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Maximum Likelihood, Fisher Information, Cramer Rao Inequality - Omkar Ranadive

omkar-ranadive.github.io · 494 words · saved by 2 readers

In simple terms, maximum likelihood is estimating a distribution using a likelihood function (which is made up some parameters) such that the likelihood of observed data being a part of that distribution is maximized. For example: The red points are the observed data samples. We can see that if we estimate the distribution using parameter alpha, then only in the middle diagram the distribution correctly fits the data (i.e, joint probability of all the observed samples under that distribution is high). The likelihood function can be defined as follows: Likelihood is not probability and the integral of likelihood will have no interpretation at all. If we assume that the set of samples are i.i.d then the likelihood function can be written as a joint product of probabilities: Where f(xi, theta) is the probability to observe xi in the interval x + dx. How to maximize the likelihood? To maximize, we can take the first order derivative of the likelihood function and set it to 0. In practice i

Credits: All images used in this post are courtesy of Prof. Michael Schmitt and Ben Lambert Maximum likelihood: In simple terms, maximum likelihood is estimating a distribution using a likelihood function (which is made up some parameters) such that the likelihood of observed data being a part of that distribution is maximized. For example: The red points are the observed data samples. We can see that if we estimate the distribution using parameter alpha, then only in the middle diagram the distribution correctly fits the data (i.e, joint probability of all the observed samples under…

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