# Bayesian Inference III – Probabilistic Predictions

We are sometimes more interested in the probability that an outcome occurs than in the probability of the hypotheses. We then talk about probabilistic predictions.

## Probabilistic Predictions

Probabilistic predictions are when we talk about things like: “It will rain tomorrow with a probability of 60%, and not rain with a probability of 40%”

We often use probabilistic predictions for medical treatment outcomes, sports betting, elections and stock price predictions.

Probabilistic predictions assign each outcome a probability.

Example: Suppose we have four coins of three types in a drawer:

• Type A has probability of 0.5 for tails
• Type B has probability of 0.6 for tails
• Type C has probability of 0.9 for tails

In the drawer are two of type A, one of type B and one of Type C. We then have the following Bayesian Updating Table:

 Hypothesis Prior Likelihood unnormalised Posterior normalised Posterior H P(H) P(D|H) P(D|H)P(H) P(H|D) A 0.5 0.5 0.25 0.4 B 0.25 0.6 0.15 0.24 C 0.25 0.9 0.225 0.36 Total 1 0.625 1

What are the prior and posterior probabilistic predictions?

Answer: We calculate the probabilistic predictions with the Law of Total Probability. We therefore have for the prior and posterior predictive probabilities:

$P(tails)=0.5\cdot 0.5 + 0.25\cdot 0.6 + 0.25 \cdot 0.9 = 0.625$

$P(tails|D) = 0.4\cdot 0.5 + 0.24 \cdot 0.6 + 0.36 \cdot 0.9 = 0.668$

P(heads) and P(heads|D) are just the complement probabilities of P(tails) and P(tails|D), respectively.

Remember: Predictive Probabilities tell us the probabilities for outcomes not for hypotheses.