Obtaining the likelihood of an individual observation In Likelihood and All That Ben Bolker states " the joint likelihood of the whole data set is the product of the likelihoods of each individual observation". Is there a way that someone could explain to me, in simple terms, how we obtain the likelihood of an individual observation? 
 A: The likelihood is what you get when you evaluate the appropriate probability density function (using the model and parameter values of interest) at the value of the observation.
A: Let's play a game that you might solve by maximum likelihood:
Assume that a someone is in a room, you can not see the person.  The person has a die and a coin and decides to try one of these.  
If he trows the die and observes a '1' then he says you that he has '1' else he says you he has zero.  If he throws the coin then he says that he has '1' when it ends head up and '0' otherwise.
He does not tell you whether he threw the coin or the die, you can not see it neither. The only thing that he tells you is whether he has one or zero. 
Assume that he says it is 1. What would you guess, did he throw the die or the coin ?
What is the likelihood of this observation ?


*

*if the coin has been thrown, then the probability that you have a one, given that is was the coin that was tossed, is equal to 0.5 or 50%, or the likelihood of having a coin, given that he says it is '1', is 50%  (note that this is a value of the parameter of the coin)

*If the die has been thrown, then the probability that he says '1', given that the die was thrown is 1/6=16.666%.  Or the likelihood that it was a die given that he says '1' is 16.666% (note that this is a value of the parameter of the die).  


If you have to take a decision based on the maximum likelihood principle then you decide for the one with the highest likelihood, or you decide that he has thrown a coin (because the likelihood of the coin given that he says '1' is 50%, which is higher than the likelihood that he threw a die given that he says 1 (16.6666%)).
A: Suppose there are three observations: 3, 4, 8 (and they are independent). In this example, each of the three numbers is an individual observation. The whole data set is represented by the three observations. 
Because, I think, a specific example makes it easier to understand... If we assume that the data (3, 4, and 8) came from a Poisson distribution with mean 5. The likelihoods of the observations (3, 4, and 8), respectively, are dpois(3,lambda=5), dpois(4,lambda=5), dpois(8,lambda=5).
The joint likelihood of the whole data set is
dpois(3,lambda=5)*dpois(4,lambda=5)*dpois(8,lambda=5)

The example is consistent with the statement. dpois(3,lambda=5), dpois(4,lambda=5), and dpois(8,lambda=5) are the likelihood of an individual observation.
