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.

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