Pearson Correlation

Context

• In BayesiaLab's approach to learning and analyzing Bayesian networks, statistical concepts play a secondary role compared to concepts from the field of Information Theory.

• Nevertheless, statistical measures, such as correlation, can provide certain insights that are unavailable from non-statistical measures.

Definition

The Pearson Correlation Coefficient $r$ between two nodes $X$ and $Y$ is defined as the covariance of the two corresponding variables divided by the product of their standard deviations:

$$R = \frac{{{\mathop{\rm cov}} (X,Y)}}{{{\sigma _X}{\sigma _Y}}}$$

Where the covariance is defined by:

$${\mathop{\rm cov}} (X,Y) = \sum\limits_{x,y} {p(x,y) \times ({V_x} - {v_X})} \times ({V_y} - {v_Y})$$

And the standard deviation:

$${\sigma _X} = \sqrt {{{\sum\limits_x {{p_x} \times ({V_x} - {v_X})} }^2}}$$

• ${{V_x}}$ is the value that is associated with the state $x$.

• ${{v_X}}$ is the Expected Value of the node $X$

• ${{p_x}}$ is the marginal probability of state $x$ returned by the Bayesian network

• ${p(x,y)}$ is the joint probability of states $x$ and $y$ returned by the Bayesian network

Special Considerations

• For calculating the Pearson Correlation $R$, BayesiaLab must use the values of node states.

• In BayesiaLab, there are Discrete Nodes and Continuous Nodes with discretized numerical states. As a result, the value of a node's state may not always be apparent:

  • For Discrete Nodes that have states with integer or real values, BayesiaLab uses these numerical values directly.

  • For Discrete Nodes that have states without values, e.g., {red, green, blue}, BayesiaLab uses the indices of the states as values, i.e., {red, green, blue} would have the values {0, 1, 2} for the purpose of calculating $R$. Note that the index of states starts at 0.

  • For Continuous Nodes, BayesiaLab uses these mean values of each interval.

• Please see Mean, Value, and Standard Deviations for a detailed discussion.