2019年1月7日 · MLE = Maximum Likelihood Estimation. MAP = Maximum a posteriori. MLE is intuitive/naive in that it starts only with the probability of observation given the parameter (i.e. the likelihood function) and tries to find the parameter best accords with the observation.

2021年3月16日 · MLE comes from frequentist statistics where practitioners let the likelihood "speak for itself." Whereas MAP comes from Bayesian statistics where prior beliefs (usually informed by domain knowledge of parameters) effectively regularize the point estimate. Note: MAP, while Bayesian, is atypical of Bayesian philosophy.

Hence, the MLE is just a special version of the MAP, using a uniform distribution. EM also assumes that the given data is uniformly distributed, thus it uses the MLE. Having written that, Dempster et al. [1] point out multiple times, that the EM "algorithm" can easily be altered to use MAP, meaning a different probability distribution of the data.

From the point of view of Bayesian inference, MLE is a special case of maximum a posteriori estimation (MAP) that assumes a uniform prior distribution of the parameters. For details please refer to this awesome article: MLE vs MAP: the connection between Maximum Likelihood and Maximum A Posteriori Estimation.

As I understand it, MLE, MAP, and conditional MLE all attempt to find the best parameters, $\theta$, given the data by maximizing the left hand side by maximizing a subset of terms on the right. For MLE, we maximize the likelihood term, $\prod_i P_i(X_i, Y_i | \theta)$.

$\begingroup$ So for a binary dependent variable, linear regression using OLS estimator is no longer identical to using MLE estimator, is my understanding correct? $\endgroup$ – Finley Huaxin Commented Jun 4, 2022 at 18:16

Going through the Wikipedia article on Maximum a posteriori estimation, it got confusing after reading this:. It is closely related to the method of maximum likelihood (ML) estimation, but employs an augmented optimization objective which incorporates a prior distribution (that quantifies the additional information available through …

2020年11月22日 · Naive Bayes is not the best example in here, hence your confusion. Naive Bayes is the whole classification algorithm, that tells us how to make classifications given the data, by calculating the conditional probabilities and combining them, by making the naive assumption of independence.

2018年3月2日 · The essence of it is that, as the sample size grows, the relative information contained in the prior and in the data shifts in favor of the data, so the posterior becomes more concentrated around the data-only estimate of the MLE, and the peak actually converges to the MLE (with the usual caveat that certain assumptions have to be met.)

2023年2月16日 · Personally, I would usually take the mean of the posterior distribution, as the MAP and MLE do not correspond to a loss function and so seem difficult to justify. I might start with a different prior, such as a Jeffreys' $\operatorname{Beta}(\frac12,\frac12)$ prior.