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: [math: β=(−ln⁡(2),ln⁡(5),−ln⁡(7))]\beta = (-\ln(2), \ln(5), -\ln (7)). You then receive a new document [math: x∗=(1,−1,−1)]x_* = (1,-1,-1). Compute [math: P(y∗=0|x∗)]P(y_*=0|x_*).

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BlackTom AI · Accepted Answer

We start by restating the problem and the given quantities to ensure clarity: the weight vector is β = (-ln(2), ln(5), -ln(7)) and the new feature vector is x* = (1, -1, -1). The logistic model gives P(y=1|x) = sigmoid(β^T x) where sigmoid(z) = 1 / (1 + exp(-z)). We need P(y* = 0 | x*) which is 1 - P(y* = 1 | x*) = 1 - sigmoid(β^T x*). So our first step is to compute the inner product z = β^T x*.

Compute z component-wise:
- β1 * x1 = (-ln 2) * 1 = -ln 2
- β2 * x2 = (ln 5) * (-1) = -ln 5
- β3 * x3 = (-ln 7) * (-1) = +ln 7

Sum them: z = -ln 2 - ln 5 + ln 7 = ln 7 - ln(2) - ln(5) = ln(7 / (2*5)) = ln(7/10) = ln(0.7).

Numerically, ln(0.7) ≈ -0.3567.

Next, compute the probability for y=1 given x*: P(y=1|x*) = sigmoid(z) = 1 / (1 + exp(-z)). Since z ≈ -0.3567, we have exp(-z) = exp(0.3567) ≈ 1.428. Therefore P(y=1|x*) ≈ 1 / (1 + 1.428) ≈ 1 / 2.428 ≈ 0.412.

Finally, P(y=0|x*) = 1 - P(y=1|x*) ≈ 1 - 0.412 ≈ 0.588. This matches the latter option among the choices.

Now evaluate each option against the computed value:
- Option a: 0.6814 — This would imply P(y=0|x*) ≈ 0.6814, which contradicts the calculated probability; likely a miscalculation in z or in the sigmoid step would be needed to reach this value.
- Option b: 0.3343 — This is significantly lower than the computed 0.588; it would correspond to a larger positive z (since P(y=0) decreases as z increases).
- Option c: 0.3453 — This is also far from the computed 0.588, indicating a substantial discrepancy in the inner product or the sigmoid evaluation.
- Option d: 0.5882 — This aligns with the numeric result from the sigmoid of z = ln(0.7) and is consistent with P(y=0|x*) = 1 / (1 + exp(z)) when z is negative.

In summary, the key steps were calculating z = β^T x* = ln(7/10) ≈ -0.3567, then P(y=1|x*) = sigmoid(z) ≈ 0.412, and finally P(y=0|x*) = 1 - 0.412 ≈ 0.588, which corresponds to the value in option d (0.5882) and is inconsistent with the other options.

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