Conditional Independence vs. Collinearity I know that when two predictor variables are correlated then that increases error and distorts the results, which is called multicollinearity. But what is conditional independence, is it the same as multicollinearity? Can you please give an example?
 A: Conditional independence means that two random variables are independent given knowledge of another, but not necessarily independent absent that knowledge.
A simple example would be to first roll a die, and then depending on the number that comes up flip two separate coins that number of times and count how many tails you get.  Call the value of the die $X$ and the number of tails from coin one and two $Y_1$ and $Y_2$.  Both $Y_1$ and $Y_2$ are i.i.d. binomial random variables given $X$, but they are not unconditionally independent since knowing the value of one carries information about the value of $X$, which in turn carries information about the other.
In a modeling context it's often assumed that the response variables are conditionally independent given the values of the predictors.  There isn't really much connection between this and multicollinearity as far as I can tell.
A: From Wikipedia:

Multicollinearity refers to a situation in which two or more
  explanatory variables in a multiple regression model are highly
  linearly related

From the definition, there is no involvement of predictor in the correlations among these explanatory variables.

Intuitively, two random variables X and Y are conditionally
  independent given Z if, once Z is known, the value of Y does not add
  any additional information about X.

From the definition, two conditional independent explanatory variables need involvement of predictor variable.
Example: Y = aX1 + bX2 + c
If X1 and X2 are multi-collinear implies they have linear relationship.
    So X1 will increase or decrease as X2 changes, no matter how Y
    changes. 
This definitely means they are not independent. But they may or may not conditional independent on Y. Because independence and conditional independence can not imply each other, see this question.
