Bayes Thorem
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Bayes' Theorem
The mathematical method known as Bayes' Theorem, which is named after the British mathematician Thomas Bayes from the 18th century, is used to calculate conditional probability. Conditional probability is the probability that an event will occur dependent on how similar the circumstances were to an earlier outcome. Given fresh or more evidence, Bayes' Theorem offers a method for updating probabilities, which are predictions or theories that have already been put out.
The risk of providing money to potential borrowers can be rated in the finance industry using Bayes' Theorem. The theorem, which forms the basis of the study of Bayesian statistics, is often known as Bayes' Rule or Bayes' Law.
ESSENTIAL NOTES
By adding additional data, you can adjust the expected probabilities of an occurrence using Bayes' Theorem.
After the 18th-century mathematician Thomas Bayes, the Bayes Theorem was named.
In finance, it is frequently used to compute or update risk assessment.
Due to the large amount of computing power needed to carry out the theorem's transactions, it was ignored for two centuries.
Comprehending the Bayes Theorem
The Bayes Theorem has several applications outside of the financial sector. For instance, by taking into account the general accuracy of the test and the likelihood that a particular individual will have an illness, Bayes' theorem can be used to assess the accuracy of medical test findings. In order to produce posterior probabilities, Bayes' theorem depends on incorporating prior probability distributions.
Prior probability, as used in Bayesian statistical inference, is the likelihood that an event will transpire prior to the collection of fresh evidence. Put another way, it stands for the most logical estimation of the likelihood of a specific result based on available information prior to the conduct of an experiment.
The possibility of an event happening again after taking the new knowledge into account is called posterior probability. Using the Bayes theorem, posterior probability is computed by updating the prior probability. The posterior probability, expressed statistically, is the likelihood that event A will occur after event B has occurred.
Particular Points to Remember
Bayes' In light of fresh information that is or may be connected to an occurrence, the theorem provides the probability of that happening.
The method can also be used to calculate the potential impact of hypothetical new information on the probability of an event occurring, assuming that the new information proves to be accurate.
Think about drawing one card, for example, from a whole 52-card deck.
Given that there are four kings in the deck, the likelihood that the card is a king is calculated by dividing four by 52, yielding a probability of 1/13, or roughly 7.69%. Let's say it turns out that the chosen card is a face card.Given that the chosen card is a face card—there are 12 face cards in a deck—the likelihood that it is a king is four divided by twelve, or roughly 33.3%.
Theorem of Bayes formula
where P(A) is the likelihood that A will occur.
P(B) is the likelihood that B will occur.
The probability of A provided B is P(A∣B).
The probability of B given A is P(B∣A).
P(A⋂B)= The likelihood that A and B will occur together
An Empirical Illustration of Bayes' Theorem
Consider a drug test that is 98% accurate. This means that, 98% of the time, it will provide a real positive result for drug users and a true negative result for nonusers. This is an example of numerical accuracy in action.
Assume next that 0.5% of people utilise the medication. The likelihood that an individual chosen at random who tests positive for the substance is indeed a user of the drug may be calculated using the formula below, where the terms are:
A = Likelihood that a positive test outcome is accurate
B = The percentage of drug users
The likelihood that a positive test result is true (1 - A) is equal to A x B.
This is how the formula would appear:
The formula is (A x B ) / [ (A x B ) + { ( 1-A ) x ( 1-B ) } ]. = Likelihood of Using the Substance
The computation yields the following results when using the values:
( 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% of all the Baileys According to the theorem, even in this case, there is a 19.76% chance that the person takes the medicine and an 80.24% chance that they don't.
A = Likelihood that a positive test outcome is accurate
B = The percentage of drug users
The likelihood that a positive test result is true (1 - A) is equal to A x B.
When Is It Appropriate to Apply the Bayes Theorem?
The Bayes theorem is used when calculating the likelihood of an event given the existence of another circumstance that may affect the event's likelihood.
The Final Word
In its most basic form, the Bayes Theorem connects a test result to the conditional probability of that result in light of further relevant occurrences. In cases of high probability false positives, the theorem provides a more rational probability of a specific result.