# Bayesian statistics example problems

The general law of addition is used to find the probability of the union of two events. The expression denotes the probability of X occurring or Y occurring or both X and Y occurring. If the two events are mutually exclusive, the probability of the union of the two events is the probability of the first event plus the probability of the second event.

Since mutually exclusive events do not intersect, nothing has to be subtracted. If X and Y are mutually exclusive, then the addition law of probability is given by.

The probability of the intersection of two events is called joint probability. The notation is the intersection of two events and it means that both X and Y must happen. When two events X and Y are independent. If X and Y are independent then the multiplication law of probability is given by. Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page. Addition Law of Probability The general law of addition is used to find the probability of the union of two events. If X and Y are mutually exclusive, then the addition law of probability is given by Multiplication Law of Probability The probability of the intersection of two events is called joint probability.

The Multiplication Law of Probability is given by The notation is the intersection of two events and it means that both X and Y must happen. You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics.

Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.This course introduces you to sampling and exploring data, as well as basic probability theory and Bayes' rule.

You will examine various types of sampling methods, and discuss how such methods can impact the scope of inference. A variety of exploratory data analysis techniques will be covered, including numeric summary statistics and basic data visualization. You will be guided through installing and using R and RStudio free statistical softwareand will use this software for lab exercises and a final project.

The concepts and techniques in this course will serve as building blocks for the inference and modeling courses in the Specialization. The tutor makes it really simple. The given examples really helped to understand the concepts and apply it to a wide range of problems. Thank you for this. Wish I could complete the assignments too.

Great course! Explained the concepts so clear and crisp and the exercises with R are great. The project reinforces all the concepts.

All in all, a great course for beginners in statistics and R. Welcome to Week 3 of Introduction to Probability and Data! Last week we explored numerical and categorical data. Thank you for your enthusiasm and participation, and have a great week! Loupe Copy. Introduction to Probability and Data. Course 1 of 5 in the Statistics with R Specialization. Enroll for Free. From the lesson. Conditional Probability Probability Trees Bayesian Inference Examples of Bayesian Inference Taught By. Try the Course for Free. Explore our Catalog Join for free and get personalized recommendations, updates and offers. Get Started. All rights reserved.Frequentist vs Bayesian statistics- this has been an age-old debate, seemingly without an end in sight.

Both these methods approach the same problem in different ways, which is why there is so much talk about which is better.

### Bayesian inference problem, MCMC and variational inference

This is particularly important because proponents of the Bayesian approach blame the Frequentist approach for the reproducibility crisis in scientific studies. For instance, a team at biotech company Amgen found that it could not replicate 47 out of the 53 cancer studies it had analyzed. Many experts believe this is because of the use of frequentist statistics and that the Bayesian approach is an alternative that could solve this crisis. These include:. The frequentist approach follows from the first definition of probability.

According to the frequentist definition of probability, only events that are both random and repeatable, such as flipping of a coin or picking a card from a deck, have probabilities.

These probabilities are equal to the long-term frequencies of such events occurring. The frequentist approach does not attach probabilities to any hypothesis or to any values that are fixed but not known. The Bayesian approach, on the other hand, is rooted in the second and third definitions described above.

Therefore, the Bayesian approach views probability as a more general concept; thereby allowing the assigning of probabilities to events which are not random or repeatable. For example, Bayesians would find it perfectly okay to assign a probability to an event like Donald Trump winning the election.

Say, the problem involves estimating the average height of all men who are currently in or have ever attended college. We assume that the height has a normal distribution and that the standard deviation is available. Therefore, all we need to estimate is the mean. A frequentist would reason that since the mean height is an actual number, they cannot assign a random probability to it being equal to, less than, or greater than a certain value.

Therefore, a Frequentist would collect some sample data from the universal data and estimate the mean as the value which is most consistent with the actual mean. This is known as a maximum likelihood estimate. When the distribution is normal, this estimate is simply the mean of the sample. A Bayesianon the contrary, would reason that although the mean is an actual number, there is no reason not to assign it a probability.

The Bayesian approach will do so by defining a probability distribution based on possible values of the mean. This distribution will then be updated using data from the sample.

Bayes' Theorem of Probability With Tree Diagrams & Venn Diagrams

Based on our understanding from the above Frequentist vs Bayesian example, here are some fundamental differences between Frequentist vs Bayesian ab testing. The use of prior probabilities in the Bayesian technique is the most obvious difference between the two. Frequentists believe that there is always a bias in assigning probabilities which makes the approach subjective and less accurate.

Bayesians, on the other hand, believe that not assigning prior probabilities is one of the biggest weaknesses of the frequentist approach. Bayesians, on the other hand, have a complete posterior distribution over possible parameter values. This allows them to account for the uncertainty in the estimate by integrating the entire distribution, and not just the most likely value.

The Bayesian approach to mitigating uncertainty is by treating it probabilistically. The estimate derived from sample data can, and is often, wrong. In order to mitigate this uncertainty, Frequentists use two techniques. Frequentist vs Bayesian Statistics — The Differences. The Frequentist approach has held sway in the world of statistics through most of the 20th century.Steve Miller wrote an article a couple weeks ago on using Bayesian statistics for risk management.

He describes his friend receiving a positive test on a serious medical condition and being worried. I never studied statistics, nor do I plan to.

But I am interested in the concepts behind statistics, so I can understand probabilities better. And I can do basic math. The disease occurs infrequently in the general population. But what are his actual chances of having the disease? Steve presents the math in his article. To gain an intuitive understanding of the problem, I translated from abstract probabilities to actual numbers of people. This allows us to normalize the percentage rates so we can compare them.

Okay, people have the disease. How many of these people tested positive or negative? Now is where it gets interesting. How many people tested positive versus negative in our entire group? So 5, people tested positive, but we know only 99 of those actually have the disease. The probability of actually having the disease if you test positive is then:.

Clear and convincing evidence that demos work, right? December 15, December 30, at am UTC I thought this was extremely easy to read. I have been looking for ways to get a groups I train warmed up for understanding Bayesian Networks. I will certainly move on to the original article as well.

February 16, at am UTC November 11, at am UTC This is not what the numbers tell us. No doubt. He would not have been given the test unless someone already hypothesized that he had it or he would not have had it.

You and I have a 1. November 11, at pm UTC The assumption provided in the math above is that allpeople are tested.

The 1. The chance of having the disease before you are tested is 0. The chance of having a positive test raises your chances of having the disease by 20 times. But the absolute chance is still small.This post was co-written with Baptiste Rocca. Bayesian inference is a major problem in statistics that is also encountered in many machine learning methods. For example, Gaussian mixture models, for classification, or Latent Dirichlet Allocation, for topic modelling, are both graphical models requiring to solve such a problem when fitting the data.

Meanwhile, it can be noticed that Bayesian inference problems can sometimes be very difficult to solve depending on the model settings assumptions, dimensionality, …. In large problems, exact solutions require, indeed, heavy computations that often become intractable and some approximation techniques have to be used to overcome this issue and build fast and scalable systems.

In this post we will discuss the two main methods that can be used to tackle the Bayesian inference problem: Markov Chain Monte Carlo MCMCthat is a sampling based approach, and Variational Inference VIthat is an approximation based approach.

In the first section we will discuss the Bayesian inference problem and see some examples of classical machine learning applications in which this problem naturally appears.

Finally in the third section we will introduce Variational Inference and see how an approximate solution can be obtained following an optimisation process over a parametrised family of distributions.

Notice also that in this post p. In this section we present the Bayesian inference problem and discuss some computational difficulties before giving the example of Latent Dirichlet Allocation, a concrete machine learning technique of topic modelling in which this problem is encountered. Statistical inference consists in learning about what we do not observe based on what we observe. In other words, it is the process of drawing conclusions such as punctual estimations, confidence intervals or distribution estimations about some latent variables often causes in a population, based on some observed variables often effects in this population or in a sample of this population.

In particular, Bayesian inference is the process of producing statistical inference taking a Bayesian point of view. A classical example is the Bayesian inference of parameters.

Then, when data x are observed, we can update the prior knowledge about this parameter using the Bayes theorem as follows. The Bayes theorem tells us that the computation of the posterior requires three terms: a prior, a likelihood and an evidence.

The first two can be expressed easily as they are part of the assumed model in many situation, the prior and the likelihood are explicitly known. However, the third term, that is a normalisation factor, requires to be computed such that. Although in low dimension this integral can be computed without too much difficulties, it can become intractable in higher dimensions.

In this last case, the exact computation of the posterior distribution is practically infeasible and some approximation techniques have to be used to get solutions to problems that require to know this posterior such as mean computation, for example.

We can notice that some other computational difficulties can arise from Bayesian inference problem such as, for example, combinatorics problems when some variables are discrete. Among the approaches that are the most used to overcome these difficulties we find Markov Chain Monte Carlo and Variational Inference methods.

Bayesian inference problem naturally appears, for example, in machine learning methods that assume a probabilistic graphical model and where, given some observations, we want to recover latent variables of the model. In topic modelling, the Latent Dirichlet Allocation LDA method defines such a model for the description of texts in a corpus.

Thus, given the full corpus vocabulary of size V and a given number of topics T, the model assumes:. The purpose of the method, whose name comes from the Dirichlet priors assumed in the model, is then to infer the latent topics in the observed corpus as well as the topic decomposition of each documents.

Here, beyond the fact that the normalisation factor is absolutely intractable due to a huge dimensionality, we face a combinatoric challenge as some variables of the problem are discrete that require to use either MCMC or VI to get an approximate solution.

The reader interested by topic modelling and its specific underlying Bayesian inference problem can take a look at this reference paper on LDA. As we mentioned before, one of the main difficulty faced when dealing with a Bayesian inference problem comes from the normalisation factor. In this section we describe MCMC sampling methods that constitute a possible solution to overcome this issue as well as some others computational difficulties related to Bayesian inference.

The idea of sampling methods is the following.Thomas Bayes Wikipedia article died in by which time he had written an unpublished note about the binomial distribution and what would now be called Bayesian inference for it using a flat prior. The note was found by a friend and read to the Royal Society of London in and published in its Philosophical Transactions in thus becoming widely known.

Bayesian inference was the first form of statistical inference to be developed. The notion of maximum likelihood is a twentieth century notion, invented by R.

Fisher in and given its fundamental theory by him in This connection between least squares and the normal distribution is why the normal distribution is often called the Gaussian distributiondespite its discovery as the limit in the central limit theorem by de Moivre in GossetR.

FisherE. PearsonJerzy NeymanAbraham Waldand many others. It should be called samplingdistributionist if English made words that way, because, of course, it is statistics based on sampling distributions. More on this below. Bayesianism was resuscitated as philosophy by B.

SavageD. LindleyG. Boxand others between andbut it remained impractical. The handful of very simple Bayesian models that one learns to analyze by hand in a theory course like STAT — are just about all the Bayesian inference one can do by hand. Bayesian inference was revolutionized in when the connection was made with Markov chain Monte Carlo MCMC which allowed Bayesianism to be applied universally in principle.

The Monte Carlo method is a cute name for computer simulation of probability distributions and calculating probabilities and expectations by averaging over the simulations more on this later. At the time the term was invented gambling was illegal everywhere in the USA but Nevada and was still a small industry there.

The casino at Monte Carlo in the country of Monaco was the most famous in the world; gambling has something to do with probability; hence the name. It has now become a colorless technical term, used with no thought of its original motivation.Suppose there are two full bowls of cookies.

Bowl 1 has 10 chocolate chip and 30 plain cookies, while bowl 2 has 20 of each. Our friend Fred picks a bowl at random, and then picks TWO cookies at random. We may assume there is no reason to believe Fred treats one bowl differently from another, likewise for the cookies.

BOTH the cookies turns out to be a plain one. How probable is it that Fred picked them out of Bowl 1? I just wanted to know how to solve the problem of the cookies if in the above cookies problem, if two cookies were selected at a timethen what is the probability that both the cookies are from bowl 1?? Hi Nikhil, Good question. Maybe I will post an update with a longer answer, but here's the short version.

You can extend the analysis shown above to handle this data by computing the likelihood of the data under each hypothesis. Nice post! To answer the Elvis question, you actually still need some additional information from About. Also just to be perfectly thorough identical twins are certain to have the same gender. I agree that we need more information, but I am pretty sure that answer from About. A biometric security device using fingerprints erroneously refuses to admit 1 in 1, authorized persons from a facility containing classified information.

The device will erroneously admit 1 in 1, unauthorized persons. Assume that 95 percent of those who seek access are authorized.

## Frequentist vs Bayesian- Which Approach Should You Use?

If the alarm goes off and a person is refused admission, what is the probability that the person was really authorized? Please help me solve this problem as soon as possible. If you Google this problem, you'll find lots of online solutions. If you are taking a class and don't know how to solve this, ask questions until you do!