Consider the following dataset with features X, Y and Z
[table]
X | Y | Z
0 | 0 | 0 0 | 1 | 1
1 | 0 | 0 1 | 1 | 1
2 | 0 | 1 2 | 1 | 1
[/table] Compute the entropy of [math: XYZ+1]\frac{XY}{Z+1}

Question

BlackTom AI · Accepted Answer

We start by restating the problem and the provided choices: compute the entropy of the dataset feature XY/(Z+1) for the given six observations.
First, we compute the value of XY and Z+1 for each row and then the ratio XY/(Z+1):
- Row 1: X=0, Y=0, Z=0 → XY=0, Z+1=1 → ratio = 0
- Row 2: X=0, Y=1, Z=1 → XY=0, Z+1=2 → ratio = 0
- Row 3: X=1, Y=0, Z=0 → XY=0, Z+1=1 → ratio = 0
- Row 4: X=1, Y=1, Z=1 → XY=1, Z+1=2 → ratio = 0.5
- Row 5: X=2, Y=0, Z=1 → XY=0, Z+1=2 → ratio = 0
- Row 6: X=2, Y=1, Z=1 → XY=2, Z+1=2 → ratio = 1
Thus, the observed values are {0, 0, 0, 0.5, 0, 1}.
Next, we treat these as a discrete distribution by counting frequencies: 0 occurs 4 times, 0.5 occurs once, and 1 occurs once. So the empirical distribution is:
- P(0) = 4/6 = 2/3
- P(0.5) = 1/6
- P(1) = 1/6
Now we compute the Shannon entropy in bits: H = - [ (2/3) log2(2/3) + (1/6) log2(1/6) + (1/6) log2(1/6) ].
Evaluating the terms: log2(2/3) ≈ -0.5850, so (2/3) log2(2/3) ≈ -0.39. Also log2(1/6) ≈ -2.585, so (1/6) log2(1/6) ≈ -0.431, and since there are two such terms, their sum is ≈ -0.862. Adding these contributions inside the brackets gives ≈ -1.2517. The outer negative sign yields H ≈ 1.2517 bits.
Comparing with the answer choices, option a. 1.2516 matches the computed entropy (rounding differences are expected).
Why the other options are not correct:
- b. 1: The entropy is not exactly 1 bit here because the distribution is not uniform over two outcomes; we have three distinct observed values with a 2/3 probability on 0 and 1/6 on each of the others.
- c. Undefined: The entropy is well-defined for a discrete distribution with finite samples, so this choice is incorrect.
- d. 2.11328: This would correspond to a significantly different distribution or base of log, but with the computed probabilities the entropy is nearer 1.25 bits, not this value.
In summary, the empirical distribution leads to an entropy of approximately 1.2516 bits, which aligns with option a.

COMP9417-Machine Learning & Data Mining - T3 2025

Consider the following dataset with features X, Y and Z [table] X | Y | Z 0 | 0 | 0 0 | 1 | 1 1 | 0 | 0 1 | 1 | 1 2 | 0 | 1 2 | 1 | 1 [/table] Compute the entropy of [math: XYZ+1]\frac{XY}{Z+1}

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