What's the difference between probability and likelihood?
Discrete Random Variables
Suppose that you have a stochastic process that takes discrete values (e.g., outcomes of tossing a coin 10 times, number of customers who arrive at a store in 10 minutes etc). In such cases, we can calculate the probability of observing a particular set of outcomes by making suitable assumptions about the underlying stochastic process (e.g., probability of coin landing heads is pp and that coin tosses are independent).
Denote the observed outcomes by OO and the set of parameters that describe the stochastic process as θθ. Thus, when we speak of probability we want to calculate P(O|θ)P(O|θ). In other words, given specific values for θθ, P(O|θ)P(O|θ) is the probability that we would observe the outcomes represented by OO.
However, when we model a real life stochastic process, we often do not know θθ. We simply observe OO and the goal then is to arrive at an estimate for θθ that would be a plausible choice given the observed outcomes OO. We know that given a value of θθ the probability of observing OO is P(O|θ)P(O|θ). Thus, a 'natural' estimation process is to choose that value of θθ that would maximize the probability that we would actually observe OO. In other words, we find the parameter values θθ that maximize the following function:
L(θ|O)=P(O|θ)L(θ|O)=P(O|θ)
L(θ|O)L(θ|O) is called as the likelihood function. Notice that by definition the likelihood function is conditioned on the observed OO and that it is a function of the unknown parameters θθ.
Continuous Random Variables
In the continuous case the situation is similar with one important difference. We can no longer talk about the probability that we observed OO given θθ because in the continuous case P(O|θ)=0P(O|θ)=0. Without getting into technicalities, the basic idea is as follows:
Denote the probability density function (pdf) associated with the outcomes OO as: f(O|θ)f(O|θ). Thus, in the continuous case we estimate θθ given observed outcomes OO by maximizing the following function:
L(θ|O)=f(O|θ)L(θ|O)=f(O|θ)
In this situation, we cannot technically assert that we are finding the parameter value that maximizes the probability that we observe OO as we maximize the pdf associated with the observed outcomes OO.