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Sem 5 Prep
← BI & Data ScienceBayes' theorem and Bayesian networks

Formulas for this chapter

Conditional probability

P(A | B) = P(A and B) / P(B) P(A and B) = P(A | B) x P(B)

Whenever the question says "given that". Read the numerator off the joint cell and the denominator off the B total.

P(A and B)
Probability both happen, the joint cell over the grand total
P(B)
Probability of the condition, the new denominator

Bayes' theorem

P(A | B) = P(B | A) x P(A) / P(B) P(B) = P(B | A) x P(A) + P(B | not A) x P(not A)

When you are given the likelihood the wrong way round: a test's sensitivity, a machine's defect rate, a per-class table. Build the denominator over every cause first.

P(A)
Prior, belief before the evidence
P(B | A)
Likelihood, how well A explains the evidence
P(A | B)
Posterior, belief after the evidence
P(B)
Evidence, the total probability of what was observed

Naive Bayes score

score(class) = P(class) x P(x1|class) x P(x2|class) x ... x P(xn|class) P(class | x) = score(class) / sum of scores over all classes

Classifying a row with several categorical predictors. The scores are not probabilities until you divide by their total.

P(class)
Prior, that class's share of the training rows
P(xi|class)
Count of that level within the class, over the class size
n
Number of predictors, each contributing one factor

Conditional probability table in Excel

P(level | class) = COUNTIFS(target, class, predictor, level) / COUNTIF(target, class) Laplace smoothing: add 1 to every count before dividing

Building the tables from raw rows. Each predictor's entries must add to 1 within a class.

target
The class column, here Subscriber
predictor
One categorical column, here Age, Gender, Income or Location
Step 1 of 19
The ideaTheory

A rare thing stays rare, even after a good test

A disease affects one person in a hundred. A test catches 99 % of those who have it, and wrongly flags 5 % of those who do not.

Your test comes back positive. Most people guess your chance of being ill is about 99 %.

It is one in six. Out of 1,000 people, 10 are ill and about 10 of them test positive; 990 are well and about 50 of them also test positive. So 10 positives out of 60 are real.

Bayes' theorem is the arithmetic that stops you making that mistake.

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