Terms

• Probability: The chance of the occurrence of an event
• Probability Mass: The chance of the occurrence of an random discrete event
• Probability Density: The probability that a continues event variable is exactly equal to some value
• Probability Mass Function (PMF): A function which gives the probability that a discrete random variable is exactly equal to some value.
• Probability Density Function (PDF): A function which gives the probability that a continues random variable is exactly equal to some value.

Predictive Distribution

Now with Bayesian Model, we can get the prior/posterior probability distribution of an extra sample $\tilde y$.

$$ P(\tilde y | y,n,M) = \int_{0}^{1} P(\tilde y | \theta, y, n, M) P(\theta | y, n, M) d\theta \ = \int_{0}^{1} \theta P(\theta | y, n, M) d\theta \ = E(\theta | y) $$

where $n$ represents the number of experiment that have been taken already, $y$ stands for the event, and $M$ denotes the prior model we assume.

With a binomial prior $M$, we can have:

$$E(\theta | y) = \frac{y+1}{n+2}$$