Ajay Sarangam

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Even among gifted analysts, the study of Bayesian Statistics continues to be a vastly challenging field. But why use Bayesian Statistics in the first place? Well, the answer to that lies right in front of us. The tech-savvy world that we have transitioned into today has created an immense awareness about the unending possibilities of Machine Learning and artificial intelligence that exist all around us.

In the process, we have gotten so increasingly swayed by the apparent black box charm of the fascinating applications of various Machine Learning operations that we have become painfully ignorant of the inherent force that drives it forward- Statistics.

**Bayesian Statistics For Beginners: A Step By Step Approach****A Student’s Guide To Bayesian Statistics****Understanding Frequentist Statistics****Criticism Of Frequentist Statistics****What Are Bayesian Statistics?****What Is Conditional Probability And When To Use Bayesian Statistics?****Bayes’ Theorem And Why Is Bayesian Statistics Important****Understanding Bayesian Inference****Understanding The Significance Test**

Although Machine Learning and its applied concepts enable us to solve a horde of real-time problems, we need to understand that it is not the only means of tackling these problem statements. Occasionally, Machine Learning fails to address issues even when it involves a substantial amount of data. In such situations, comprehensive and structured knowledge of Statistics resolves these complex analytical problems effectively.

Moreover, it allows them to infer powerful insights from their database, irrespective of their volume. Here, basic Bayesian Statistics and Bayes theorem Bayesian Statistics come into the equation. Bayesian Statistics explained to beginners in simple English aims to acquaint readers with all the necessary terminologies in this study.

Bayesian Statistics for beginners a step by step approach consists of understanding both Bayes’ theorem and Bayesian Statistics. The Bayes’ theorem and the history of Bayesian Statistics trace themselves to the 1770s English statistician and philosopher Thomas Bayes. Centuries hence, the theorem has not lost any shred of its importance. Understanding Bayesian Statistics continues to be an essential requisite for paving the path towards pivotal scientific developments.

Thus this article has been written to provide students and beginners with a simplified intro to Bayesian Statistics. In the process, they will discover how to learn Bayesian Statistics effectively and come across various applications of Bayesian Statistics. Although it is not mandatory to possess a prior familiarity with the basic concepts of Statistics and Probability, it is desirable to grasp the fundamental Bayesian Statistics definition and related Bayesian Statistics examples.

Before we delve into the various topics and concepts that Bayesian Statistics encompasses, the reader must develop a detailed understanding of Frequentist Statistics. This version is the most popular and standard version of Statistics that most of us would have come across in our academic careers. Any introduction to Bayesian Statistics remains incomplete without first understanding the concept of Frequentist Statistics. The argument between Bayesian Statistics and Frequentist Statistics is one that is as intricate as it is long-standing. Hence, it becomes crucial to properly distinguish the differences between the two and spot the all-important line of distinction subsequently.

Frequentist Statistics assumes importance not only because it is the most extensively used inference technique in the Statistical world but also because it is the mainstream of thought that a person encounters upon entering the world of Statistics. At the outset, it tests and determines whether a given event or hypothesis has occurred or not. If the experiment is conducted repetitively under the same set of conditions, then Frequentist Statistics is used to determine the event’s probability in the long run.

The sampling distributions taken here are of a fixed size. Subsequently, the experiment repeats for an infinite period in theory, but it is conducted in practicality to bring it to an eventual halt. Gradually, for adopting the Frequentist approach, one comes across the inherent flaw in this particular technique- a tendency of the result of an experiment to directly depend on the number of instances that the experiment gets reiterated.

The criticism of Frequentist Statistics draws mostly from the list of flaws inherent to this particular approach. The 20th century saw a massive rise in the application of Frequentist Statistics to Numerical and Mathematical models to determine the difference between various samples. However, the approach was not without its own set of flaws. These shortcomings largely undermined its fundamental interpretation and design for their application in real-life problem statements. They have been listed below for the reader’s reference –

- If two individuals operate on the same data set and possess different stopping intentions, an undesirable eventuality arises when they might have two separate p-values.
- Like the p-value, the Confidence Interval (C.I) also relies heavily on the dataset’s sample size. As a result, the stopping potential ends up becoming absurd since the product must be consistent, irrespective of the number of people conducting tests on that particular dataset.
- Since Confidence Intervals (C.I) are not the same as probability distributions, they cannot offer the most suitable value for a given parameter and, subsequently, the most appropriate and probable values as well.

These three drawbacks form the primary reasons why an individual shuns the Frequentist approach and adopt the Bayesian approach to Statistics instead.

Bayesian Statistics for dummies is a Mathematical phenomenon that revolves around applying probabilities to various problems and models in Statistics. It offers individuals with the requisite tools to upgrade their existing beliefs to accommodate all instances of data that is new and unprecedented.

To elaborate, let’s take the aid of a suitable example. Consider two racers who took part in a series of four races. Assume that one of them has won three races while the other has won just one. In this situation, it is logical for you to choose the first racer over the other should you be asked to predict the upcoming race winner. However, the introduction of new data stands to alter the route of your prediction completely. Suppose you were told that each of the drivers had registered their wins on days that had witnessed rainfall and that it was reasonably sure that there would be rain during their upcoming race.

It would become substantially more difficult for you to make your prediction. To deal with these kinds of situations and prevent taking decisions intuitively, one uses various Bayesian Statistics exercises. However, to understand all relevant Bayesian Statistics problems, one must first wrap one head around some underlying concepts that find their relevance in both simple and advanced Bayesian Statistics. Additionally, applied Bayesian Statistics also draw their inferences from these concepts.

To properly define Bayesian Statistics, first, one needs to understand the concept of Conditional Probability. Theoretically, Conditional Probability is the probability of an event A with respect to event B is the same as the probability of both those events taking place together and divided altogether by event B’s Probability.

It is mathematically denoted as:

Understanding Conditional Probability is crucial for understanding Bayesian Statistics and its applications, for it lies at the heart of Bayesian Inference.

Put merely, Bayes’ theorem refers to the manner of determining the probability of a particular event when specific other and related Probabilities are known to us.

Mathematically, it is denoted as:

P(A|B) = P(A) P(B|A)

___________________

P(B)

Essentially, this formula informs us of the frequency of the occurrence of event A, given that event B takes place when we are already aware of the frequency of the occurrence of event B, given that event A takes place, the likelihood of event A occurring on its own, and the possibility of event B taking place on its own as well.

Bayesian inference refers to the manner of Statistical Inference wherein one uses Bayes’s theorem to update the likelihood of a particular hypothesis in the event of additional information and evidence becoming available for it. As a technique, it is imperative in the domain of Mathematical Statistics. Additionally, it finds widespread relevance in engineering, sports, philosophy, science, and law.

Further, while carrying out the Dynamic Analysis of a given sequence of data, Bayesian updating assumes significant importance. Bayesian inference draws striking parallels with Subjective Probability that falls under the philosophy of Decision Theory. T=Often, this probability gets referred to as Bayesian Probability. Numerous functions exist to lend support to the Bayes theorem’s presence. Some leading ones are Prior Belief Distribution, Bernoulli Likelihood Function, and Posterior Belief Distribution.

Bayesian Statistics take the help of various methods to implement a practical test for significance. They’re enlisted below for the reference of the user.

**P-value**

In this method, we first compute a particular sample’s t-score taken from a sampling distribution of fixed size, and then we predict the p-values for the same set subsequently.

**Confidence Interval**

Since confidence intervals do not constitute a Probability Distribution by themselves, they suffer from the same flaw as p-values. One obtains different p-values and t-scores for the sampling distribution of various sizes.

**High-Density Intervals**

We implement the High-Density Intervals and Bayes factor to carry out significance tests as well.

Overall, a sound understanding of significance tests helps us draw a structured criticism of Bayesian Statistics and visualize and map Bayesian Statistics from concept to Data Analysis.

The segments above all strive to answer one question in unison- why Bayesian Statistics? Well, the advantages of Bayesian Statistics lie in the fact that its applications can extend to all the real-time settings around us and not just revolve around Computer Scientists, hardcore Mathematics, and philosophers.

Additionally, it gives you the advantage of approaching a problem from both a Bayesian and non-Bayesian viewpoint, not only broadening your horizons but also letting you arrive sooner at your solution. For a beginner, wrapping their heads around these concepts can be quite overwhelming. This scenario is where Jigsaw Academy’s Postgraduate Diploma In Data Science comes to your rescue. This 11-month in-person program has ranked second among the ‘Top 10 Full-Time Data Science Courses in India’ in 2019, 2018, 2017. Enroll now and build a successful career in Data Science.

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