# Maximum Likelihood Estimation: difference between probability and likelihood

In Maximum Likelihood Estimation (MLE), we want to maximize the probability $$P(x_1, x_2, x_3,.. x_n | \theta)$$, where $$x_1, x_2, x_3,.. x_n$$ are the datapoints and $$\theta$$ is the parameter vector. Conceptually, since our goal is to find $$\theta$$, I find more intuitive to think of MLE as $$P(\theta | x_1, x_2, x_3,.. x_n)$$. Indeed, the datapoints are given and we are trying to find a specific value for $$\theta$$. I know that this is not right since all the books and courses I have been reading on the subject formalize it as $$P(x_1, x_2, x_3,.. x_n | \theta)$$, but I don't understand why. I guess it concerns the conceptual difference between likelihood and probability, but I would be grateful if someone could explain it to me.

Your intuition is correct. Actually, making decision based on $$P(\theta |x_1,\cdots ,x_n)$$ is called Maximum Aposteriori Probability (MAP) Decision Making which is the start point of a variety of decision making rules in detection and communication theory. Both MAP and ML translate to the same decision making rule when all the possibilities of $$\theta$$ occur equally likely, since $$P(\theta |x_1,\cdots ,x_n)={P(\theta)\cdot P(x_1,\cdots ,x_n|\theta)\over P(x_1,\cdots ,x_n)}$$which is equivalent to $$P(x_1,\cdots ,x_n|\theta)$$ since $$P(x_1,\cdots ,x_n)$$ is independent of $$\theta$$ and $$P(\theta)$$ is equal based on our assumption. However, ML does have an advantage. You can interpret decision-making as choosing the region in which the observation occurs. For example, if $$\theta$$ were to choose from $$-1,1$$ equally likely over whole the $$\Bbb R$$, an observation $$y>0$$ would be translated to $$\theta=1$$ (since it has fallen in the positive numbers region) and an observation $$y<0$$ would be translated to $$\theta=-1$$ (since it has fallen in the negative numbers region).