De Finetti's Theorem - Example

Example

Here is a concrete example. Suppose p = 2/3 with probability 1/2 and p = 9/10 with probability 1/2. Suppose the conditional distribution of the sequence

given the event that p = 2/3, is described by saying that they are independent and identically distributed and X1 = 1 with probability 2/3 and X1 = 0 with probability 1 − (2/3). Further, the conditional distribution of the same sequence given the event that p = 9/10, is described by saying that they are independent and identically distributed and X1 = 1 with probability 9/10 and X1 = 0 with probability 1 − (9/10). The independence asserted here is conditional independence, i.e., the Bernoulli random variables in the sequence are conditionally independent given the event that p = 2/3, and are conditionally independent given the event that p = 9/10. But they are not unconditionally independent; they are positively correlated. In view of the strong law of large numbers, we can say that

\lim_{n\rightarrow\infty} \frac{X_1+\cdots+X_n}{n} = \begin{cases} 2/3 & \text{with probability }1/2, \\ 9/10 & \text{with probability }1/2. \end{cases}

Rather than concentrating probability 1/2 at each of two points between 0 and 1, the "mixing distribution" can be any probability distribution supported on the interval from 0 to 1; which one it is depends on the joint distribution of the infinite sequence of Bernoulli random variables.

The conclusion of the first version of the theorem above makes sense if the sequence of exchangeable Bernoulli random variables is finite, but the theorem is not generally true in that case. It is true if the sequence can be extended to an exchangeable sequence that is infinitely long. The simplest example of an exchangeable sequence of Bernoulli random variables that cannot be so extended is the one in which X1 = 1 − X2 and X1 is either 0 or 1, each with probability 1/2. This sequence is exchangeable, but cannot be extended to an exchangeable sequence of length 3, let alone an infinitely long one.

Read more about this topic:  De Finetti's Theorem

Famous quotes containing the word example:

    “Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
    —Marcel Proust (1871–1922)