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A statistical theorem given by the English statistician and philosopher Thomas Bayes in the 1700s continues to be a guiding light for scientists and analysts across the world. Today, Bayesian thinking finds application in medicine, science, technology, and several other disciplines and continues to influence our worldview and resultant actions strongly.  Thomas’ Bayes idea was strikingly simple. According to Bayes, the probability of a hypothesis being true depends on two conditions: how reasonable it is based on what we already know (the prior knowledge) and how well it fits new evidence. Thus, Bayesian thinking differs from traditional hypothesis testing in that the former includes the prior knowledge before jumping to conclusions. With the preliminary introduction in mind, let us dive into a bit more detail about Bayesian statistics. Bayesian Statistics In simple terms, Bayesian statistics apply probabilities to statistical problems to update prior beliefs in light of the evidence of new data. The probability expresses a degree of belief in a specific event. The degree of belief may be based on previous knowledge about the event based on personal assumptions or results of prior experiments. Bayesian statistics use the Bayes’ Theorem to compute probabilities. The Bayes’ Theorem, in turn, describes the conditional probability of an event based on new evidence and prior information related to the event. With that in mind, let us brush up on the fundamental concept of conditional probability before we understand Bayes’ Theorem in depth. Conditional Probability Conditional probability can be defined as the likelihood of an event or outcome based on the occurrence of a previous event or outcome. It is calculated by multiplying the probability of the prior event by the probability of the subsequent or conditional event.  Let’s take a look at an example to understand the concept better. Event A is that a family planning an outing will go on a picnic. There is an 80% chance that the family will go on the picnic. Event B is that it will rain on the day the family goes out on a picnic. The weather forecast says that there is a 60% chance of precipitation on the picnic day. Hence, the probability (P) that the family goes on the picnic and it rains is calculated as follows: P (Picnic and rain) = P (Rain | Picnic) P (Picnic) = (0.60) * (0.80) = 0.48 In the above example, conditional probability looks at the two events A and B in relationship with one another, that is, the probability that the family does go to the picnic and it also rains on the same day. Hence, conditional probability differs from unconditional probability because the latter refers to the likelihood of occurrence of an event regardless of whether any other event or events have taken place or any other conditions are present. The formula for conditional probability The formula for conditional probability comes from the probability multiplication rule : P (A and B) or P (A U B) = P ( B given A) or P (B | A) * P (A)  In the above equation, P (A and B) is the joint probability, referring to the likelihood of two or more events occurring simultaneously. It is also written as P (A,B). Here’s how to deduce the conditional probability equation from the multiplication rule: Step 1: Write down the multiplication rule. P (A and B) = P (B | A) * P (A)  Step 2: Divide both sides of the equation by P (A). P (A and B) / P (A) = P (B | A) * P (A) / P (A) Step 3: Cancel P (A) on the right side of the equation. P (A and B) / P (A) = P (B | A) Step 4: Rewrite the equation. P (A and B) = P (B | A) / P (A) Thus, the formula for conditional probability is given as: P (A and B) = P (B | A) / P (A) Bayes’ Theorem Using Bayes’ Theorem, we can update our beliefs and convictions based on new and relevant pieces of evidence. For instance, if we are trying to figure out the probability of a given person having cancer, we would generally assume it to be the percentage of the population that has cancer. However, if we introduce extra evidence, such as the person in question is a regular smoker, we can update our perception (and hence the probability) since the probability of having cancer is higher if an individual is a smoker. Hence, we utilize both our prior knowledge and the additional evidence to improve our estimations. The formula for Bayes’ Theorem Source The above equation is the Bayes’ rule. Now, let us look into the stepwise derivation of the Bayes’ Theorem equation. Step 1: Consider two events, A and B. A is the event whose probability we want to calculate and B is the additional evidence that is related to A. Step 2: Write down the relationship between the joint probability and conditional probability of events A and B. P (A,B) = P (A | B) * P(B) = P (B,A) = P (B | A) * P(A) Step 3: Set the two conditional probability terms equal to each other. P (A | B) * P(B) = P (B | A) * P(A) Step 4: Divide both sides of the equation by P (B). P (A | B) * P(B) / P (B) = P (B | A) * P(A) / P (B) Step 5: Cancel P (B) on the left side of the equation. P (A | B) = P (B | A) * P(A) / P (B) Thus, we get the formula of Bayes’ Theorem as follows: P (A | B) = P (B | A) * P(A) / P (B) Understanding the terms in the Bayes’ Theorem equation P (A | B) = P (B | A) * P(A) / P (B) P (A | B) is called the posterior probability or the probability we are trying to estimate. Based on the previous example, the posterior probability would be the probability of the person having cancer, given that the person is a regular smoker. P (B | A) is called the likelihood, referring to the probability of detecting the additional evidence, given our initial hypothesis. In the above example, the likelihood is the probability of the person being a smoker, given that the person has cancer. P (A) is the prior probability or the probability of our hypothesis without any additional evidence or information. In the above example, the prior probability is the probability of having cancer. P (B) is the marginal likelihood or the total probability of observing the evidence. In the context of the above example, the marginal likelihood is the probability of being a smoker. A Simple Example To Understand Bayes’ Theorem Using some hypothetical numbers in the previous example, we will see the effect of applying the Bayes’ Theorem. Suppose the probability of having cancer is 0.06, that is, 6% of the people have cancer. Now, say that the probability of being a smoker is 0.20 or 20% of people are smokers, and 30% of people with cancer are smokers. So, P (Smoker | Cancer) = 0.30. Initially, the probability of having cancer is simply 0.06 (prior). But using the new evidence, we can calculate P (Cancer | Smoker) = P ((Smoker | Cancer) * P (Cancer)) / P (Smoker) = (0.30*0.06) / (0.20) = 0.09. Learn data science courses online from the World’s top Universities. Earn Executive PG Programs, Advanced Certificate Programs, or Masters Programs to fast-track your career. Way Forward: Master the Concepts of Statistics for a Career in Data Science or Machine Learning upGrad’s higher EdTech learning platform has impacted over 500,000 working professionals worldwide with its plethora of courses and immersive learning experiences. With a 40,000+ learner base spread across 85+ countries, upGrad’s industry-relevant courses are guaranteed to advance your career in your field of choice. Master of Science in Data Science is an 18-months course imparting key skills in Statistics, Predictive Analysis, Machine Learning, Big Data Analytics, Data Visualization, etc. Program Highlights: Master’s Degree from Liverpool John Moores University and Executive PGP from IIIT Bangalore 500+ hours of content, 60+ case studies and projects, 20+ live sessions, 14+ programming languages and tools Industry networking, doubt resolution sessions, and learning support Advanced Certificate Program in Machine Learning and Deep Learning is a rigorous 6-months course with peer networking opportunities, hands-on projects, industry mentorship, and 360-degree career assistance. Program Highlights: Prestigious recognition from IIIT Bangalore 240+ hours of content, 5+ case studies, and projects, 24+ live sessions, coverage of 12 programming languages, tools, and libraries 1:8 group coaching sessions and 1:1 mentorship sessions with industry experts Conclusion  Bayesian thinking underpins several areas of human thinking, inquiry, and belief, even though most of us are unaware of it. From cancer screening and global warming to monetary policy and risk assessment and insurance, Bayesian thinking is fundamental. Even the famous British mathematician Alan Turing is believed to have employed the Bayesian approach to crack the German Enigma Code during the Second World War. Sign up with upGrad and further your knowledge of key statistical concepts and more!

by Karan Kurani

04 Sep 2021

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5078

This is an excerpt from the e-book ‘self.debug – An Emotional Guide To Being A World-Class Software Engineer’ written by Karan Kurani. It’s a guide to increasing your skill set as a software engineer to the next level by debugging yourself and your emotions. Karan Kurani is the co-founder and CTO of DoctorC, which is the leading marketplace in India connecting consumers with easy, affordable diagnostic & lab tests. DoctorC’s mission is to make healthcare simple, transparent and affordable. Karan is an alumnus of Cornell University with an overall experience of 8+ years. He has worked with GREE as a Lead Software engineer and founded two startups Shoutt and DoctorC. Interview with Karan Kurani, Co-founder & CTO, DoctorC Here’s an excerpt from the chapter ‘Hacking Skillz – Jack of more than 1 trade’ it talks about how being average in more than 1 thing is easier and more valuable than being excellent at just 1 thing. Check out our free courses to get an edge over the competition  Note – This guideline is most useful for the bottom 99% of the practising software engineers. So if you are part of the remaining 1% – you can safely skip this post. There is a surprisingly easy hack that you can apply to increase your value generally in life. Pick up more skills. It sounds obvious when put on paper but there is one subtlety which makes it a “hack” in my opinion. For this point, we turn to the excellent Scott Adams’ book – “How to fail at almost everything and still win big”. Speaking about being successful, he says – “… I tell them there’s a formula for it. You can manipulate your odds of success by how you choose to fill out the variables in the formula. The formula, roughly speaking, is that every skill you acquire doubles your odds of success.” He goes on to mention that the level of proficiency for a skill is not mentioned because – “… you can raise your market value by being merely good – not extraordinary – at more than one skill.” “To put the success formula into its simplest form: Good + Good > Excellent” upGrad’s Exclusive Software Development Webinar for you – SAAS Business – What is So Different? document.createElement('video'); https://cdn.upgrad.com/blog/mausmi-ambastha.mp4   Check out upGrad’s Java Bootcamp This subtlety makes it very easy to execute. You don’t need to be extraordinary, you just have to be ordinary/average. Hence, if you are an average software engineer and you have any of the skills mentioned below – Good at drawing Public speaking Managing people Have a knack to pick up people’s emotions An eye for design in product Spot problems in operational processes Shoot, edit and/or make videos Make original music Sing Writing essays/blog posts Or anything else Explore Our Software Development Free Courses Fundamentals of Cloud Computing JavaScript Basics from the scratch Data Structures and Algorithms Blockchain Technology React for Beginners Core Java Basics Java Node.js for Beginners Advanced JavaScript Check out upGrad’s Full Stack Development Bootcamp (JS/MERN)  Learn Software development courses online from the World’s top Universities. Earn Executive PG Programs, Advanced Certificate Programs, or Masters Programs to fast-track your career. You are immediately more valuable to yourself and your organisation vs someone is who is a very good software engineer only. So if you are a software engineer who can think of product ideas and execute independently (remember you only have to be of an average skill in it) – you have more than doubled your value. Leverage that skill. What Does A Software Developer Do? Explore our Popular Software Engineering Courses Master of Science in Computer Science from LJMU & IIITB Caltech CTME Cybersecurity Certificate Program Full Stack Development Bootcamp PG Program in Blockchain Executive PG Program in Full Stack Development View All our Courses Below Software Engineering Courses In-Demand Software Development Skills JavaScript Courses Core Java Courses Data Structures Courses Node.js Courses SQL Courses Full stack development Courses NFT Courses DevOps Courses Big Data Courses React.js Courses Cyber Security Courses Cloud Computing Courses Database Design Courses Python Courses Cryptocurrency Courses Just like programming, it’s an adventure to delve into your own mind. You can debug yourself – this book shows you how.   If you’re interested to learn more about full stack software development, check out upGrad & IIIT-B’s Executive PG Programme in Software Development – Specialization in Full Stack Development which is designed for working professionals and offers 500+ hours of rigorous training, 9+ projects and assignments, IIIT-B Alumni status, practical hands-on capstone projects & job assistance with top firms. Read our Popular Articles related to Software Development Why Learn to Code? How Learn to Code? How to Install Specific Version of NPM Package? Types of Inheritance in C++ What Should You Know?

by Karan Kurani

06 Sep 2018