# 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.  Bayesia USA
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