# MLE (maximum likelihood estimation) confusion

I am a bit confused how the MLE (maximum likelihood estimation) formulation works.

Using Bayes theorem, we can see that

$$\mathbb P(\theta|X)=\mathbb P(\theta)\mathbb P(X|\theta)/\mathbb P(X)$$

$$\mathbb P(X)$$ is independent of model choice, so we can simply maximize $$\mathbb P(\theta)\mathbb P(X|\theta)$$. Assuming a uniform prior, we are then simply maximizing $$\mathbb P(X|\theta)$$.

Now, here is where my confusion starts. For the simple case, let's assume that $$X=\{x_1\}$$ and $$\theta$$ has parameters $$\mu$$ and $$\sigma$$ for a normal distribution. So then $$\mathbb P(X|\theta)$$ is just $$\mathbb P(z\sigma+\mu=x_1)$$. Given that the normal distribution is continuous, the probability mass of a single point is exactly zero for all points.

Now, common sense tells us that (holding the variance equal) a normal distribution with parameter $$\mu$$ close to $$x_1$$ is a better fit than a normal distribution with a parameter $$\mu$$ far from $$x_1$$.

However, $$\mathbb P(z\sigma+\mu=x_1)=0$$ for all $$\mu$$ and $$\sigma$$, so how is this reconciled mathematically?

• Welcome to MSE. Please refer to this page in future when you need help formatting questions. – Cardioid_Ass_22 May 6 at 19:30

1. With a discrete prior probability mass function for $$\theta$$ of $$\mathbb P_0(\theta)$$ and a discrete conditional probability for $$X$$ given $$\theta$$, you have (much as you stated) a posterior $$\mathbb P(\theta \mid X=x) \propto \mathbb P_0(\theta) \mathbb P( X=x \mid \theta)$$ and you can divide by $$\sum_{\theta'} \mathbb P_0(\theta') \mathbb P( X=x \mid \theta')$$ to turn this into an actual probability
2. With a prior density for $$\theta$$ of $$\pi_0(\theta)$$ and a discrete conditional probability for $$X$$ given $$\theta$$, you have a posterior density $$\pi(\theta \mid X=x) \propto \pi_0(\theta) \mathbb P( X=x \mid \theta)$$ and you can divide by $$\int_{\theta'} \pi_0(\theta') \mathbb P( X=x \mid \theta') d\theta'$$ to turn this into a probability density
3. With a prior density for $$\theta$$ of $$\pi_0(\theta)$$ and a conditional probability density for $$X$$ given $$\theta$$, you have a posterior density $$\pi(\theta \mid X=x) \propto \pi_0(\theta) f(x \mid \theta)$$ and you can divide by $$\int_{\theta'} \pi_0(\theta') f(x \mid \theta') d\theta'$$ to turn this into a probability density
In the third case, you can find the $$\theta$$ which maximises $$f(x \mid \theta)$$ to give the maximum likelihood estimate of $$\theta$$. A Bayesian could find the $$\theta$$ which maximises $$\pi(\theta \mid X=x)$$ to find the mode of the posterior distribution (the so-called maximum a posteriori probability estimate) though, if being forced to give a point estimate, might prefer to take a loss function into account and give the value which minimises the expected loss based on the whole posterior distribution