For continuous random variables, evaluating p(x) for a specific value of x is always 0 as show ,  and . So when we're calculating the likelihood for a random variable X that is represented by a continuous distribution we have:  $$ P(𝜃 \mid X) = P(x_1, x_2 ..., x_n \mid 𝜃) = \prod\limits_{i=1}^n = p(x_i \mid 𝜃) $$  There are two ways I consider thinking about.  1) The first is the intuitive definition of likelihood, e.g. how probable is a set of n draws for random variable X. Intuitively the likelihood seems like it calculates the probability of n draws occuring which, for any continuous random variable would be zero. This does not make much sense to me.  2) If we consider a gaussian distribution of µ = 0 and 𝜎 = 1 then p(0) = 0.4. This number doesn't represent anything concrete, but when used in the likelihood calculation, can serve as a relative reference for which model parameters 𝜃 fit our data better. In this scenario, likelihood is a bit of a misnomer as it isn't actually the probability of getting our observed data, but is a number that only draws meaning in reference to other likelihood calculations.    makes some really good points about interpreting a density function as density and not probability, which suggests that likelihood is indeed a bit of a misnomer. It's also somewhat confusing that the likelihood is expressed in terms of p(x), which I read as probability of x occurring.
I am trying to understand Maximum likelihood estimation but it looks like I am missing something rather elementary.          suppose we have an iid random sample $X_1, X_2,..., X_n$ for which the probability density function of each $X_i$ is $f(x_i; \theta)$ where $\theta$ is an unknown parameter. Then, the joint probability density function of $X_1, X_2,..., X_n$ is given by:          $f(X_1=x_1,X_2=x_2,\cdots,X_n=x_n)=\prod_{i=1}^{n} f(x_i, \theta)$      In Maximum Likelihood estimation, we try to maximize $f$ as a function of $\theta$.  Question: Why do we maximize probability density instead of probability? In what way does it make sense?  Edit: As explained in the answers, the probability of choosing a finite number of sample points from a continuous probability distribution is zero, so maximizing probability doesn't make sense, but how does maximizing probability density make sense?
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