# Tag Info

### A and B are independent. Does P(A ∩ B|C) = P(A|C) · P(B|C) hold?

No this is not in general true, as you can see from a simple counter example: Toss two independent coins. Event $A$ is coin 1 head. $P(A)=0.5$ Event $B$ is coin 2 head. $P(B)=0.5$ Event $C$ is either ...
• 2,054
Accepted

### How to incorporate prior knowledge after ML training?

You can either use an ensemble of multiple classifiers that will improve performance when your ML classifier breaks down[1], or implement boosting for your ML classifier assuming that it's a "...
Accepted

### Can a ML classifier's prediction be understood as a probability?

That would be desirable, but it is not guaranteed to make as much sense as we might like. First, you could make an argument that any predicted $p(\mathcal C_k|\mathbf x_i)\in[0,1]$ is a probability in ...
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### A seeming paradox regarding estimation of the number of buttons

This, however, seems quite counter-intuitive to me, because (for small 𝑘) our inference only depends on our will regarding how many data points we want to generate! Yes, but only because we're ...
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### A and B are independent. Does P(A ∩ B|C) = P(A|C) · P(B|C) hold?

A related question is If $X, Y$ are independent of $Z$, is $P(X|Y, Z) = P(X|Y)$? The example there, $C = XOR(A,B)$, is a simple counter example to this question as well ...
• 75.6k
1 vote
Accepted

### Exercise involving Bayes' Theorem

The introduction of event $C$, which describes the inspector entering the room and observing rule-breaking, does not fundamentally change the calculation. Since the observation of students asking ...
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1 vote

### Probability Question on Dice with condition

Assuming there is at most one pearl per box and the values are uniformly distributed between {4,5,6,7,8}. A general R function to calculate the probability: ...
• 1,436
1 vote
Accepted

### A seeming paradox regarding estimation of the number of buttons

You could sketch this situation more easily when $N \leq 2$ and we can model the two potential means $\mu_1, \mu_2$ with a uniform prior on a square. An equality $\mu_1 = \mu_2$ corresponds to the ...
• 75.6k
1 vote
Accepted

### Understanding three prisoners (Statistical Inference - Cassella and Berger)

The wikipedia page for the problem helps pinpoint the reason for the 1/6 probability which I quote below. It also articulates the problem setting in a bit of a more verbose way which helps understand ...
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1 vote

### A seeming paradox regarding estimation of the number of buttons

This one already has a few correct answers, but I think the intuition could be stronger. To start, let's set aside the noise temporarily, and assume that we can observe the $\mu_i$ values directly. ...
• 8,867
1 vote

### A seeming paradox regarding estimation of the number of buttons

We can predictively manipulate our inference regardless of what data we will get. If I decide to bug the person in the room one more time, I know beforehand that my best estimate would be N=k+1 . So I ...
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