Bayes’ Theorem

Bayes’ Theorem allows us to calculate conditional probabilities. It relates the conditional probability $P(D_n\mid E)$ (the probability of event $D_n$ occurring given event $E$) to the reverse conditional probability $P(E\mid D_n)$, and it can be expressed as:

$$P(D_n\mid E)=\frac{P(D_n\cap E)}{P(E)}=P(D_n)\times P(E\mid D_n)$$

Explanation:

• $P(D_n\mid E)$ is the probability that event $D_n$ occurs given that event $E$ has occurred.
• $P(D_n\cap E)$ is the joint probability that both events $D_n$ and $E$ occur.
• $P(E)$ is the total probability that event $E$ occurs.
• $P(D_n)$ is the prior probability of event $D_n$.
• $P(E\mid D_n)$ is the likelihood, or the conditional probability of $E$ given $D_n$.

This formula allows us to update our beliefs about the probability of $D_n$ based on the new evidence $E$.

Extended Form:

If there are multiple events $D_1,D_2,\dots ,D_n$, then Bayes’ Theorem can be generalized as:

$$P(D_n\mid E)=\frac{P(D_n)\cdot P(E\mid D_n)}{{\sum }_{i=1}^{n}P(D_i)\cdot P(E\mid D_i)}$$

In this case, we consider all possible events $D_1,D_2,\dots ,D_n$ and compute the sum of their contributions to the evidence $E$.

Applications

1. Medical Diagnosis: Updates disease probabilities based on test results.
2. Spam Filtering: Classifies emails as spam using word frequencies.
3. Machine Learning: Used in Naive Bayes classifiers for text classification.
4. Finance: Assesses market risks based on new data.
5. Quality Control: Predicts product defects from inspection results.
6. Forensic Science: Evaluates evidence strength in criminal cases.
7. Search Engines: Ranks search results based on relevance.
8. Weather Forecasting: Updates predictions with new data.
9. NLP: Used in language translation and speech recognition.
10. Decision Making: Informs decisions under uncertainty.

References

Information

• date: 2025.02.03
• time: 09:23