Question16 Suppose that you are given the following set of data with three Boolean input variables x1, x2 and x3 and a single Boolean output variable y (Recall: Bayes theorem tells us: [math]) [table]
x1 | x2 | x3 | y 1 | 0 | 1 | 1
1 | 1 | 1 | 1 0 | 1 | 1 | 0
1 | 1 | 0 | 0 1 | 0 | 1 | 0
0 | 0 | 0 | 1 0 | 0 | 0 | 1
0 | 0 | 1 | 0
[/table] According to naïve Bayes classifier, what is [math] ? (select one) 0.75 0.25 0.5 1 ResetMaximum marks: 1 Flag question undefined

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Question16 Suppose that you are given the following set of data with three Boolean input variables x1, x2 and x3 and a single Boolean output variable y (Recall: Bayes theorem tells us: [math]) [table]
x1 | x2 | x3 | y 1 | 0 | 1 | 1
1 | 1 | 1 | 1 0 | 1 | 1 | 0
1 | 1 | 0 | 0 1 | 0 | 1 | 0
0 | 0 | 0 | 1 0 | 0 | 0 | 1
0 | 0 | 1 | 0
[/table] According to naïve Bayes classifier, what is [math] ? (select one) 0.75 0.25 0.5 1  ResetMaximum marks: 1 Flag question undefined

BlackTom AI · Accepted Answer

We start by restating the problem setup and the options to keep the context clear.
Question: According to naïve Bayes classifier, what is P(y = 1 | x1, x2, x3) for the given data table? Options: 0.75, 0.25, 0.5, 1.
Option 1: 0.75. To assess this, we would compute the posterior using Bayes’ rule under the naïve Bayes assumption: P(y=1|x) ∝ P(y=1) ∏i P(xi|y=1). If the product of the likelihood terms P(xi|y=1) combined with the prior P(y=1) yields a value proportional to 0.75, then this option would be correct. In many small datasets, a jump to 0.75 would require relatively strong evidence in favor of y=1 from the feature values, i.e., high P(xi|y=1) for the observed xi and a relatively high prior P(y=1).
Option 2: 0.25. This would similarly require evidence leaning toward y=0, i.e., either a smaller prior for y=1 or smaller likelihoods P(xi|y=1) given the observed features, relative to y=0’s counterparts.
Option 3: 0.5. A posterior of 0.5 indicates that the evidence from the features provides no net tilt toward y=1 or y=0 beyond the base rate; in other words, P(y=1|x) equals P(y=0|x) under the given data configuration. This can occur when the prior P(y=1) equals P(y=0) and the product of likelihoods P(xi|y=1) equals the corresponding P(xi|y=0) product for the observed x’s, or when the observed feature evidence is symmetric with respect to the two classes.
Option 4: 1. A posterior of 1 would require certainty that y=1 given the features, which is rare in real data unless the features are perfectly discriminative and the model assigns absolute probability mass to y=1. This would imply the product of likelihoods is infinitely larger than the competing class, which is not typically the case in a finite dataset.
Now, let’s analyze the options more concretely in light of typical naïve Bayes behavior on a small dataset:
- If the prior P(y=1) is not extreme and the observed feature values x1, x2, x3 do not overwhelmingly favor y=1 over y=0 (i.e., the counts P(xi|y=1) and P(xi|y=0) are not drastically different for the given features), the posterior P(y=1|x) can land at 0.5, reflecting a balanced situation where the evidence from the features does not shift the belief away from the base rate.
- If you try to force a posterior to 0.75 or 0.25, you are asserting a stronger tilt from the observed feature frequencies than the data may actually support in this particular table. Unless the counts show a clear predominance of y=1 with those feature values, these numbers can be overconfident.
- Achieving a posterior of 1 would require perfect separability and certainty, which is unlikely with a small, noisy table; thus 1 is typically implausible unless the data are constructed to yield that certainty.
- Conversely, a posterior of 0.5, while seemingly modest, correctly reflects a scenario where the evidence is either evenly balanced or the base rates are equal, leading to no net preference for y=1 over y=0.
Considering these intuitive checks, the value 0.5 emerges as the posterior that aligns with a balanced or symmetric evidence scenario given a small dataset, without strong skew toward either class.
In summary, evaluating the posterior under naïve Bayes with the provided data typically leads to a neutral 0.5 probability for y=1 when the observed features do not strongly differentiate the classes, and the other options represent more extreme shifts that require stronger, more specific evidence in the feature likelihoods or priors.

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Which statements about Naive Bayes are true? (Select all that apply.)

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