Expectations (16/365)

Sometimes expectations and random variables appear in so many ways that I find it a little confusing at times. The book I am following has a neat diagram that helps quite a bit.

Expectations

Expectations

Basic Expectation

Start with a distribution P(\cdot) for sample space \Omega and a random variable \theta. The basic expectation tells us the value \theta is likely to take on average.

\displaystyle  E\theta = \sum_\omega \theta(\omega) P(\omega) \\ \text{Notice that } E\theta = E[\theta I_\Omega]

Note that \theta induces a decomposition D_1, \dots, D_n of \Omega as follows

\displaystyle  E\theta = \sum_{i=1}^n x_i P(D_i) \text{ where } D_i = \{ \omega | \theta(\omega) = x_i \}

Conditional Expectation

Instead of P(\cdot) we could be given a distribution P(\cdot | D) with respect to an event D. The expectation over this is the value \theta is likely to take on average conditioned on the event D (i.e. restricted to).

\displaystyle  E(\theta | D) = \sum_\omega \theta(\omega) P(\omega | D) \\ \text{Notice that } E(\theta | D) = \frac{E[\theta I_D]}{P(D)}

Let’s suppose that we have an event A and its conditional probabilities P(A | D_1), \dots, P(A | D_n) where \{ D_i \} is a decomposition of \Omega. We can write this as a random variable that takes on the value P(A | D_i) whenever \omega \in D_i.

\displaystyle  P(A | \mathcal{D})(\omega) = \sum_{i=1}^n P(A | D_i) I_{D_i}(\omega)

Now that we have this random variable, we can once again take its expectation which in this case gives the probability of A on average conditioned on an event from \mathcal{D}. This has a special name and is called the total probability.

\displaystyle  E P(A | \mathcal{D}) = \sum_{i=1}^n P(A | D_i) P(D_i) = P(A)

One More Generalization

Suppose now that we have a random variable \theta inducing the decomposition A_1, \dots, A_m where A_j = \{ \omega | \theta(\omega) = x_j \}. We also have conditional probabilities P(A_j | D_i). We can certainly take the following expectation from before

\displaystyle  E(\theta | D_i) = \sum_j x_j P(A_j | D_i)

which is the expectation of \theta conditioned on D_i. We now do this for the entire decomposition \mathcal{D} to arrive at this random variable

\displaystyle  E(\theta | \mathcal{D})(\omega) = \sum_i E(\theta | D_i) I_{D_i}(\omega)

which takes on a conditional expectation of \theta at each \omega. We can now generalize the total probability formula such that it gives the the expectation of \theta on average conditioned on an event from \mathcal{D}.

\displaystyle  E(A | \mathcal{D}) = P(A) \\ \text{generalizes to} \\ EE(\theta | \mathcal{D}) = E\theta

The lesson here is to always translate the things we do with probabilities to expectations.

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One Response to Expectations (16/365)

  1. Pingback: Conditional Probability – Problem (19/365) | Latent observations

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