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A new market recently appeared on PredictIt, for the chance of Trump winning 50% of the vote in each of the April 26th primaries.

This market seemed overvalued to me. I tried to get a handle on how overvalued it was by calculating the conjunct probability of Trump winning majorities in all five states (explanation of my method here).

In doing this, I assumed that the primaries were statistically independent. My reasoning was that, because they are occurring simultaneously, there is no way they could affect each other, so it's safe to assume independence.

A couple of people have pointed out that assuming this independence is incorrect. I'm realizing that I don't have a good grasp of what independence means.

Some questions I have about this:

  • Why are the primaries not independent? Are they not independent because they are correlated somehow? (i.e. if events are correlated, can they not be independent?)
  • When two variables are influenced by a third, confounding variable, can we assume that the variables are independent because they don't effect each other, even though they correlate?
  • If the primaries are not independent, how can I calculate the conjunct probability that Trump wins >50% in each? Is it unsafe to assume independence for the purpose of getting an estimate? Is there a method for calculating the conjunct probability of dependent events?
  • For future cases, how can I assess whether events are independent? If there isn't a clear mechanism linking two events, should I assume independence?
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Independence is not an easy concept. Independence is an assumption which is very often violated and it is hard to prove. It mostly rely on your knowledge about phenomena your are observing.

  • non-zero correlation directly proves dependence. see Does non-zero correlation imply dependence?

  • There is a concept of conditional independence. Let's say 2 variables (X,Y) are influenced by a third one (Z). You can say that the 2 variables are conditionally independent if when you know the value of Z then knowing the value of one of them (X or Y) do not influence the probability of the other (Y or X). You can look up the mathematical definition.

  • When events are not independent you need to use conditional probabilities and bayes theorem is generally very useful in those cases.

  • Generally it is hard to verify independence but you can use your knowledge about what you are observing and some tests (like correlation or product of the 2 densities...) to get a flavor of how badly the assumption might be violated.

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