Local Likelihood & other models, Kernel Density Estimation & Classification, Radial Basis Functions.

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Presentation transcript:

Local Likelihood & other models, Kernel Density Estimation & Classification, Radial Basis Functions

Local Likelihood and Other Models Other parametric models can be made local Any such model that supports observation weights is easily adapted.

Local Likelihood and Other Models Example: Likelihood The probability of the parameters fitting the data, as opposed to the other way around [Wikipedia]

Local Likelihood and Other Models Example: Likelihood

Local Likelihood and Other Models Example: Likelihood

Local Likelihood and Other Models Example: Likelihood Many likelihood models involve the covariates in a linear fashion. This lets you take advantage of this while only requiring local linearity of the model.

Local Likelihood and Other Models Example: Log Likelihood

Local Likelihood and Other Models Example: Log Likelihood

Local Likelihood and Other Models Example: Log Likelihood Log is monotonically increasing Same local max/min etc as original function Often Easier to find these maximas from log likelihood

What are the differences among density estimation, classification, and regression? What are the uses of kernels in each of them? (- Grace) What is Kernel density? (-Yifang)

Kernel Density Estimation and Classification

Kernel Density Estimation

Natural local estimate

Kernel Density Estimation Parzen estimate

Kernel Density Estimation Parzen Estimate, choice of K_{lambda}

Kernel Density Estimation Parzen Estimate, choice of K_{lambda}

Kernel Density Estimation Parzen Estimate, choice of K_{lambda} How do we get [the second equation above]? (-Brendan)

Kernel Density Classification

Idiot’s Bayes

What is the relationship between Naive bayes classifier and kernel functions that we study in this chapter? (Grace)

Radial Basis Functions Basis Functions [image from brnt.eu]

Radial Basis Functions Basis Functions Kernel Methods

Radial Basis Functions