Despite criticisms from frequentist scholars, Bayesian methods has been used by scholars in the Allies in World War II, such as Alan Turing, in an algorithm to break coded messages in the Enigma machine that the German Navy used to communicate. However, because of the more complex mathematics involved in Bayesian statistics, Bayesian statistics is limited to straight-forward problems and theoretical discussions until the early s, when computing speed increases tremendously and makes Markov Chain Monte Carlo —the major algorithm for Bayesian estimation in modern Bayesian statistics—feasible.
With the help of increased computing speed, Bayesian statistics has come back and been used as an alternative way of thinking, especially given growing dissatisfaction towards the misuse of frequentist statistics by some scholars across disciplines. Bayesian estimation methods have also been applied to many new research questions where frequentist approaches work less well, as well as in big data analytics and machine learning.
Based on my personal experience, Bayesian methods is used quite often in statistics and related departments, as it is consistent and coherent , as contrast to frequentist where a new and probably ad hoc procedure needed to be developed to handle a new problem.
Social and behavioral scientists are relatively slow to adopt the Bayesian method, but things have been changing. In a recently accepted paper by van de Schoot et al. The other reasons included the flexibility of Bayesian methods for complex and nonstandard problems, and the use of techniques traditionally attached to Bayesian such as missing data and model comparisons.
The rise of Bayesian methods is also related to the statistical reform movement in the past two decades. Bayesian is no panacea to the problem. Indeed, if misused it can give rise to the same problems as statistical significance. I see this as the most important mission for someone teaching statistics. There are multiple perspectives for understanding probability. Although, in my opinions, the impact of these differences in interpretations of probability on statistical practices is usually overstated, understanding the different perspectives on probability is helpful for understanding the Bayesian framework.
This is an earlier perspective, and is based on counting rules. For example, when one throws a die, one does not think that a certain number is more likely than another, unless one knows that the die is biased.
The frequentist interpretation states that probability is essentially the long-term relative frequency of an outcome. What is the secret that the adherents of Bayes know? What is the light that they have seen? Soon you will know. Soon you will be one of us. Or is he?
I trust kind readers will, as usual, point out any errors. Got that? Medical testing often serves to demonstrate the formula. P B , the probability that you have cancer prior to getting tested, is one percent, or. So is P E , the probability that you will test positive. If you test positive, you definitely have cancer, and vice versa. In the real world, tests are rarely if ever totally reliable. That is, 99 out of people who have cancer will test positive, and 99 out of who are healthy will test negative.
If your test is positive, how probable is it that you have cancer? Most people assume the answer is 99 percent, or close to it. P B is still. P E B , the probability of testing positive if you have cancer, is now. This is the probability that you will get a true positive test, which shows you have cancer. What about the denominator, P E? Here is where things get tricky. P E is the probability of testing positive whether or not you have cancer. In other words, it includes false positives as well as true positives.
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. 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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