X \mid \theta$$ averaged over all possible values of $$\theta$$.
This can be simply written using expectation: \begin{align*} p(x) &= \mathbb{E}_\theta \Big [ p(x \mid \theta) \Big ] \\ &= \int_\Theta p(x \mid \theta) \pi(\theta) d\theta. \end{align*}

### The posterior predictive distribution

The interpretation is the same than for the prior predictive distribution, is it the marginal distribution of $$X \mid \theta$$ averaged over all values of $$\theta$$.
But this time the "weighting" function to be used is not $$\pi(\theta)$$ but our updated knowledge about $$\theta$$ after observing data $$X^*$$: $$\pi(\theta \mid X^*)$$.
Using the known known Bayes theorem we have: $$\pi(\theta \mid X^*) = \frac{p(X^* \mid \theta) \pi(\theta)}{p(X^*)}$$ And thus, the marginal distribution of $$X \mid (X^*,\theta)$$ averaged over $$\Theta$$ is: $$p(x \mid X^*) = \int_\Theta p(x \mid \theta) \pi(\theta \mid X^*)d\theta$$

## Example: Gamma-Poisson mixture.

Suppose our observations are made of counts, $$X$$, and we define a Poisson model that is: $$X \mid \lambda \sim \mathcal{P}(\lambda)$$.
From a Bayesian perspective, we also define a prior distribution for $$\lambda$$.
For mathematical reasons, it is appealing to use a Gamma distribution, $$\lambda \sim \mathcal{G}(a,b)$$.

### The prior predictive distribution

One particulariy of this Gamma-Poisson mixture, is that the marginal distribution will be distribution as a Negative-Binomial random variable.
That is, if $$X \mid \lambda \sim \mathcal{P}(\lambda)$$ and $$\lambda \sim \mathcal{G}(a,b)$$ then, $$X \sim \mathcal{NB}\big (a,\frac{b}{b+1} \big )$$.
Thus the prior predictive distribution of $$X$$ is a Negative Binomial distribution $$\mathcal{NB}\big (a,\frac{b}{b+1} \big )$$.

### The posterior predictive distribution

Now, say we have observed $$n$$ counts $$X =(X_1,\dots,X_n)$$.
First, thanks to the choice of a Gamma prior for $$\lambda$$, the posterior distribution of $$\lambda$$ can be easily derived as being also a Gamma distribution: $$\lambda \mid X \sim \mathcal{G} \bigg ( a + \sum_{i=1}^n X_i , b+n \bigg)$$

From what we saw for the prior predictive distribution, the posterior predictive distribution of $$X$$ will also be a Negative-Binomial: $$\mathcal{NB} \bigg ( a + \sum_{i=1}^n X_i, \frac{b+n}{b+1+n} \bigg )$$

Here is an example where $$a=100$$, $$b=2$$ and we observe the vector of counts $$X=(85,80,70,65,71,92)$$ : Here is the R code to produce the plot :

### Gamma-Poisson mixture: prior and posterior predictive distributions :
require(ggplot2)
# Parameters of the prior distribution of lambda
a<-100
b<-2
x<-0:150
y1<-dnbinom(x,a,b/(b+1))

# Vector of observations and posterior predictive distribution
X<-c(85,80,70,65,71,92)
n<-length(X)
XS<-sum(X)
x<-0:150
au<-a+XS
bu<-b+n
y2<-dnbinom(x,size=au,prob=bu/(bu+1))

plot1<-ggplot() + aes(x=x,y=y1,colour="Prior") + geom_line(size=1)+
geom_line(aes(x=x,y=y2,colour="Post"))+
scale_colour_manual(breaks=c("Prior",    "Post"),
values=c("#cd7118","#1874cd"),labels=c("Prior Predictive",
"Postererior    Predictive"))+
ggtitle("Prior and posterior predictive distributions for a=100 and b=2")+
labs(x="x", y="Density") +
theme(
panel.background = element_rect(fill = "white",
colour = "white",
size = 0.5, linetype = "solid"),

axis.line = element_line(size = 0.2, linetype = 'solid',
colour = "black"),
axis.text = element_text(size=10),
axis.title = element_text(size=10),
legend.title = element_blank(),
legend.background = element_blank(),
legend.key = element_blank(),
legend.position = c(.7,.5)
)
plot1

• Thank you for the detailed answer. So the idea in the prior predictive is, since we are uncertain about $\theta$, we can predict the $p(x)$ as the average of $p(x|\theta)$ in order to take into account all possible values of $\theta$ right? And the same logic is extended to the posterior predictive distribution. – Thomas Nov 27 '19 at 17:31
• @Thomas Yes that's the idea. If you want to study some probabilities about the upcoming observations (for example for the gamma-poisson, the probability that the next count will be between 60 and 70) it's better to consider all possible values of $\theta$ and average using the probability (or density) of these values of $\theta$ than just using a fixed value for $\theta$. – winperikle Nov 27 '19 at 17:58
• Alright, thanks a lot. Have a good day! And thanks for the code as well. – Thomas Nov 27 '19 at 18:10