Multiplication rule

As explained in Grokking Machine Learning

A rule stating that the probability of independent events occurring together equals the product of their individual probabilities.

The multiplication rule finds the probability that two events occur together. For events E and F, P(E ∩ F) = P(E) · P(F|E). If they are independent, this becomes P(E ∩ F) = P(E) · P(F). section 8.2

Estimating emails with two words

If 20% of emails contain “lottery” and 10% contain “sale,” assuming the words are independent gives an estimate of 2% containing both: 0.20 · 0.10. section 8.2

Figure 8.11 Say 20% of the emails contain the word lottery, and 10% of the emails contain the word sale. We make the naive assumption that these two words are independent of each other. Under this assumption, the percentage of emails that contain both words can be estimated as 2%, namely, the product of 20% and 10%.

See how the independence assumption turns the two word probabilities into an estimate for their joint probability. section 8.2

How Grokking Machine Learning covers it

How deep it goes

Works through conditional probabilities with an email tree and applies independence to estimate joint word probabilities.

Examples

Calculates email probabilities using spam, ham, and the words “lottery” and “sale.”

What you can do after

Evaluate joint email probabilities and estimate them using an independence assumption.

How it connects

Connects joint probability with conditional probability, independence, and naive Bayes.

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Also explained in

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  Statistics Every Programmer NeedsThe multiplication rule of counting determines the number of outcomes formed by successive choices by multiplying the numbers of options at each stage.

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Related concepts

• large language modelused by · A large language model is a machine-learning model, typically neural, trained on extensive text data to process and generate language.
• independenceuses · Two events or random quantities are independent when knowing the outcome or value of one does not change the probability or distribution of the other.
• language modelused by · A language model is a computational model that represents patterns in linguistic tokens to estimate sequence probabilities and predict or generate language from context.
• binomial distributionused by · A binomial distribution is the probability distribution of the number of successes in a fixed number of independent trials, each with the same probability of success.
• Bayesian networkused by · A Bayesian network is a probabilistic graphical model whose directed acyclic graph represents random variables and their conditional dependencies.
• addition rulecontrasts with · The addition rule determines the total count or probability of mutually exclusive alternatives by summing their individual counts or probabilities.

This explainer was generated by Manning's knowledge graph from the text of Grokking Machine Learning. The section links point to the parts of the book it draws on.