贝叶斯定理：理解条件概率与信息更新

Bayes’ Theorem: Understanding Conditional Probability and Information Update

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Bayes' theorem, named after the 18th-century British mathematician Thomas Bayes, is a mathematical formula used to determine conditional probabilities. Conditional probability refers to the likelihood of a certain outcome occurring under similar circumstances. Bayes' theorem provides a way to revise existing predictions or theories (update probabilities) when new evidence or additional information becomes available.

In finance, Bayes' theorem can be used to assess the risk of lending to potential borrowers. This theorem, also known as Bayes' rule or Bayes' law, is the basis of Bayesian statistics.

Key Points

• Bayes' theorem allows you to update the predicted probability of an event by introducing new information.
• Bayes' theorem is named after the 18th century mathematician Thomas Bayes.
• It is often used in the financial sector to calculate or update risk assessments.
• Bayes' theorem went unused for two centuries due to the massive computing power required.

Understand Bayes’ Theorem

Bayes' theorem has a wide range of applications and is not limited to the financial field. For example, Bayes' theorem can be used to evaluate the accuracy of medical test results by considering the likelihood that a person has the disease and the general accuracy of the test. Bayes' theorem relies on the introduction of a prior probability distribution to generate a posterior probability.

In Bayesian statistical inference, the prior probability refers to the probability of an event occurring before new data is collected. In other words, it represents the best rational assessment of the probability of a particular outcome based on current knowledge before the experiment is conducted.

The posterior probability is the revised probability of an event occurring after taking into account new information. The posterior probability is calculated by updating the prior probability using Bayes' theorem. From a statistical perspective, the posterior probability is the probability of event A occurring, given that event B has occurred.

Special Considerations

Therefore, Bayes' theorem provides the probability of an event occurring based on new information related to that event. The formula can also be used to determine how hypothetical new information affects the probability of an event, if the hypothetical new information turns out to be true.

For example, consider drawing a card from a full deck of 52 cards.

There are four kings in the deck, so the probability of drawing a king is four divided by 52, which equals 1/13, or about 7.69%. Now, suppose it is revealed that the selected card is the face card. Based on the condition that the face card is the king, the probability that the selected card is the king is four divided by 12, which is about 33.3%, because there are 12 face cards in the deck.

Formula of Bayes’ Theorem

Example of Bayes’ Theorem

Below are two examples of Bayes' theorem. The first example shows how to derive the formula using a stock investment example in Amazon.com Inc. (AMZN). The second example applies Bayes' theorem to drug testing.

Bayes' theorem simply comes from the axiom of conditional probability, which is the probability of an event given the occurrence of another event. For example, a simple probability question might be: "What is the probability that Amazon's stock price will fall?" Conditional probability further asks: "Given the previous decline in the Dow Jones Industrial Average (DJIA), what is the probability that AMZN's stock price will fall?"

The conditional probability A, given the occurrence of B, can be expressed as:

If A is: "AMZN price fell", then P(AMZN) is the probability of AMZN falling; B is: "The Dow Jones Index has fallen", P(DJIA) is the probability of the Dow Jones Index falling; then the conditional probability expression is "The probability of AMZN falling, given that DJIA falls is equal to the joint probability of AMZN price falling and DJIA falling divided by the probability of the DJIA index falling."

P(AMZN and DJIA) is the probability that A and B occur simultaneously. This is multiplied by the probability of A happening, given the probability of B happening, that is, P(AMZN) x P(DJIA|AMZN). The fact that these two expressions are equal led to the formulation of Bayes' theorem, expressed as:

P(AMZN) and P(DJIA) are the probabilities that Amazon and the Dow Jones will decline independently.

This formula explains the relationship between the probability P(AMZN) assumed before seeing the evidence and the probability P(AMZN|DJIA) assumed after the evidence is obtained, assuming that it is based on the Dow Jones proof.

As a numerical example, suppose there is a drug test that is 98% accurate, meaning that the test shows the correct positive result for drug users 98% of the time and the correct negative result for non-users 98% of the time.

Next, assume that 0.5% of the population uses the drug. If a randomly selected person tests positive, the following calculation can be made to determine the probability that this person actually uses the drug, in terms of:

• A = probability that a positive test result is true
• B = Proportion of people using drugs
• A x B = probability that a positive test result is true
• (1 - A) x (1 - B) = probability that a negative test result is true

The formula will look like this:

Using these values, the calculation results are as follows:

Bayes' theorem shows that even if a person tests positive in this situation, the probability that they will use the drug is only 19.76%, and the probability of not using the drug is 80.24%.

What is the purpose of Bayes' rule?

Bayes' rule is used to update the probabilities with updated conditional variables. Investment analysts use it to predict probabilities in the stock market, but it has applications in many other industries as well.

Why is Bayes’ Theorem so powerful?

Mathematically, it shows that the two probabilities are equal. Whether in statistics, investing, or other industries, this enables you to view conditional probabilities.

How to tell when to use Bayes’ Theorem?

If you need to determine the probability of a certain situation occurring, given the presence of other conditions that could affect that situation, you use Bayes' theorem.

Conclusion

At its simplest, Bayes' theorem relates a test result to the conditional probability of that test result given other relevant events. For a high probability of false positives, this theorem provides the possibility that some outcome is more reasonable.

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