Bayes’ Theorem – Simple Problems

Conditional probability may be calculated using Bayes’ formula. It is named after the 18th-century British mathematician Thomas Bayes. It is more likely that a certain result will transpire if a primary outcome has resulted under comparable conditions.

For example, suppose we want to find the probability of finding a parking spot at a conference. The time and location of parking, as well as the conferences that are taking place at the time, will all have an impact on the odds of securing a parking spot. In a nutshell, Bayes’ theorem uses test data to compute the real chance of an event.

What is Bayes’ theorem?

The Bayes theorem determines the probability of an occurrence based on data that is or could be connected to it. The formula can also see how hypothetical new information affects the probability of an event occurring, assuming the new information is confirmed. 

Consider a single card chosen at random from a deck of 52.The card’s chances of being a king are 4 divided by 52, or 1/13, or around 7.69 percent. Keep in mind that there are 4 kings on the deck. Suppose the chosen card turns out to be a face card. Because there are 12 face cards in a deck, the probability that the picked card is a king is 4 divided by 12, or nearly 33.3 percent.

The Bayes’ theorem formula

The Bayes’ theorem (sometimes called Bayes’ Rule) is a straightforward method for calculating conditional probability. The rule’s formal definition is as follows:

Bayes’s theorem

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

P (A|B) = P(AB)P(B)

P (B|A) = P(AB)P(A)

In most cases, we must first determine the “tests” and “events” before entering values into an equation.Bayes’ theorem calculates p(A|B) and  p(B|A) for two events, A and B.

Terms associated with the Bayes’ theorem

Let us comprehend the definitions of a few phrases linked to the notion that is used in Bayes’ theorem.

Conditional likelihood: The possibility of an event A occurring based on the existence of another event B. P(A|B) denotes the likelihood of A occurring provided that event B already has occurred.

Random variables: A random variable is a real-valued variable decided by chance.

Experimental probability: It refers to the probability of certain variables.

Posterior probability: A posterior probability is a measure of an event estimated after all relevant data has been taken into account. Conditional probability is another name for it.

Prior probability: It is the probability of an event estimated before additional information is taken into account. Before the experiment, the probability of a particular outcome is calculated based on current information.

Bayes’ theorem simple problems

Consider a drug test that is 98% accurate, meaning it produces an actual positive outcome for someone who is using the drug 98% of the time and a genuinely negative outcome for non-addicts of the drug 98% of the time. Assume that 0.5% of the population uses the drug. If a person randomly chosen tested positive for the substance, the below calculation can determine the likelihood of a drug user.

(0.98 x 0.005) / [(0.98 x 0.005) + ((1- 0.98) x (1 – 0.005)))] = 0.0049 / (0.0049 + 0.0199) = 19.76%

Even if a person tests positive in this circumstance, Bayes’ theorem shows that the person is considerably more likely not to be a drug user.

Applications of Bayes’ theorem

To estimate posterior probabilities, Bayes’ theorem relies on mixing prior probability distributions. In Bayesian statistical reasoning, pre-probability is the probability of an event before additional evidence is collected. By evaluating how probable each specific person is to have an illness and the test’s overall accuracy, we may apply Bayes’ theorem to define the accuracy of medical test results. To estimate posterior probabilities, Bayes’ theorem relies on mixing prior probability distributions. In Bayesian statistical reasoning, pre-probability is the probability of an event before additional evidence is collected.

Important points about Bayes’ theorem 

• The Bayes theorem is used to calculate the probability of a given event.

• P(A|B) = P(A), and P(B|A) = P(A) when the two occurrences are independent (B)

• The Bayes theorem for continuous random variables may compute the conditional probability.

Conclusion

Bayes’ theorem is a basic mathematical method used for determining conditional probabilities. It strongly influences subjectivist or Bayesian approaches to epistemology, statistics, and inductive reasoning. 

Bayes’ theorem is a way of determining the probability of a hypothesis based on conditions using the data-hypothesis relationship in machine learning.

Data science classification problems and using the Naive Bayes classifier are also the initial steps in learning true positive, false positive, true negative, and false negative notions.

The rules of probability regulate subjectivists, who hold that rational belief, focus extensively on conditional probabilities in their theories of evidence and practical learning models.