Prediction

Outline

Topics

• Prediction using decision trees
• Example

Rationale

Often we do not care so much about “parameters” but instead about predicting future observations.

Example: coins in a bag

Consider the setup from last week with 3 coins and 3 flips with \(Y = (1, 1, 1)\) (in the following, let \(\boldsymbol{1}\) denote a vector of 1’s).

Question: given you have see 3 heads, what is the probability that the next one is also heads?

Mathematically: \(\mathbb{P}(Y_4 = 1 | Y_{1:3} = \boldsymbol{1})\). This is known as “prediction”.

General approach

Key message: In Bayesian statistics, prediction and parameter estimation are treated in the exact same way!

Idea: Add \(Y_4\) to the unobserved random variables, i.e. set \(\tilde X = (X, Y_4)\).

Then, to compute \(\mathbb{P}(Y_4 = 1 | Y_{1:3} = \boldsymbol{1})\) use same techniques as last week (decision tree, chain rule, axioms of probability).

Example, continued

Use the following picture to help you computing \(\mathbb{P}(Y_4 = 1 | Y_{1:3} = \boldsymbol{1})\).