To show that \(\Pr(Y_i = 1 | X_i) = \Pi_i = \frac{1}{1 + e^{-Z_i}}\) is valid, we typically use the context of a logistic regression model, where \(Z_i\) is a linear combination of the predictors \(X_i\) (i.e., \(Z_i = \beta_0 + \beta_1 X_{i1} + \beta_2 X_{i2} + ... + \beta_k X_{ik}\)).
Proof Outline
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Define the Logistic Model: We define the logistic function as: \[ \sigma(Z_i) = \frac{1}{1 + e^{-Z_i}} \] This function is continuous and maps any real number \(Z_i\) to the interval (0, 1).
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Interpret \(\Pr(Y_i = 1 | X_i)\): In the context of logistic regression, we model the probability that the outcome \(Y_i = 1\) given predictors \(X_i\) as: \[ \Pr(Y_i = 1 | X_i) = \sigma(Z_i) \]
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Relate to Parameters: Often \(Z_i\) is defined as \(Z_i = \beta^T X_i\), where \(\beta\) is a vector of coefficients. Hence, we can write: \[ \Pr(Y_i = 1 | X_i) = \frac{1}{1 + e^{-(\beta^T X_i)}} \]
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Conclusion: This shows that the probability of the outcome \(Y_i\) given the features \(X_i\) can be expressed as: \[ \Pr(Y_i = 1 | X_i) = \frac{1}{1 + e^{-Z_i}} \] where \(Z_i\) is determined by a linear predictor. Therefore, it validates the expression: \[ \Pr(Y_i = 1 | X_i) = \Pi_i = \frac{1}{1 + e^{-Z_i}} \]
This demonstrates that under logistic regression, the probability of the binary outcome influenced by the explanatory variables is represented by the logistic function.