An illustration of the Product Rule of probability: The probability of the joint events on the right end of the diagram is obtained by multiplying the probabilities along the path that leads to it. The paths indicate where and how we are progressively splitting the initial probability into smaller subsets. A suggested exercise to test understanding and gain familiarity with the rules is to construct the equivalent path diagram i.

The event A is that it rains today. The event B is that it rains tomorrow. Sum across rows to find P A , sum down columns to find P B. Together [the Sum and Product Rules] solve the problem of inference, or, better, they provide a framework for its solution. Applications of Bayesian inference are creative ways of looking at a problem through the lens of these two rules. The rules form the basis of a mature philosophy of scientific learning proposed by Dorothy Wrinch and Sir Harold Jeffreys Jeffreys, , ; Wrinch and Jeffreys, ; see also Ly et al. Together, the two rules allow us to calculate probabilities and perform scientific inference in an incredible variety of circumstances.

We begin by illustrating one combination of the two rules that is especially useful for scientific inference: Bayesian hypothesis testing. In such a scenario, it is important to keep in mind that we cannot make inferential statements about any model not included in the set. Likelihoods can be thought of as how strongly the data X are implied by an hypothesis.

It is important to distinguish Bayes factors from posterior probabilities. Both are useful in their own role—posterior probabilities to determine our total belief after taking into account the data and to draw conclusions, and Bayes factors as a learning factor that tells us how much evidence the data have delivered.

Consider as an extreme example Bem who presented data consistent with the hypothesis that some humans can predict future random events. Since Bayes factors quantify statistical evidence, they can serve two closely related purposes. These practical considerations often left implicit are formalized by utility loss functions in Bayesian decision theory. We will not go into Bayesian decision theory in depth here; introductions can be found in Lindley or Winkler , and an advanced introduction is available in Robert The rule allows us to update our belief regarding an hypothesis in response to data.

Our beliefs after taking account the data are captured in the posterior probability , and the amount of updating is given by the Bayes factor. We now move to some applied examples that illustrate how this simple rule pertains to cases of inference. However, it has turned out that one in a thousand codacle plants is afflicted with a mutation that changes its effects: Consuming those rare plants causes unpleasant side effects such as paranoia, anxiety, and spontaneous levitation.

In order to evaluate the quality of her crops, Professor Sprout has developed a mutation-detecting spell. When Professor Sprout presents her results at a School colloquium, Trelawney asks two questions: What is the probability that a codacle plant is a mutant, when your spell says that it is? And what is the probability the plant is a mutant, when your spell says that it is healthy? It is instructive to consider some parallels of this admittedly fictional example to current practices in social science.

In the example above the rate at which we falsely reject the null hypothesis i. The rate at which we correctly reject the null hypothesis i. However, even with a low false alarm rate and a very high correct rejection rate, a null hypothesis rejection may not necessarily provide enough evidence to overcome the low prior probability an alternative hypothesis might have.

At the start of every school year, new Hogwarts students participate in the centuries-old Sorting ceremony, during which they are assigned to one of the four Houses of the School: Gryffindor, Hufflepuff, Ravenclaw, or Slytherin. For hundreds of years the Sorting Hat has assigned students to houses with perfect accuracy and in perfect balance one-quarter to each House. Unfortunately, the Hat was damaged by a stray curse during a violent episode at the School.

Excellent S E. Outstanding S O. Acceptable S A. Poor S P. The Sorting Hat example introduces two extensions from the first. The extension that allows for the evaluation of multiple hypotheses did not require the ad hoc formulation of any new rules, but relied entirely on the same basic rules of probability. The example additionally underscores an inferential facility that we believe is vastly underused in social science: we selected between models making use of two qualitatively different sources of information.

Again, this extension is novel only in that we had not yet considered it — the fact that information from multiple sources can be so combined requires no new facts and is merely a consequence of the two fundamental rules of probability. In Bayesian parameter estimation, both the prior and posterior distributions represent, not any measurable property of the parameter, but only our own state of knowledge about it.

The width of the [posterior] distribution… indicates the range of values that are consistent with our prior information and data, and which honesty therefore compels us to admit as possible values. An example of a probability density function PDF.

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PDFs express the relative plausibility of different values and can be used to determine the probability that a value lies in any interval. Gaussian distribution with mean and standard deviation In this distribution, the filled region to the left of 81 has an area of approximately 0. Similarly, the narrow shaded region on the right extends from to and also has an area of 0. Obtaining the posterior density involves the evaluation of Eq. The numerator on the right-hand side of Eq. For this reason, many authors prefer to ignore the denominator of Eq.

We do not, because this conceals the critical role the denominator plays in a predictive interpretation of Bayesian inference.

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The target audience is Hogwarts students who need an excuse to leave class and enjoy making terrible messes. From scattered customer reports, George believes the expulsion rate to be between three to five events per hour, but he intends to collect data to determine the rate more precisely.

At the start of this project, George has no distinct hypotheses to compare—he is interested only in estimating the expulsion rate. Top row: An example Poisson distribution. The height of each bar indicates the probability of that particular outcome e. Before collecting further data, the Weasleys make sure to specify what they believe to be reasonable values based on the reports George has heard.

In the second panel of Fig. Three volunteers are easily found, administered one puking pastille each, and monitored for 1 h. When priors and likelihoods are conjugate, three main advantages follow. First, it is easy to express the posterior density because it has the same form as the prior density as seen in Eq.

Second, it is straightforward to calculate means and other summary statistics of the posterior density. Third, it is straightforward to update the posterior distribution sequentially as more data become available. Social scientists estimate model parameters in a wide variety of settings. The puking pastilles example illustrates how Bayesian parameter estimation is a direct consequence of the rules of probability theory, and this relationship licenses a number of interpretations that the New Statistics does not allow.

Specifically, the basis in probability theory allows George and Ginny to 1 point at the most plausible values for the rate of expulsion events and 2 provide an interval that contains the expulsion rate with a certain probability e. The applications of parameter estimation often involve exploratory settings: no theories are being tested and a distributional model of the data is assumed for descriptive convenience. In the social sciences most measurements have a natural reference point of zero, so this type of hypothesis will usually be in the form of a directional prediction for an effect.

Carefully inspecting these equations can be instructive. If the data are noisy compared to the prior i. If the data are relatively precise i. The above effect is often known as shrinkage because our sample estimates are pulled back toward prior estimates i. Since Bayesian estimates are automatically shrunk according to the relative precision of the prior and the data, incorporating prior information simultaneously improves our parameter estimates and protects us from being otherwise misled by noisy estimates in small samples. Quoting Gelman , p. Another way to interpret these weights is to think of the prior density as representing some amount of information that is available from an unspecified number of previous hypothetical observations, which are then added to the information from the real observations in the sample.

In studies for which obtaining a large sample is difficult, the ability to inject outside information into the problem to come to more informed conclusions can be a valuable asset. A common source of outside information is estimates of effect sizes from previous studies in the literature. As the sample becomes more precise, usually through increasing sample size, W 2 will continually decrease, and eventually the amount of information added by the prior will become a negligible fraction of the total see also the principle of stable estimation, described in Edwards et al.

While typically not very aggressive, a startled Murtlap might bite a human, causing a mild rash, discomfort in the affected area, profuse sweating, and some more unusual symptoms. Anecdotal reports dating back to the s indicate that Muggles non-magical folk suffer a stronger immunohistological reaction to Murtlap bites. The Ministry of Magic keeps meticulous historical records of encounters between wizards and magical creatures that go back over a thousand years, so Scamander has a great deal of information on wizard reactions to Murtlap bites. Specifically, the average duration of the ensuing sweating episode is 42 hours, with a standard deviation of 2.

Due to the large amount of data available, the standard error of measurement is negligible. From his informal observations, Scamander believes that the mean difference between wizards and Muggles will probably not be larger than 15 hours. Scamander covertly collects information on a representative sample of 30 Muggles by exposing them to an angry Murtlap. The left panel shows the location of the fixed value 42 in the body of the prior and posterior distributions. The right panel is zoomed in on the density in the area around the fixed value. Comparing the prior density to the posterior density at the fixed value reveals that very little was learned about this specific value: the density under the posterior is close to the density under the prior and amounts to a Bayes factor of approximately 3 supporting a deviation from the fixed value.

In summary, the probability that the reaction to Murtlap bites in the average Muggle is greater than in the average wizard increases from exactly 50 to The conclusion of a Bayesian estimation problem is the full posterior density for the parameter s. That is, once the posterior density is obtained then the estimation problem is complete. However, researchers often choose to report summaries of the posterior distribution that represent its content in a meaningful way. One common summary of the posterior density is a posterior credible interval.

Credible intervals have a unique property: as Edwards et al. This property is made possible by the inclusion of a prior density in the statistical model Rouder et al. It is important not to confuse credible intervals with confidence intervals , which have no such property in general Morey et al. Thus, when Scamander reports that there is a It is important to note that there is no unique interval for summarizing the posterior distribution; the choice depends on the context of the research question. It is sometimes considered a paradox that the answer depends not only on the observations but on the question; it should be a platitude.

Consider the following theoretical questions. Is participant performance different than chance? Does this gene affect IQ? Does stimulus orientation influence response latency? For each of these questions the researcher has a special interest in a particular parameter value and entertains it as a possibility. That is, with a continuous probability distribution, probability only exists within a given range of the parameter space; the probability of any single point within the distribution is zero.

This is inconsistent with our belief that a specified parameter value might hold true. Moreover, this poses a problem for any research question that focuses on a single value of a continuous parameter, because if its prior probability is zero then no amount of data can cause its posterior probability to become anything other than zero. The solution involves applying the sum and product rules across multiple independent statistical models at once. As before, this scenario can be approached with the product and sum rules of probability. The setup of the problem is captured by Fig. This layer of analysis is called the model space , since it deals with the probability of the models.

This layer of analysis is called the parameter space since it specifies what is known about the parameters within a model, and it is important to note that each model has its own independent parameter space. Each model also makes predictions about what data will occur in the experiment i. We then condition on the data we observe, which allows us to update each layer of the analysis to account for the information gained.

Below is a step-by-step account of how this is done, but we remind readers that they should feel free to skip this technical exposition and jump right into the next examples. In our development above there is only one parameter so this condition is automatically satisfied.

In practice, Bayes factors can be difficult to compute for more complicated models because one must integrate over possibly very many parameters to obtain the marginal likelihood Kass and Raftery ; Wasserman Recent computational developments have made the computation of Bayes factors more tractable, especially for common scenarios Wagenmakers, Love, et al. However, it should be emphasized that for the purposes of inference these alternative methods can be suboptimal.

Proud of his work on Murtlap bite sensitivity, Newt Scamander from Example 4 decides to present his results at a conference on magical zoology held in Carcassonne, France. As required by the International Decree on the Right of Access to Magical Research Results, he has made all his data and methods publicly available ahead of time and he is confident that his findings will withstand the review of the audience at this annual meeting.

He delivers a flawless presentation that culminates in his conclusion that Muggles are, indeed, slightly more sensitive to Murtlap bites than magical folk are. The evidence, he claims, is right there in the data. In other words, you are putting the cart before the horse because you estimate a population difference before establishing that evidence for one exists. Instead, if you please, let us ascertain how much more stock we should put in your claim over the more parsimonious claim of no difference between the respective population means.

A Bayes factor of not even three favors your hypothesis. What has happened here? At first glance, it appears that at first Scamander had strong evidence that Muggles are more sensitive than magical folk to Murtlap bites, and now through some sleight of hand his evidence appears to have vanished. To resolve the paradox of le Cornichonesque, it is important to appreciate a few facts. The paradox of le Cornichonesque occurs in part because of a confusion between the hypotheses being considered.

Implicitly, there are four different models being considered in all. Indeed, because we know that the probability that Muggles are more vs. In odds notation, the prior odds were increased from 1 to ; the Bayes factor, found by taking the ratio of posterior to prior odds, is in this case equal to the posterior odds.

As a rule, inference must be limited to the hypotheses under consideration: No method of inference can make claims about theories not considered or ruled out a priori. Moreover, the answer we get naturally depends on the question we ask. The example that follows involves a very similar situation, but the risk of the paradox of le Cornichonesque is avoided by making explicit all hypotheses under consideration. In the wizarding world, the Ministry of Magic distinguishes between two types of living creatures.

Beings , such as witches, wizards, and vampires, are creatures who have the intelligence needed to understand laws and function in a peaceful society. By contrast, Beasts are creatures such as trolls, dragons, and grindylows, which do not have that capacity. Recently, the classification of house-elves has become a matter of contention. On one side of the debate is the populist wizard and radio personality Edward Runcorn, who claims that house-elves are so far beneath wizard intelligence that they should be classified as Beasts; on the other side is the famed elfish philosopher and acclaimed author Doc, who argues that elves are as intelligent as wizards and should be classified as Beings, with all the rights and responsibilities thereof.

Karin Bones of the Magical Testing Service to decide whether house-elves are indeed as intelligent as wizards. Bones knows she will be asked to testify before W. However, the junior members of W. I would like you to distribute these tokens over these three cups. You should distribute them proportionally to how strongly you believe in each hypothesis. In this scenario, the three models do not overlap in the parameter space: no parameter value is supported by more than one model.

However, this is merely a convenient feature of this example and not a requirement of Bayesian model selection—it is entirely possible and common for two different models to support the same parameter value. Right: The predicted observed difference in a sample with a standard error of estimation of 1. Here, the predictive distribution for each model has been multiplied by the prior probability for that model.

This representation has the interesting property that the posterior ratio between two models, given some observed difference, can be read from the figure as the ratio between the heights of the two corresponding densities. If the distributions had not been scaled by the prior probability, these height ratios would give the Bayes factor. Prior and posterior probabilities for each hypothesis and each committee member. Probabilities are updated with Eq.

Probability theory allows model comparison in a wide variety of scenarios. In this example the psychometrician deals with a set of three distinct models, each of which was constructed ad hoc—custom-built to capture the psychological intuition of the researcher and a review panel. As this example illustrates, the practical computation of posterior probabilities will often rely on calculus or numerical integration methods; several papers in this special issue deal with computational software that is available Wagenmakers, Love, et al.

An interesting aspect to this example is the fact that the analyst is asked to communicate to a diverse audience: three judges who hold different prior notions about the crucial hypotheses. That is, they hold different notions on the prior probability that each hypothesis is true. This is comparable to the situation in which most researchers find themselves: there is one data set that brings evidence, but there are many—possibly diverse—prior notions.

Given that prior probabilities must be subjective, how can researchers hope to reasonably communicate their results if they can only report their own subjective knowledge? One potential strategy is the one employed by the psychometrician in the example. The strategy relies on the realization that we can compute posterior probabilities for any rational person as soon as we know their prior probabilities. Because the psychometrician had access to the prior probabilities held by each judge, she was able to determine whether her evidence would be compelling to this particular audience.

Social scientists who present evidence to a broad audience can take a similar approach by formulating multiple prior distributions — for example, some informative priors motivated by theory, some priors that are uninformative or indifferent in some ways, and some priors that might be held by a skeptic. Such a practice would be a form of sensitivity analysis or robustness analysis.

If the data available are sufficiently strong that skeptics of all camps must rationally come to the same conclusion, then concerns regarding the choice of priors are largely alleviated. Of course, data is often noisy and the evidence may in many cases not be sufficient to convince the strongest skeptics. In such cases, collecting further data may be useful. Otherwise, the researcher can transparently acknowledge that reasonable people could reasonably come to different conclusions. An alternative option is to report the evidence in isolation. Especially when the ultimate claim is binary—a discrimination between two models—one might report only the amount of discriminating evidence for or against a model.

By reporting only the amount of evidence, in the form of a Bayes factor, every individual reader can combine that evidence with their own prior and form their own conclusions. This is now a widely-recommended approach e. Every four years, the wizarding world organizes the most exhilarating sporting event on earth: the Quidditch World Cup.

However, the Cup is often a source of controversy. From these data, Johnson reasons as follows. Top: The model space shows the contending models. In this case, both Johnson and Cuffe are comparing two models. The prior probabilities for the models are left unspecified. Bottom: The sample space shows what each model predicts about the data to be observed i. The Bayes factor is formed by taking the ratio of the probability each model attached to the observed data, which was four wins in four coin tosses.

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Johnson concludes that these data afford only a modest amount of evidence—certainly not enough evidence to support a controversial and consequential recommendation—and decides to return to tallying quidditch-related nose fractures instead. As might be expected, the Irish quidditch controversy did not fail to pique interest throughout the wizarding world. Cuffe was promoted; his colleague dismissed. As it happens, Cuffe is an accomplished statistician, and he reasons in much the same way as Angelina Johnson the junior statistician at the Ministry.

If there is cheating, the winning probability should be higher. The shape of this density function is depicted in the right half of Fig. The fact that the latter notation of the prior does not include mention of y serves to illustrate that densities and probabilities are often implicitly conditional on sometimes informal background knowledge. Note, for instance, that the entire calculation above assumes that felix felicis was taken, but this is not made explicit in the mathematical notation. This final, two-part example served mostly to illustrate the effects of prior knowledge on inference.

This is somewhat in contrast to Example 6, where the prior information was overwhelmed by the data. In the two scenarios here, the Ministry junior statistician and the Prophet editor are both evaluating evidence that discriminates between two models. The Ministry statistician, having no particular knowledge of the luck doping potion, considers all better-than-chance values equally plausible, whereas the Prophet editor can quantify and insert relevant prior information that specifies the expected effects of the drug in question to greater precision.

As illustrated in the bottom row of Fig. This example illustrates a general property in Bayesian model comparison: A model that makes precise predictions can be confirmed to a much stronger extent than a model that makes vague predictions, while at the same time the precision of its predictions makes it easier to disconfirm.

In sum, the ability to incorporate meaningful theoretical information in the form of a prior distribution allows for more informed predictions and hence more efficient inferences Lee and Vanpaemel this issue.

The Bayesian approach is a common sense approach. It is simply a set of techniques for orderly expression and revision of your opinions with due regard for internal consistency among their various aspects and for the data.

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In our opinion, the greatest theoretical advantage of Bayesian inference is that it unifies all statistical practices within the consistent formal system of probability theory. Indeed, the unifying framework of Bayesian inference is so uniquely well suited for scientific inference that these authors see the two as synonymous. Inference is the process of combining multiple sources of information into one, and the rules for formally combining information derive from two simple rules of probability.

As we have illustrated, common statistical applications such as parameter estimation and hypothesis testing naturally emerge from the sum and product rules. However, these rules allow us to do much more, such as make precise quantitative predictions about future data. This intuitive way of making predictions can be particularly informative in discussions about what one should expect in future studies — it is perhaps especially useful for predicting and evaluating the outcome of a replication attempt, since we can derive a set of new predictions after accounting for the results of the original study e.

The practical advantages of using probability theory as the basis of scientific and statistical inference are legion.

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## Bayesian Inference in the Social Sciences

Bayesian inference also gracefully handles so-called nuisance parameters. In most of our present examples there has been only a single quantity of interest—in order to help keep the examples simple and easy to follow. In real applications, however, there are typically many parameters in a statistical model, some of which we care about and some of which we do not. The latter are called nuisance parameters because we have little interest in them: we only estimate them out of necessity.

Similarly, in Examples 7 and 7b, the exact win rate from a luck-doped coin toss is not of primary interest, only whether the coin tossed in the four games was plausibly fair or not. Here, the bias parameter of the coin can be seen as a nuisance parameter. Dealing with nuisance parameters in a principled way is a unique advantage of the Bayesian framework: except for certain special cases, frequentist inference can become paralyzed by nuisance parameters. Note that model averaging is in a sense the flip-side of model selection: In model selection, the identity of the model is central while the model parameters are sometimes seen as nuisance variables to be integrated away.

The flexibility to perform model averaging across any variable we care to name e. Finally, Bayesian analysis allows for immense freedom in data collection because it respects the likelihood principle Berger and Wolpert The likelihood principle states that the likelihood function of the data contains all of the information relevant to the evaluation of statistical evidence. What this implies is that other properties of the data or experiment that do not factor into the likelihood function are irrelevant to the statistical inference based on the data Lindley ; Royall Adherence to the likelihood principle means that one is free to do analyses without needing to adhere to rigid sampling plans, or even have any plan at all Rouder Note that we did not consider the sampling plan in any of our examples above, and none of the inferences we made would have changed if we had.

Leahy [emphasis original]. The goal of this introduction has been to familiarize the reader with the fundamental principles of Bayesian inference. Other contributions in this special issue Dienes and McLatchie this issue; Kruschke and Liddell this issue focus on why and how Bayesian methods are preferable to the methods proposed in the New Statistics Cumming The Bayesian approach to all inferential problems follows from two simple formal laws: the sum and product rules of probability.

Taken together and in their various forms, these two rules make up the entirety of Bayesian inference—from testing simple hypotheses and estimating parameters, to comparing complex models and producing quantitative predictions. The Bayesian method is unmatched in its flexibility, is rooted in relatively straightforward calculus, and uniquely allows researchers to make statements about the relative probability of theories and parameters — and to update those statements with more data.

That is, the laws of probability show us how our scientific opinions can evolve to cohere with the results of our empirical investigations. For these reasons, we recommend that social scientists adopt Bayesian methods rather than the New Statistics , and we hope that the present introduction will contribute to deterring the field from taking an evolutionary step in the wrong direction. For example, this would apply to values that follow a normal distribution. Strictly speaking, this integral is the probability that a is less than or equal to 81, but the probability of any single point in a continuous distribution is 0.

Recall that x! Unlike a factorial, however, the Gamma function is more flexible in that it can be applied to non-integers. To ease readability we use Greek letters for the parameters of a likelihood function and Roman letters for the parameters of prior posterior distributions. Note that we will now be using probabilities and probability densities side-by-side. In general, if the event to which the measure applies i. In the case of a joint event in which at least one component has an infinite set of possibilities, the joint event will also have an infinite set of possibilities and we will use probability densities there also.

Since these distributions are truncated, they must be multiplied by a suitable constant such that they integrate to 1 i. This fact is used in moving from the second step to the third. Note that here and below, we make use of a convenient approximation: 0. Making the calculation exact is not difficult but requires a rather unpleasant amount of space.

Also note that the indicator function from the prior density carries over to the posterior density. The authors would like to thank J. Rowling for the Harry Potter universe. Skip to main content Skip to sections. Advertisement Hide. Download PDF. Introduction to Bayesian Inference for Psychology. Article First Online: 04 April What is probability? Why this matters We argue that the distinction above is directly relevant for empirical psychology. Understanding the Sum Rule of Probability requires one further concept: the disjoint set. A disjoint set is nothing more than a collection of mutually exclusive events.

To simplify the exposition, we will also assume that exactly one of these events must be true although that is not part of the common definition of such a set. An illustration of the Product Rule of Probability is shown by the path diagram in Fig. Every fork indicates the start of a disjoint set, with each of the elements of that set represented by the branches extending out. The lines indicate the probability of selecting each element from within the set. Starting from the left, one can trace this diagram to find the joint probability of, say, A and B.

At the Start fork there is a probability of. Then there is a probability of. Hence, of the initial. The probability of any joint event at the end of a path can be found by multiplying the probabilities of all the forks it takes to get there. Open image in new window. For example, adding up all of the joint probabilities across the row denoted A gives P A. Adding up all of the joint probabilities down the column denoted B gives P B. This can also be seen by noting that in Fig. Table 1 An illustration of the Sum Rule of Probability. Lindley Bayesian inference is the application of the product and sum rules to real problems of inference Applications of Bayesian inference are creative ways of looking at a problem through the lens of these two rules.

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## BAYES’ THEOREM (Social Science)

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