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What is the difference between maximum likelihood ML and maximum a posteriori MAP estimation?
MLE gives you the value which maximises the Likelihood P(D|θ). And MAP gives you the value which maximises the posterior probability P(θ|D). This is the difference between MLE/MAP and Bayesian inference. MLE and MAP returns a single fixed value, but Bayesian inference returns probability density (or mass) function.
How does maximum likelihood relate to OLS Under what conditions are they the same?
The OLS method is computationally costly in the presence of large datasets. The maximum likelihood estimation method maximizes the probability of observing the dataset given a model and its parameters. In linear regression, OLS and MLE lead to the same optimal set of coefficients.
When does the posterior reach the maximum likelihood?
In this case, even though the likelihood reaches the maximum when p (head)=0.7, the posterior reaches maximum when p (head)=0.5, because the likelihood is weighted by the prior now. By using MAP, p (Head) = 0.5. However, if the prior probability in column 2 is changed, we may have a different answer.
What’s the difference between MLE and maximum likelihood estimation?
Comparing the equation of MAP with MLE, we can see that the only difference is that MAP includes prior in the formula, which means that the likelihood is weighted by the prior in MAP. In the special case when prior follows a uniform distribution, this means that we assign equal weights to all possible value of the Θ.
How to write posterior as a product of likelihood and prior?
Recall, we could write posterior as a product of likelihood and prior using Bayes’ rule: In the formula, p (y|x) is posterior probability; p (x|y) is likelihood; p (y) is prior probability and p (x) is evidence.
How to replace the Mle with the posterior?
In order to get MAP, we can replace the likelihood in the MLE with the posterior: Comparing the equation of MAP with MLE, we can see that the only difference is that MAP includes prior in the formula, which means that the likelihood is weighted by the prior in MAP.