MIT15_097S12_lec15

Because the logarithm is monotonic it does not aect

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: (3) θ As a practical matter, when computing the maximum likelihood estimate it is often easier to work with the log-likelihood, R(θ) := log p(y |θ). Because the logarithm is monotonic, it does not affect the argmax: ˆ θML ∈ arg max R(θ). θ 3 (4) The ML estimator is very popular and has been used all the way back to Laplace. It has a number of nice properties, one of which is that it is a consistent estimator. Let’s explain what that means. Definition 1 (Convergence in Probability). A sequence of random variables X1 , X2 , . . . is said to converge in probability to a random variable X if, ∀E > 0, lim n→∞ 1 (|Xn − X | ≥ E) = 0. P We denote this convergence as Xn − X . → Definition 2 (Consistent estimators). Suppose the data y1 , . . . , ym were gen­ ˆ erated by a probability distribution p(y |θ0 ). An estimator θ is consistent if P ˆ→ it converges in probability to the true value: θ − θ0 as m → ∞. We said that maximum likelihood is consistent. This means that if the distri­ bution that generated the data belongs to the family defined by our likeliho...
View Full Document

This note was uploaded on 03/24/2014 for the course MIT 15.097 taught by Professor Cynthiarudin during the Spring '12 term at MIT.

Ask a homework question - tutors are online