Laplace formalized the Bayes concept and is now viewed by economists as the individual who should share the credit for developing what's known as the "Bayesian probability.
Alan Turing, a British mathematician, used Bayes Theorem to assess the translations culled from the Enigma encryption machine used to crack the German messaging code. Applying probability models, Turing and his staff were able to break down the almost infinite number of possible translations based on the messages that were most likely to be translatable, and ultimately crack the German Enigma code.
Interestingly, there is no known portrait of Bayes in existence, and nobody really knows what he looked like there is a sketched image floating around the internet, but it's never been officially confirmed as the "real Bayes. But exist he did, and his theory on conditional probability remains widely praised today by mathematicians, businesses and even poker players , all over the world. Bayes Theorem is a mathematic model, based in statistics and probability, that aims to calculate the probability of one scenario based on its relationship with another scenario.
Largely defined, conditional probability is the likelihood of an event transpiring, due to its association with another scenario. For instance, your likelihood of playing a round of golf within four hours depends on the time of previous rounds played, the time of day, the course you're playing , how many other people you're golfing with, and where and how often you hit your golf ball.
Or, consider this scenario: your son is coming home from college for a long weekend and tells that he's bringing a friend with him. Then your son texts you and says "Oh, you remember my friend with the long blond hair. In Bayes' line of thinking, events are actually tests that indicate the probability of something happening. Bayes saw tests as a way to measure the probability of an event occurring, even though tests really are not events, and results from tests are invariably flawed.
Using testing models and equations, Bayes plugged in previous information plus data formulations to predict multiple probabilities in a given situation. P A and P B are the probabilities of A and B occurring independently of one another the marginal probability. To calculate the probability of a false positive, you multiply the rate of false positives, which is one percent, or.
The total comes to. Yes, your terrific, percent-accurate test yields as many false positives as true positives. To get P E , add true and false positives for a total of. So once again, P B E , the probability that you have cancer if you test positive, is 50 percent. If you get tested again, you can reduce your uncertainty enormously, because your probability of having cancer, P B , is now 50 percent rather than one percent.
But if the reliability of your test is 90 percent, which is still pretty good, your chances of actually having cancer even if you test positive twice are still less than 50 percent. Check my math with the handy calculator in this blog post. Most people, including physicians , have a hard time understanding these odds, which helps explain why we are overdiagnosed and overtreated for cancer and other disorders.
This example suggests that the Bayesians are right: the world would indeed be a better place if more people—or at least more health-care consumers and providers--adopted Bayesian reasoning. If so, this introduction has entirely succeeded in its purpose. In short, beware of false positives. Here is my more general statement of that principle: The plausibility of your belief depends on the degree to which your belief--and only your belief--explains the evidence for it.
The more alternative explanations there are for the evidence, the less plausible your belief is. Your evidence might be erroneous, skewed by a malfunctioning instrument, faulty analysis, confirmation bias, even fraud. Your evidence might be sound but explicable by many beliefs, or hypotheses, other than yours. It boils down to the truism that your belief is only as valid as its evidence. Garbage in, garbage out. In the real world, experts disagree over how to diagnose and count cancers.
Your prior will often consist of a range of probabilities rather than a single number. In many cases, estimating the prior is just guesswork, allowing subjective factors to creep into your calculations.
You might be guessing the probability of something that--unlike cancer—does not even exist, such as strings, multiverses, inflation or God. You might then cite dubious evidence to support your dubious belief.
Scientists often fail to heed this dictum, which helps explains why so many scientific claims turn out to be erroneous. Bayesians claim that their methods can help scientists overcome confirmation bias and produce more reliable results , but I have my doubts.
And as I mentioned above, some string and multiverse enthusiasts are embracing Bayesian analysis. The prominent Bayesian statistician Donald Rubin of Harvard has served as a consultant for tobacco companies facing lawsuits for damages from smoking.
Bayes' theorem is also called Bayes' Rule or Bayes' Law and is the foundation of the field of Bayesian statistics.
Applications of the theorem are widespread and not limited to the financial realm. As an example, Bayes' theorem can be used to determine the accuracy of medical test results by taking into consideration how likely any given person is to have a disease and the general accuracy of the test.
Bayes' theorem relies on incorporating prior probability distributions in order to generate posterior probabilities. Prior probability, in Bayesian statistical inference, is the probability of an event before new data is collected. This is the best rational assessment of the probability of an outcome based on the current knowledge before an experiment is performed.
Posterior probability is the revised probability of an event occurring after taking into consideration new information. Posterior probability is calculated by updating the prior probability by using Bayes' theorem. In statistical terms, the posterior probability is the probability of event A occurring given that event B has occurred. Bayes' theorem thus gives the probability of an event based on new information that is, or may be related, to that event.
The formula can also be used to see how the probability of an event occurring is affected by hypothetical new information, supposing the new information will turn out to be true. For instance, say a single card is drawn from a complete deck of 52 cards. Remember that there are four kings in the deck. Now, suppose it is revealed that the selected card is a face card. The probability the selected card is a king, given it is a face card, is four divided by 12, or approximately Below are two examples of Bayes' theorem in which the first example shows how the formula can be derived in a stock investing example using Amazon.
The second example applies Bayes' theorem to pharmaceutical drug testing. Bayes' theorem follows simply from the axioms of conditional probability. Conditional probability is the probability of an event given that another event occurred. For example, a simple probability question may ask: "What is the probability of Amazon. The conditional probability of A given that B has happened can be expressed as:.
The fact that these two expressions are equal leads to Bayes' theorem, which is written as:. Next, assume 0. If a person selected at random tests positive for the drug, the following calculation can be made to see whether the probability the person is actually a user of the drug.
Bayes' theorem shows that even if a person tested positive in this scenario, it is actually much more likely the person is not a user of the drug.
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