13
Oct 18

Law of iterated expectations: informational aspect

Law of iterated expectations: informational aspect

The notion of Brownian motion will help us. Suppose we observe a particle that moves back and forth randomly along a straight line. The particle starts at zero at time zero. The movement can be visualized by plotting on the horizontal axis time and on the vertical axis - the position of the particle. W(t) denotes the random position of the particle at time t.

Unconditional expectation

Figure 1. Unconditional expectation

In Figure 1, various paths starting at the origin are shown in different colors. The intersections of the paths with vertical lines at times 0.5, 1 and 1.5 show the positions of the particle at these times. The deviations of those positions from y=0 to the upside and downside are assumed to be equally likely (more precisely, they are normal variables with mean zero and variance t).

Unconditional expectation

“In the beginning there was nothing, which exploded.” ― Terry Pratchett, Lords and Ladies

If we are at the origin (like the Big Bang), nothing has happened yet and EW(t)=0 is the best prediction for any moment t>0 we can make (shown by the blue horizontal line in Figure 1). The usual, unconditional expectation EX corresponds to the empty information set.

Conditional expectation

Conditional expectation

Figure 2. Conditional expectation

In Figure 2, suppose we are at t=2. The dark blue path between t=0 and t=2 has been realized. We know that the particle has reached the point W(2) at that time. With this knowledge, we see that the paths starting at this point will have the average

(1) E(W(t)|W(2))=W(2), t>2.

This is because the particle will continue moving randomly, with the up and down moves being equally likely. Prediction (1) is shown by the horizontal light blue line between t=2 and t=4. In general, this prediction is better than EW(t)=0.

Note that for different realized paths, W(2) takes different values. Therefore E(W(t)|W(2)), for t<2, is a random variable. It is a function of the event we condition the expectation on.

Law of iterated expectations

Law of iterated expectations

Figure 3. Law of iterated expectations

Suppose you are at time t=2 (see Figure 3). You send many agents to the future t=3 to fetch the information about what will happen. They bring you the data on the means E(W(t)|W(3)) they see (shown by horizontal lines between t=3 and t=4). Since there are many possible future realizations, you have to average the future means. For this, you will use the distributional belief you have at time t=2. The result is E[E(W(t)|W(3))|W(2)]. Since the up and down moves are equally likely, your distribution at time t=2 is symmetric around W(2). Therefore the above average will be equal to E(W(t)|W(2)). This is the Law of Iterated Expectations, also called the tower property:

(2) E[E(W(t)|W(3))|W(2)]=E(W(t)|W(2)).

The knowledge of all of the future predictions E(W(t)|W(3)), upon averaging, does not improve or change our current prediction E(W(t)|W(2)).

For a full mathematical treatment of conditional expectation see Lecture 10 by Gordan Zitkovic.

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