Your dataset consists of documents, each of which may be represented as a 3 dimensional feature vector. You decide to fit a logistic regression to the data, and derive the following estimates for your weight vector: \beta = (-\ln(2), \ln(5), -\ln (7)). You then receive a new document x_* = (1,-1,-1). Compute P(y_*=0|x_*).

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BlackTom AI · Accepted Answer

We are given a logistic regression model with weight vector β = (-ln2, ln5, -ln7) and a new document x* = (1, -1, -1). The probability model is P(y=1|x) = sigmoid(β^T x).

First, compute the linear predictor: β^T x* = (-ln2)*1 + (ln5)*(-1) + (-ln7)*(-1).
	= -ln2 - ln5 + ln7.
Use log rules: -ln2 - ln5 = -(ln2 + ln5) = -ln(2*5) = -ln10, so β^T x* = ln7 - ln10 = ln(7/10) = ln(0.7).
Numerically, ln(0.7) ≈ -0.3566749439.

Next, compute the sigmoid of η = β^T x*: sigmoid(η) = 1 / (1 + e^{-η}). Since η ≈ -0.3567, e^{-η} = e^{0.3567} ≈ 1.428571..., so sigmoid(η) ≈ 1 / (1 + 1.428571) ≈ 1 / 2.428571 ≈ 0.4118.
This is P(y=1|x*) ≈ 0.4118.
Therefore, P(y=0|x*) = 1 - P(y=1|x*) ≈ 1 - 0.4118 ≈ 0.5882.

Now compare to the provided options: a) 0.6814, b) 0.3343, c) 0.3453, d) 0.5882. The 0.5882 value corresponds to option d, while the other values do not match the calculated probability. The discrepancy for those options arises from miscomputing the linear predictor or the sigmoid, since the key step is β^T x* = ln(0.7) and the resulting P(y=0|x*) = 0.5882.

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