Logistic Regression-MaximumLikelihood

Logistic Regression - Maximum Likelihood

Concept

https://youtu.be/vN5cNN2-HWE

https://youtu.be/BfKanl1aSG0

Introduction

Click here for the full note

Logistic regression is a classification problem

It is choosing class 'k' of highest $Pr(Y=k | X=x)$

Q. How to estimate $Pr(Y=k | X=x)$ ?

• Option 1: Indirectly estimate using Bayes rule. e.g. LDA

• Option 2: Directly estimate using Logistic Regression

Logistic Regression Model of Posterior Probability Function

From now on, we will assume it is binary classification, K=[0,1], number of K=2

Let function $p(x)$ is

$$p(x)=Pr(Y=1|X=x)$$

Definition of odds are

• Odds: $p(x)/(1-p(x))$

• log-odds: $log[p(x)/(1-p(x))]$

Let Log-odds is a Linear Function of data X

\begin{array}{c} p(x) = \frac{{{e^{{{\bf{\beta }}^T}{\bf{x}}}}}}{{1 + {e^{{{\bf{\beta }}^T}{\bf{x}}}}}}\\ {\rm{where,}}\,\,\,\,{{\bf{x}}_i} = \left[ {\begin{array}{*{20}{c}} 1\\ {{x_{i1}}}\\ {..}\\ {{x_{ip}}} \end{array}} \right],{\bf{\beta }} = \left[ {\begin{array}{*{20}{c}} {{\beta _0}}\\ {{\beta _1}}\\ {..}\\ {{\beta _p}} \end{array}} \right] \end{array}

image-20220120120416955

Q) How to estimate the parameter $\beta_i$ ? Use maximum likelihood

Estimate Posterior Probability Function with Likelihood Function

Given: Training dataset: $(z_1,y_1)~(z_N,y_N)$

The probability of the observed data: product of probability that Y=1 for z_i of k=1 and probability that Y=0 for z_i of k=0

$$l(\beta ) = \prod\limits_{i:yi = 1} {\Pr (Y = 1|X = {z_i})} \prod\limits_{i:yi = 0} {\Pr (Y = 0|X = {z_i})}$$

This is also known as Likelihood function.

Let function $p(z_i)$ is

$$p(z_i)=Pr(Y=1|X=z_i)$$

Then, Likelihood function is expressed as

$$l(\beta ) = \prod\limits_{i:yi = 1} {p({z_i})} \prod\limits_{i:yi = 0} {(1-p({z_i}))}$$

image-20220120121204377

Maximizing Likelihood

Goal: estimate parameter $\beta$ that maximizes likelihood function.

Maximizing likelihood is also maximizing log-likelihood

$$L(\beta)=log(l(\beta))$$

\begin{array}{l} L({\bf{\beta }}) = \sum\limits_{i = 1}^N {[{{\bf{y}}_i}{{\bf{\beta }}^T}{{\bf{z}}_i} - \log (1 + {e^{{{\bf{\beta }}^{\bf{T}}}{{\bf{z}}_i}}})]} \\ {\rm{where,}}\,\,\,\,{{\bf{z}}_i} = \left[ {\begin{array}{*{20}{c}} 1\\ {{z_{i1}}}\\ {..}\\ {{z_{ip}}} \end{array}} \right],{\bf{\beta }} = \left[ {\begin{array}{*{20}{c}} {{\beta _0}}\\ {{\beta _1}}\\ {..}\\ {{\beta _p}} \end{array}} \right] \end{array}

image-20220120122406662

Estimating parameter $\beta$ for Maximizing Log-Likelihood

$${{\bf{\beta }}_{t + 1}} = {{\bf{\beta }}_t} - {{\bf{H}}^{ - 1}}\nabla L({{\bf{\beta }}_t})$$

image-20220120123452524

image-20220120123524244

Expressing with Matrices

$$\begin{array}{c} {{\bf{\beta }}_{t + 1}} = {{\bf{\beta }}_t} - {{\bf{H}}^{ - 1}}\nabla L({{\bf{\beta }}_t})\\ {{\bf{\beta }}_{t + 1}} = {{\bf{\beta }}_t} - {{\bf{(}}{{\bf{Z}}^{\bf{T}}}{\bf{WZ)}}^{{\bf{ - 1}}}}{{\bf{Z}}^{\bf{T}}}{\bf{(y - p)}}\\ = {{\bf{(}}{{\bf{Z}}^{\bf{T}}}{\bf{WZ)}}^{{\bf{ - 1}}}}({{\bf{Z}}^{\bf{T}}}{\bf{W}}){\bf{v}}\\ {\bf{v}} = {\bf{Z}}{{\bf{\beta }}_t} - {{\bf{W}}^{ - 1}}{\bf{(y - p)}} \end{array}$$

image-20220120124311939

Iterative Reweighted Least Squares (IRLS)

Solving iteratively with updated values of W,p

$$\begin{array}{c} {{\bf{\beta }}_{t + 1}} = {{\bf{(}}{{\bf{Z}}^{\bf{T}}}{\bf{WZ)}}^{{\bf{ - 1}}}}({{\bf{Z}}^{\bf{T}}}{\bf{W}}){\bf{v}}\\ {\bf{v}} = {\bf{Z}}{{\bf{\beta }}_t} - {{\bf{W}}^{ - 1}}{\bf{(y - p)}}\\ p(x) = \frac{{{e^{{{\bf{\beta }}^T}{\bf{z_i}}}}}}{{1 + {e^{{{\bf{\beta }}^T}{\bf{z_i}}}}}} \end{array}$$

image-20220120132840280

Further Reading

Q) Comparison with Logistic Regression with Sigmoid Function (see Andrew Ng's lecture)

Note:

$$1/(1+e^{-x})==e^x/(1+e^x)$$

• Maximizes the likelihood vs Minimizes the loss function

image-20220120133134874

Example

Step 1

image-20220120132917461

Step 2

image-20220120132940999

Step 3

image-20220120133007539

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Next

1. Multinomial Logistic Regression:

   MATLAB Example

2. Fitting Data with Logistic Regression:

   MATLAB Example