## 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. denotes the random position of the particle at time .

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

### Unconditional expectation

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

If we are at the origin (like the Big Bang), nothing has happened yet and

### Conditional expectation

In Figure 2, suppose we are at

(1)

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

Note that for different realized paths,

### Law of iterated expectations

Suppose you are at time **Law of Iterated Expectations**, also called the **tower property**:

(2)

The knowledge of all of the future predictions

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

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