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

To begin, restate the setup: we have a logistic regression model with weight vector β = (-ln(2), ln(5), -ln(7)) and a new document x* = (1, -1, -1). We are asked to compute P(y* = 0 | x*).

Compute the linear predictor z = β^T x* = (-ln 2)*1 + (ln 5)*(-1) + (-ln 7)*(-1).
This simplifies to z = -ln 2 - ln 5 + ln 7 = ln 7 - ln 2 - ln 5 = ln(7/(2*5)) = ln(7/10) = ln(0.7).
Numerically, z ≈ ln(0.7) ≈ -0.3567.

In logistic regression, P(y=1 | x) = σ(z) where σ(z) = 1 / (1 + e^{-z}). Since z ≈ -0.3567, e^{-z} = e^{0.3567} ≈ 1.428. Therefore, σ(z) ≈ 1 / (1 + 1.428) ≈ 0.4118.
Thus P(y* = 0 | x*) = 1 - P(y* = 1 | x*) ≈ 1 - 0.4118 ≈ 0.5882.

Now evaluate each option:

Option a: 0.3343. This value is far from 0.5882; it would correspond to a much smaller P(y=0) given typical z values, but our calculation shows P(y=0) is about 0.588, not 0.334. This option miscomputes the probability or uses an incorrect z.

Option b: 0.3453. Similar issue as Option a; this is significantly different from the correct 0.5882 and does not align with the computed z and logistic form.

Option c: 0.5882. This matches the computed P(y=0|x*). Since P(y=0) = 1 - σ(z) and σ(z) ≈ 0.4118, 1 - 0.4118 ≈ 0.5882, so this is the correct value.

Option d: 0.6814. This is again inconsistent with the calculated probability given z ≈ -0.3567; it would require a different z or an inverted interpretation, which is not supported by the derivation above.

In summary, the key steps are: compute z = β^T x* = ln(0.7) ≈ -0.3567, apply the logistic function to get P(y=1|x*) ≈ 0.4118, and finally take P(y=0|x*) = 1 - 0.4118 ≈ 0.5882. The only option that aligns with these calculations is option c: 0.5882.

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