Likelihood Function - Example 1

Example 1

Let be the probability that a certain coin lands heads up (H) when tossed. So, the probability of getting two heads in two tosses (HH) is . If, then the probability of seeing two heads is 0.25.

In symbols, we can say the above as:

Another way of saying this is to reverse it and say that "the likelihood that, given the observation HH, is 0.25"; that is:

But this is not the same as saying that the probability that, given the observation HH, is 0.25.

Notice that the likelihood that, given the observation HH, is 1. But it is clearly not true that the probability that, given the observation HH, is 1. Two heads in a row hardly proves that the coin always comes up heads. In fact, two heads in a row is possible for any .

The likelihood function is not a probability density function. Notice that the integral of a likelihood function is not in general 1. In this example, the integral of the likelihood over the interval in is 1/3, demonstrating that the likelihood function cannot be interpreted as a probability density function for .