1. Logistic regression with ±1 labels. Logistic regression (with ±1 labels) maximizes the likelihood L(βο,β) = Π P(X;) Π (1 - p(X;)), = i:Y₁=1 i:Y₁=-1 1 e³o+BTI = 1+e-(Bo+BT) 1+eBo+3x p(x) = Show that this is equivalent to minimizing the cost function n l(Bo, B) = log(1 + exp(-Yi (Bo + BTX₂))). i=1 Hint: Maximizing the likelihood is equivalent to minimizing the negative log-likelihood.
1. Logistic regression with ±1 labels. Logistic regression (with ±1 labels) maximizes the likelihood L(βο,β) = Π P(X;) Π (1 - p(X;)), = i:Y₁=1 i:Y₁=-1 1 e³o+BTI = 1+e-(Bo+BT) 1+eBo+3x p(x) = Show that this is equivalent to minimizing the cost function n l(Bo, B) = log(1 + exp(-Yi (Bo + BTX₂))). i=1 Hint: Maximizing the likelihood is equivalent to minimizing the negative log-likelihood.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![**1. Logistic regression with ±1 labels.** Logistic regression (with ±1 labels) maximizes the likelihood
\[
L(\beta_0, \beta) = \prod_{i:Y_i=1} p(X_i) \prod_{i:Y_i=-1} (1 - p(X_i)),
\]
where
\[
p(x) \triangleq \frac{1}{1 + e^{-(\beta_0 + \beta^T x)}} = \frac{e^{\beta_0 + \beta^T x}}{1 + e^{\beta_0 + \beta^T x}}.
\]
Show that this is equivalent to minimizing the cost function
\[
\ell (\beta_0, \beta) = \sum_{i=1}^{n} \log(1 + \exp(-Y_i(\beta_0 + \beta^T X_i))).
\]
**Hint:** Maximizing the likelihood is equivalent to minimizing the negative log-likelihood.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F49880fec-aae3-4f1a-9c1e-dd709135d382%2F5f5c9f02-efc5-488e-88df-573a5bef3b1e%2F6sygkt_processed.png&w=3840&q=75)
Transcribed Image Text:**1. Logistic regression with ±1 labels.** Logistic regression (with ±1 labels) maximizes the likelihood
\[
L(\beta_0, \beta) = \prod_{i:Y_i=1} p(X_i) \prod_{i:Y_i=-1} (1 - p(X_i)),
\]
where
\[
p(x) \triangleq \frac{1}{1 + e^{-(\beta_0 + \beta^T x)}} = \frac{e^{\beta_0 + \beta^T x}}{1 + e^{\beta_0 + \beta^T x}}.
\]
Show that this is equivalent to minimizing the cost function
\[
\ell (\beta_0, \beta) = \sum_{i=1}^{n} \log(1 + \exp(-Y_i(\beta_0 + \beta^T X_i))).
\]
**Hint:** Maximizing the likelihood is equivalent to minimizing the negative log-likelihood.
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