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What is independence? A primer in probability.

April 16, 2017

Let’s suppose Bill and Ben each toss separate coins. Let A represent the variable “Bill’s coin toss outcome”, and B represent the variable “Ben’s coin toss outcome”. Both A and B have two possible values (Heads and Tails). It would be uncontroversial to assume that A and B are independent. Evidence about B will not change our belief in A. In other words, the fact that Ben’s coin lands heads does not affect the likelihood that Bill will throw heads. What happens to Bill’s coin and Ben’s coin are unrelated. They are independent.

Now suppose both Bill and Ben toss the same coin. Again let A represent the variable “Bill’s coin toss outcome”, and B represent the variable “Ben’s coin toss outcome”. Assume also that there is a possibility that the coin is biased towards heads but we do not know this for certain. In this case A and B are not independent. Observing that Ben’s coin has landed heads might cause us to increase our belief that Bill will throw a Heads.

In the second example, the variables A and B are both dependent on a separate variable C, “the coin is biased towards Heads” (which has the values True or False). Although in this case A and B are not independent, it turns out that once we know for certain the value of C then any evidence about B cannot change our belief about A.

In such a case we say that A and B are conditionally independent given C.

In many real life situations variables which are believed to be independent are actually only independent conditional on some other variable. Let’s take an example. Suppose that Ted and Ned live on opposite sides of the city and come to work by completely different means. Let’s say Ted arrives by train while Ned drives to work. Let A represent the variable “Ted late” (which has values true or false) and similarly let B represent the variable “Ned late”. At first glance, it might seem that A and B are independent. However, even if Ted and Ned lived and worked in different countries there may be factors (such as an international fuel shortage) which could affect both Ted and Ned. In that case, A and B are not independent. Again, it doesn’t seem reasonable to exclude the possibility that both Ted and Ned may be affected by a rail strike (C). Clearly the likelihood that Ted will arrive late to work will increase if the rail strike takes place; but the likelihood that Ned will arrive late to work might also increase, indirectly, because of the additional traffic on the roads caused by the rail strike. ‘Ted to be late’ and ‘Ned to be late’ are in this case conditionally independent GIVEN the rail strike.

Two events, A and B, are defined to be conditionally independent, given some other event, C, if the probability of both A occurring and B occurring, given some other event, C, is equal to the probability of A occurring given C multiplied by the probability of B occurring given C, i.e.

The notation used for this is: P(AՈB I C) = P(AIC) . P(BIC)

In the example we have just considered, the probability that Ted and Ned are late to work given the train strike equals the probability that Ted is late given the strike multiplied by the probability that Ned is late given the strike.

This takes us to a new question.

Does conditional independence, given C, imply unconditional independence?  

Say, for example, Jack is playing Jill at snooker. Jack and Jill know nothing about each other’s ability at snooker.

Now suppose Jill wins her first 5 games. This provides evidence for her to assess the strength of her opponent, Jack, and vice-versa.

But the games may be conditionally independent (Jill is equally likely to win the fifth game as the second given Jack and Jill’s relative skill at chess).

Even so, they are not independent (that would mean that winning the first five games tells you nothing about the likelihood of winning the sixth).

So the answer to the latest question is No. Conditional independence does not imply unconditional independence.

Finally, does unconditional independence imply conditional independence?

To answer this, let’s imagine an event with multiple causes.

Let A be the event that the fire alarm goes off.

Now suppose this could be caused by a genuine fire (F) or someone making popcorn (P), which sets off a false alarm.

Now let’s suppose that the probability of a fire is completely independent of the probability of someone making popcorn. But also that the probability the alarm is indicating a real fire is 100 per cent if nobody is making popcorn.

So the probability of a fire and the probability of making popcorn are independent of each other, yet the probability it’s a genuine fire if the alarm goes off is conditionally dependent on whether someone is making popcorn (you can be sure it’s a genuine fire if nobody is making popcorn).

So, does unconditional independence imply conditional independence? The answer is No.

So, in summary, events may be independent or they may be conditionally independent. Conditional independence does not, however, imply unconditional independence, and unconditional independence does not imply conditional independence.

 

Further Reading and Links

https://selectnetworks.net

From → Probability

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