EE662Sp10BayesExample - Rhea

Example: Let $ \mathbb{X} $ and $ \mathbb{Y} $ be jointly distributed discrete random variables with ranges $ X = \{0, 1, 2, 3, 4\} $ and $ Y = \{0, 1, 2\} $ respectively. Suppose that the conditional distributions $ P_{\mathbb{X}|\mathbb{Y}} $ are empirically estimated as follows:

and the marginal $ P_{\mathbb{Y}} $ is empirically estimated as:

Estimate the conditional distributions $ P_{\mathbb{Y}|\mathbb{X}} $

By definition $ P_{\mathbb{X}|\mathbb{Y}}(x|y) = \frac{P_{\mathbb{X},\mathbb{Y}}(x,y)}{P_{\mathbb{Y}}(y)} $, so the joint distribution $ P_{\mathbb{X},\mathbb{Y}}(x,y) $ can be computed.

$ P_{\mathbb{X},\mathbb{Y}}(0,0) = P_{\mathbb{X}|\mathbb{Y}}(0|0)P_{\mathbb{Y}}(0) = .175 \cdot .63 = .11 $

Computing the rest of the distribution similarly:

The marginal distribution $ P_\mathbb{X} $ can be extracted from the joint distribution as:

$ P_\mathbb{X}(x) = \sum_{y\in Y} P_{\mathbb{X},\mathbb{Y}}(x,y) $

$ P_\mathbb{X}(0) = .11 + .01 + .03 = .15 $

Computing the rest of the distribution similarly:

Finally $ P_{\mathbb{Y}|\mathbb{X}} $ can be computed by definition.

$ P_{\mathbb{Y}|\mathbb{X}}(0|0) = \frac{P_{\mathbb{X},\mathbb{Y}}(0,0)}{P_{\mathbb{X}}(0)} = \frac{.11}{.15} = .733 $

Computing the rest similarly:

Note from these $ P_{\mathbb{Y}|\mathbb{X}} $ distributions that for large $ x $ it is highly probable that $ y=1 $ and for small $ x $ it is highly probable that $ y=0 $.

--Jvaught 22:34, 29 January 2010 (UTC)