Let's say you have a piece of evidence $~$e$~$ and a set of hypotheses $~$\mathcal H.$~$ Each $~$H_i \in \mathcal H$~$ assigns some likelihood to $~$e.$~$ The function $~$\mathcal L_{e}(H_i)$~$ that reports this likelihood for each $~$H_i \in \mathcal H$~$ is known as a "likelihood function."

For example, let's say that the evidence is $~$e_c$~$ = "Mr. Boddy was killed with a candlestick," and the hypotheses are $~$H_S$~$ = "Miss Scarlett did it," $~$H_M$~$ = "Colonel Mustard did it," and $~$H_P$~$ = "Mrs. Peacock did it." Furthermore, if Miss Scarlett was the murderer, she's 20% likely to have used a candlestick. By contrast, if Colonel Mustard did it, he's 10% likely to have used a candlestick, and if Mrs. Peacock did it, she's only 1% likely to have used a candlestick. In this case, the likelihood function is

$$~$\mathcal L_{e_c}(h) = \begin{cases} 0.2 & \text{if $h = H_S$} \\ 0.1 & \text{if $h = H_M$} \\ 0.01 & \text{if $h = H_P$} \\ \end{cases} $~$$

For an example using a continuous function, consider a possibly-biased coin whose bias $~$b$~$ to come up heads on any particular coinflip might be anywhere between $~$0$~$ and $~$1$~$. Suppose we observe the coin to come up heads, tails, and tails. We will denote this evidence $~$e_{HTT}.$~$ The likelihood function over each hypothesis $~$H_b$~$ = "the coin is biased to come up heads $~$b$~$ portion of the time" for $~$b \in [0, 1]$~$ is:

$$~$\mathcal L_{e_{HTT}}(H_b) = b\cdot (1-b)\cdot (1-b).$~$$

There's no reason to normalize likelihood functions so that they sum to 1 — they aren't probability distributions, they're functions expressing each hypothesis' propensity to yield the observed evidence. For example, if the evidence was really obvious ($~$e_s$~$ = "the sun rose this morning,") it might be the case that almost all hypotheses have a very high likelihood, in which case the sum of the likelihood function will be much more than 1.

Likelihood functions carry *absolute* likelihood information, and therefore, they contain information that relative likelihoods do not. Namely, absolute likelihoods can be used to check a hypothesis for strict confusion.