Entropy of a Discrete Probability Distribution

Supposed we have a discrete set of possible events $$1,\ldots, n$$ that occur with probabilities $$(p_1, p_2, \ldots, p_n)$$. Claude Shannon asked the question

Can we find a measure of how much "choice" is involved in the selection of the event or of how uncertain we are of the outcome?

For example, supposed we have coin that lands on heads with probability $$p$$ and tails with probability $$1-p$$. If $$p=1$$, the coin always lands on heads. Since there is no uncertainty, we might want to say the uncertainty is 0. However, if the coin is fair and $$p=0.5$$, we maximize our uncertainty: it's a complete tossup whether the coin is heads or tails. We might want to say the uncertainty in this case is 1.

In general, Shannon wanted to devise a function $$H(p_1, p_2, \ldots, p_n)$$ describing the uncertainty of an arbitrary set of discrete events (i.e. a $$n$$-sided die). He thought that "it is reasonable" that $$H$$ should have three properties:

1. $$H$$ should be a continuous function of each $$p_i$$. A small change in a single probability should result in a similarly small change in the entropy (uncertainty).
2. If each event is equally likely ($$p_i=\frac{1}{n}$$), $$H$$ should increase as a function of $$n$$: the more events there are, the more uncertain we are.
3. Finally, entropy should be recursive with respect to independent events. Supposed we generate a random variable $$x$$ by the following process: Flip a fair coin. If it is heads, $$x=0$$. If the flip was tails, flip the coin again. If the second flip is heads, $$x=1$$, if tails $$x=2$$.

We can compute the entropy as $$H(p_0=1/2, p_1=1/4, p_2=1/4)$$. However, the independence property tells us that this relationship should hold:

$H\left(\frac{1}{2}, \frac{1}{4}, \frac{1}{4}\right)=H\left(\frac{1}{2}, \frac{1}{2}\right) + \frac{1}{2} H\left(\frac{1}{2}, \frac{1}{2}\right).$

As David MacKay explains, this is the general claim that

$H(\mathbf{p})=H(p_1, 1-p_1)+(1-p_1)H\left(\frac{p_2}{1-p_1}, \frac{p_3}{1-p_1}, \ldots, \frac{p_n}{1-p_1}\right).$

Shannon showed that given these three assumptions, there is unique for that $$H$$ must take:

$H\propto -\sum_{i=1}^n p_i \log p_i=\sum_{i=1}^n p_i \log \frac{1}{p_i}.$

He named this measure of uncertainty entropy, because the form of $$H$$ bears striking similarity to that of Gibbs Entropy in statistical thermodynamics.1

Shannon observes that $$H$$ has many other interesting properties:

1. Entropy $$H$$ is 0 if and only if exactly one event has probability 1 and the rest have probability 0. (Uncertainty vanishes only when we are certain about the outcomes.)
2. Entropy $$H$$ is maximized when the $$p_i$$ values are equal.
3. The joint entropy of two events is less than or equal the sum of the individual entropies. $$H(x, y)=H(x)+H(y)$$ only when $$x$$ and $$y$$ are independent events.