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# statistical rethinking notes

[…], Here I work through the practice questions in Chapter 5, “Multivariate Linear Models,” of Statistical Rethinking (McElreath, 2016). best. This is called the marginal likelihood, and to calculate it, we need to take the probability of each possible globe and multiply it by the conditional probability of seeing land given that globe; we then add up every such product: The correct answers are Option 2 and Option 4 (they are equal). I do my […], Here I work through the practice questions in Chapter 3, “Sampling the Imaginary,” of Statistical Rethinking (McElreath, 2016). $\Pr(\mathrm{twins} | B) = 0.2$ Which of the expressions below correspond to the statement: the probability of rain on Monday? D_{i} \sim \text{Normal}(\mu_{i}, \sigma) & \text{[likelihood]} \\ $\Pr(\mathrm{Earth} | \mathrm{land}) = \frac{0.15}{\Pr(\mathrm{land})}=\frac{0.15}{0.65}$. Let’s convert each expression into a statement: Option 1 would be the probability that it is Monday, given that it is raining. If the first card was the first side of BW, then there would be 2 ways for the second card to show white (i.e., the first side of WW or the second side of WW; it would not be possible for the white side of itself to be shown). P(test says A | A) / ( P(test says A | A) + P(test says A | B) ), Your email address will not be published. What is the probability that her next birth will also be twins? $\Pr(A | +) = \frac{\Pr(+ | A) \Pr(A)}{\Pr(+)} = \frac{0.8(0.5)}{0.725} = 0.552$. Let’s update our table to include the new card. If the first card was the first side of BB, then there would be 3 ways for the second card to show white (i.e., the second side of BW, the first side of WW, or the second side of WW). $\Pr(\mathrm{twins}) = \Pr(\mathrm{twins} | A) \Pr(A) + \Pr(\mathrm{twins} | B) \Pr(B) = 0.1(0.5) + 0.2(0.5) = 0.15$, We can use the new information that the first birth was twins to update the probabilities that the female is species A or B (using Bayes’ theorem on page 37): From the Bayesian perspective, there is one true value of a parameter at any given time and thus there is no uncertainty and no probability in “objective reality.” It is only from the perspective of an observer with limited knowledge of this true value that uncertainty exists and that probability is a useful device. So it can be interpreted (repeating all the previous work) as the probability of rain, given that it is Monday. Now suppose you are managing a captive panda breeding program. The rst chapter is a short introduction to statistics and probability. Here I work through the practice questions in Chapter 2, “Small Worlds and Large Worlds,” of Statistical Rethinking (McElreath, 2016). The fact that this result is smaller suggests that the test was overestimating the likelihood of species A. I think the computation for 2H4 is incorrect. Again compute and plot the grid approximate posterior distribution for each of the sets of observations in the problem just above. Stu- Pretty much everything derives from the simple state- ment that entropy is maximized. Richard McElreath (2016) Statistical Rethinking: A Bayesian Course with Examples in R and Stan. Afte we already know age at marriage, what additional value is there in also knowing marriage rate? These relative numbers indicate plausibilities of the different conjectures. $\Pr(\mathrm{rain}|\mathrm{Monday})\Pr(\mathrm{Monday})/\Pr(\mathrm{rain})=\Pr(\mathrm{rain}, \mathrm{Monday})/\Pr(\mathrm{rain})$ Option 4 is the probability of rain and it being Monday, given that it is Monday. The probability it correctly identifies a species A panda is 0.8. Suppose you have a deck with only three cards. Rather, it is named after Stanislaw Ulam (1909–1984). Learn how your comment data is processed. Lectures. We already computed this as part of answering the previous question through Bayesian updating. Option 2 would be the probability of rain, given that it is Monday. Chapman & Hall/CRC Press. Assume that each globe was equally likely to be tossed. Option 4 would be $$\Pr(\mathrm{Monday}, \mathrm{rain})$$. Using the approach from 2E1, we could show that Option 4 is equal to $$\Pr(\mathrm{Monday}|\mathrm{rain})\Pr(\mathrm{rain})$$, but that is not what we want. \beta_{R} \sim \text{Normal}(0,1) & [\text{prior for }\beta_{R}] \\ The StatisticalRethinking.jl v3 package contains functions comparable to the functions in the R package "rethinking" associated with the book Statistical Rethinking by Richard McElreath. Option 3 needs to be converted using the formula on page 36: $\Pr(A | \mathrm{single}) = \frac{\Pr(\mathrm{single}|A)\Pr(A)}{\Pr(\mathrm{single})} = \frac{0.9(1/3)}{5/6} = 0.36$. Option 5 is the same as the previous option but with the terms exchanged. I do my best […], Here I work through the practice questions in Chapter 6, “Overfitting, Regularization, and Information Criteria,” of Statistical Rethinking (McElreath, 2016). Reflecting the need for even minor programming in today's model-based statistics, the book pushes readers to perform step-by … $The target of inference in Bayesian inference is a posterior probability distribution. Species B births twins 20% of the time, otherwise birthing singleton infants. Now compute the probability that the panda we have is from species A, assuming we have observed only the first birth and that it was twins. One card has two black sides. c Rui M. Castro and Robert D. Nowak, 2017. Let’s update the table and include new columns for the prior and the likelihood. Use the counting method (Section 2 of the chapter) to approach this problem. \[\Pr(\mathrm{single}|A) = 1 – \Pr(\mathrm{twins}|A) = 1 – 0.1 = 0.9$ Option 3 would be $$\Pr(\mathrm{Monday} | \mathrm{rain})$$. Hint: Treat this like the sequence of globe tosses, counting all the ways to see each observation, for each possible first card. Discuss the globe tossing example from the chapter, in light of this statement. \beta_{A} \sim \text{Normal}(0, 1) & [\text{prior for }\beta_{A}] \\ Now we can substitute this value into the formula from before to get our answer: So we can use the same approach and code as before, but we need to update the prior. Although it will be easier to see if we rename $$w$$ to $$\mathrm{rain}$$ and $$p$$ to $$\mathrm{Monday}$$: We can use the same formulas as before; we just need to update the numbers: $\Pr(\mathrm{BB})=\frac{\mathrm{BB}}{\mathrm{BB+BW+BW}}=\frac{2+2}{2+1+0+2}=\frac{4}{5}$ $\Pr(\mathrm{rain}|\mathrm{Monday})\Pr(\mathrm{Monday}) = \Pr(\mathrm{rain}, \mathrm{Monday})$ This one got a thumbs up from the Stan team members who’ve read it, and Rasmus Bååth has called it “a pedagogical masterpiece.” The book’s web site has two sample chapters, video tutorials, and the code. Then redo your calculation, now using the birth data as well. Now we just need to count the number of ways each card could produce the observed data (a black card facing up on the table). I do my best to use only approaches and functions discussed so far in the book, as well as to name objects consistently with how the book does. Posterior probabilities state the relative numbers of ways each conjectured cause of the data could have produced the data. The Mars globe is 100% land. So the posterior probability that this panda is species A is 0.36. Suppose there are two species of panda bear. Again suppose a card is drawn from the bag and a black side appears face up. In each case, assume a uniform prior for $$p$$. \mu_{i} = \alpha + \beta_{R}R_{i} + \beta_{A}A_{i} & \text{[linear model]}\\ PROBLEM STATEMENT The determination of an MVU estimator of a deterministic scalar parameter θ is a Sort by. Using the approach detailed on page 40, we use the dbinom() function and provide it with arguments corresponding to the number of $W$s and the number of tosses (in this case 3 and 3): We recreate this but update the arguments to 3 $W$s and 4 tosses. We just have to calculate the updated marginal probability of twins. As described on pages 26-27, the likelihood for a card is the product of multiplying its ways and its prior: Now we can use the same formula as before, but using the likelihood instead of the raw counts. California Polytechnic State University, San Luis Obispo. P(test says B | B) = 0.65. Last updated on May 12, 2020 22 min read Notes, R, Statistical Rethinking. 40 comments. These functions are used in the Pluto notebooks projects specifically intended for hands-on use while studying the book or taking the course. Suppose there are two globes, one for Earth and one for Mars. $\Pr(w,p)=\Pr(w|p)\Pr(p)$ \begin{array}{lr} In this case, we can use the ifelse() function as detailed on page 40: Any parameter values less than 0.5 get their posterior probabilities reduced to zero through multiplication with a prior of zero. New comments cannot be posted and votes cannot be cast. Further suppose that one of these globes–you don’t know which–was tossed in the air and produces a “land” observation. PREREQUISITES The reader is assumed to be familiar with basic classical estimation theory as it is presented in . So the probability that the female will give birth to twins, given that she has already given birth to twins is 1/6 or 0.17. The face that is shown on the new card is white. $\Pr(\mathrm{twins}) = \Pr(\mathrm{twins} | A) \Pr(A) + \Pr(\mathrm{twins} | B) \Pr(B) = 0.1\bigg(\frac{1}{3}\bigg) + 0.2\bigg(\frac{2}{3}\bigg) = \frac{1}{6}$. Statistical rethinking with brms, ggplot2, and the tidyverse This project is an attempt to re-express the code in McElreath’s textbook. Option 4 is the same as the previous option but with division added: The probability that it is Monday, given that it is raining. Transition from descriptive to inferential statistics (Chapters 6-7) Inferential Statistics (Chapters 8-18) Statistics Descriptive Statistics (Chapters 2-5) FIGURE 1.1 A general overview of this book. This is much easier to interpret as the probability that it is raining and that it is Monday. Rethinking P-Values: Is "Statistical Significance" Useless? Select the predictor variables you want in the linear model of the mean, For each predictor, make a parameter that will measure its association with the outcome, Multiply the parameter by the variable and add that term to the linear model. Otherwise they are the same as before. Like the other BB card, it has $$2$$ ways to produce the observed data. \end{array} So it becomes immediately intuitive that the probability of test saying A but it actually is B just means the probability of test being wrong about B. This is the information you have about the test: The vet administers the test to your panda and tells you that the test is positive for species A. To begin, let’s list all the information provided by the question: $\Pr(\mathrm{land} | \mathrm{Earth}) = 1 – 0.7 = 0.3$ $\Pr(A) = 0.36$ The posterior probability of species A (using both the test result and the birth information) is 0.409. I do my best to use only approaches and functions discussed so far in the book, as well as to name objects consistently with how the book does. More mechanically, a Bayesian model is a composite of a likelihood, a choice of parameters, and a prior. $\Pr(+|A) = 0.8$ To use the previous birth information, we can update our priors of the probability of species A and B. lecture note include statistical signal processing, digital communications, information theory, and modern con-trol theory. Since BB could produce this result from either side facing up, it has two ways to produce it ($$2$$). 2 I do […], Here I work through the practice questions in Chapter 2, “Small Worlds and Large Worlds,” of Statistical Rethinking (McElreath, 2016). Again suppose that a card is pulled and a black side appears face up. As the hint suggests, let’s fill in the table below by thinking through each possible combination of first and second cards that could produce the observed data. Why things are normal. Both are equally common in the wild and live in the same place. $\Pr(A | \mathrm{twins}) = \frac{\Pr(\mathrm{twins} | A) \Pr (A)}{\Pr(\mathrm{twins})} = \frac{0.1(0.5)}{0.15} = \frac{1}{3}$ This reflects the idea that singleton births are more likely in species A than in species B. This results in the posterior distribution. They look exactly alike and eat the same food, and there is yet no genetic assay capable of telling them apart. $\Pr(\mathrm{BB})=\frac{\mathrm{BB}}{\mathrm{BB+BW+BW}}=\frac{2}{2+2+0}=\frac{2}{4}=\frac{1}{2}$ Syllabus. A common boast of Bayesian statisticians is that Bayesian inferences makes it easy to use all of the data, even if the data are of different types. $$\Pr(\mathrm{rain}|\mathrm{Monday})\Pr(\mathrm{Monday})$$, $$\Pr(\mathrm{rain}|\mathrm{Monday})\Pr(\mathrm{Monday})/\Pr(\mathrm{rain})$$, $$\Pr(\mathrm{Monday}|\mathrm{rain})\Pr(\mathrm{rain})/\Pr(\mathrm{Monday})$$. Now we can solve this like we have been solving the other questions: This early draft is free to view and download for personal use only. $\Pr(+|B) = 0.65$ These plausibilities are updated in light of observations, a process known as Bayesian updating. Here I work through the practice questions in Chapter 2, “Small Worlds and Large Worlds,” of Statistical Rethinking (McElreath, 2016). BW could only produce this with its black side facing up ($$1$$), and WW cannot produce it in any way ($$0$$). So the probability of the other side being black is indeed now 0.5. As our society increasingly calls for evidence-based decision making, it is important to consider how and when we can draw valid inferences from data. A black side is shown facing up, but you don’t know the color of the side facing down. $\Pr(\mathrm{single}) = \Pr(\mathrm{single}|A)\Pr(A) + \Pr(\mathrm{single}|B)\Pr(B) = 0.9(\frac{1}{3}) + 0.8(\frac{2}{3}) = \frac{5}{6}$ Your email address will not be published. Just in case anyone is still looking for the correct answer and has no explanation, a rewording of the statement “correctly identifies a species A panda is 0.8” helps. $\Pr(+|B) = 0.65$ Partial least squares structural equation modeling (PLS-SEM) is an important statistical technique in the toolbox of methods that researchers in marketing and other social sciences disciplines frequently use in their empirical analyses. The correct answers are thus Option 1 and Option 4. Hugo. Species A gives birth to twins 10% of the time, otherwise birthing a single infant. This is much easier to interpret as the probability that it is Monday, given that it is raining. $\Pr(A) = \frac{1}{3}$ Compute and plot the grid approximate posterior distribution for each of the following sets of observations. Compute the posterior probability that this panda is species A. We can now use algebra and the joint probability formula (page 36) to simplify this: After we already know marriage rate, what additional value is there in also knowing age at marriage? Which of the following statements corresponds to the expression: $$\Pr(\mathrm{Monday} | \mathrm{rain})$$? The probability of the other side being black is indeed 2/3. Below are my attempts to work through the solutions for the exercises of Chapter 2 of Richard McElreath's 'Statistical Rethinking: A Bayesian course with examples in R and Stan'. Statistical Rethinking is the only resource I have ever read that could successfully bring non-Bayesians of a lower mathematical maturity into the fold. $\Pr(B) = 0.5$, Next, let’s calculate the marginal probability of twins on the first birth (using the formula on page 37): $\Pr(B) = 1 – \Pr(A) = 1 – 0.36 = 0.64$, Now we just need to do the same process again using the updated values. Required fields are marked *. The Earth globe is 70% covered in water. So the final answer is 0.2307692, which indeed rounds to 0.23. $\Pr(B) = \frac{2}{3}$ Statistical Rethinking Chapter 5 Problems John Fox 2016-11-4. $\Pr(\mathrm{land} | \mathrm{Mars}) = 1$ Before looking at the other side, we draw another card from the bag and lay it face up on the table. Note that this estimate is between the known rates for species A and B, but is much closer to that of species B to reflect the fact that having already given birth to twins increases the likelihood that she is species B. What does it mean to say “the probability of water is 0.7”? The probability of the other side being black is now 4/5. Notes on Statistical Rethinking (Chapter 8 - Markov Chain Monte Carlo) Apr 19, 2018 33 min read StatisticalRethinking The Stan programming language is not an abbreviation or acronym. Statistical inference is the subject of the second part of the book. The test says B, given that it is actually B is 0.65. The probability that the female is from species A, given that her first birth was twins, is 1/3 or 0.33. Thus P(+|B) = 1 – P(-|B) = 0.35. Ultimately, statistical learning is a fundamental ingredient in the training of a modern data scientist. Now suppose all three cards are placed in a bag and shuffled. But the test, like all tests, is imperfect. Let’s simulate an experiment. Prior beliefs about Bayesian statistics, updated by reading Statistical Rethinking by Richard McElreath. Show that the posterior probability that the globe was the Earth, conditional on seeing “land” ($$\Pr(\mathrm{Earth}|\mathrm{land})$$), is 0.23. To keep things readable, I will also rearrange things to be in terms of singleton births rather than twins. After experimenting a number of times, you conclude that for every way to pull the BB card from the bag, there are 2 ways to pull the BW card and 3 ways to pull the WW card. The third card has two white sides. The purpose of this paper is to shed light on several misconceptions that have emerged as a result of the proposed “new guidelines” for PLS-SEM. That the data are grouped makes the assumption of independence among observations suspect. We can do this using the third formula on page 37. Using the test information only, we go back to the idea that the species are equally likely. Here, we describe the meaning of entropy, and show how the tenet of maximum entropy is related to time-reversal via the ergodic theorem. This equivalence can be derived using algebra and the joint probability definition on page 36: Lecture 02 of the Dec 2018 through March 2019 edition of Statistical Rethinking: A Bayesian Course with R and Stan. Specifically, if a positive test result is indication of the subject being from species A, P(+|B) should correspond to the false positive scenario where the test shows positive yet the subject is actually from species B. For bonus, to do this in R, we can do the following: Now suppose there are four cards: BB, BW, WW, and another BB. $\Pr(\mathrm{Earth}) = \Pr(\mathrm{Mars}) = 0.5$, Now, we need to use Bayes’ theorem (first formula on page 37) to get the answer: Prior beliefs about Bayesian statistics, updated by reading Statistical Rethinking by Richard McElreath. Let’s convert each statement to an expression: Option 1 would be $$\Pr(\mathrm{rain} | \mathrm{Monday})$$. $\Pr(B | \mathrm{twins}) = \frac{\Pr(\mathrm{twins} | B) \Pr (B)}{\Pr(\mathrm{twins})} = \frac{0.2(0.5)}{0.15} = \frac{2}{3}$, These values can be used as the new $$\Pr(A)$$ and $$\Pr(B)$$ estimates, so now we are in a position to answer the question about the second birth. The rules of probability tell us that the logical way to compute the plausibilities, after accounting for the data, is to use Bayes’ theorem. Show that the probability the other side is black is now 0.5. Software. Note the discreteness of the predictor groupsize and the invariance of the group-level variables within groups. This audience has had some calculus and linear algebra, and one or two joyless undergraduate courses in statistics. Statistical Thinking By Beth Chance and Allan Rossman. Show that the probability that the other side is also black is 2/3. As before, let’s begin by listing the information provided in the question: $\Pr(\mathrm{twins} | A) = 0.1$ So the total ways for the first card to be BB is $$3+3=6$$. $\Pr(\mathrm{rain},\mathrm{Monday})=\Pr(\mathrm{rain}|\mathrm{Monday})\Pr(\mathrm{Monday})$, Now we divide each side by $$\Pr(p)$$ to isolate $$\Pr(\mathrm{rain}|\mathrm{Monday})$$: ―Andrew Gelman, Columbia University "This is an exceptional book. ... Side note … The rst part of the book deals with descriptive statistics and provides prob-ability concepts that are required for the interpretation of statistical inference. Statistical Rethinking: A Bayesian Course with Examples in R and Stan builds readers' knowledge of and confidence in statistical modeling. If you find any typos or mistakes in my answers, or if you have any relevant questions, please feel free to add a comment below. So the posterior probability of species A (using just the test result) is 0.552. We can represent the three cards as BB, BW, and WW to indicate their sides as being black (B) or white (W). The second card has one black and one white side. Which of the expressions below correspond to the statement: the probability that it is Monday, given that it is raining? What we see is that any process that adds together random values from the same distribution converges to a normal distribution. $\Pr(A | +) = \frac{\Pr(+ | A) \Pr(A)}{\Pr(+)} = \frac{0.8(0.36)}{0.704} = 0.409$. $\Pr(\mathrm{rain}, \mathrm{Monday})/\Pr(\mathrm{rain})=\Pr(\mathrm{Monday}|\mathrm{rain})$ "Statistical Rethinking is a fun and inspiring look at the hows, whats, and whys of statistical modeling. Statistical physics is a beautiful subject. This is a rare and valuable book that combines readable explanations, computer code, and active learning." $\Pr(\mathrm{BB})=\frac{\mathrm{BB}}{\mathrm{BB+BW+WW}}=\frac{6}{6+2+0}=\frac{6}{8}=0.75$. Not for re-distribution, re-sale or use in derivative works. Chapter 1 A Review of Basic Statistical Concepts 5 assembled a dream team of behavioral economists to help him get elected—and then to tackle the economic meltdown. Differences to the oringal include: a preference for putting data into containers (data frames, mostly), rather than working with lose vectors. This means counting up the ways that each card could produce the observed data (a black card facing up on the table). In practice, Bayesian models are fit to data using numerical techniques, like grid approximation, quadratic approximation, and Markov chain Monte Carlo. The probability of rain, given that it is Monday. The American Statistician has published 43 papers on "A World Beyond p < 0.05." This dream team relied not on classical economic models of what people ought to do but on empirical studies of what people actually do under different conditions. Recall all the facts from the problem above. Here is a super-easy visual guide to setting up and running RStudio Server for Ubuntu 20 on Windows 10. $\Pr(+) = \Pr(+ | A) \Pr(A) + \Pr(+ | B)\Pr(B) = 0.8(0.5) + 0.65(0.5) = 0.725$ Each method imposes different trade-offs. Statistical Rethinking I just created a slack group for people who would like to do a slow read of McElreath's Statistical Rethinking. First ignore your previous information from the births and compute the posterior probability that your panda is species A. Statistical Rethinking: A Bayesian Course with Examples in R and Stan Book Description Statistical Rethinking: A Bayesian Course with Examples in R and Stan read ebook Online PDF EPUB KINDLE,Statistical Rethinking: A Bayesian Course with Examples in R and Stan pdf,Statistical Rethinking: A Bayesian Course with Examples in R and Stan read online,Statistical Rethinking: A … I agree – see https://github.com/jffist/statistical-rethinking-solutions/blob/master/ch02_hw.R. Below are my attempts to work through the solutions for the exercises of Chapter 3 of Richard McElreath’s ‘Statistical Rethinking: A Bayesian course with examples in R and Stan’. Winter 2018/2019 Instructor: Richard McElreath Location: Max Planck Institute for Evolutionary Anthropology, main seminar room When: 10am-11am Mondays & Fridays (see calendar below) In order for the other side of the first card to be black, the first card would have had to be BB. Recall the globe tossing model from the chapter. $\Pr(B) = 0.5$ $\Pr(\mathrm{single}|B) = 1 – \Pr(\mathrm{twins}|B) = 1 – 0.2 = 0.8$ $\Pr(\mathrm{land}) = \Pr(\mathrm{land} | \mathrm{Earth}) \Pr(\mathrm{Earth}) + \Pr(\mathrm{land} | \mathrm{Mars}) \Pr(\mathrm{Mars})=0.3(0.5)+1(0.5)=0.65$ $\frac{\Pr(\mathrm{rain},\mathrm{Monday})}{\Pr(\mathrm{Monday})} = \Pr(\mathrm{rain}|\mathrm{Monday})$. The $$\Pr(\mathrm{Monday})$$ in the numerator and denominator of the right-hand side cancel out: \sigma \sim \text{Uniform}(0,10) & [\text{prior for }\sigma] Posted Mar 22, 2019 $\Pr(\mathrm{BB})=\frac{\mathrm{BB}}{\mathrm{BB+BW+BW}}=\frac{2}{2+1+0}=\frac{2}{3}$. 99% Upvoted. 3.9 Statistical significance 134 3.10 Confidence intervals 137 3.11 Power and robustness 141 3.12 Degrees of freedom 142 3.13 Non-parametric analysis 143 4 Descriptive statistics 145 4.1 Counts and specific values 148 4.2 Measures of central tendency 150 4.3 Measures of spread 157 4.4 Measures of distribution shape 166 4.5 Statistical indices 170 So there are three total ways to produce the current observation ($$2+1+0=3$$). $\Pr(A) = 0.5$ If the first card was the second side of BB, then there would be the same 3 ways for the second card to show white. $\Pr(\mathrm{Earth} | \mathrm{land}) = \frac{\Pr(\mathrm{land} | \mathrm{Earth}) \Pr(\mathrm{Earth})}{\Pr(\mathrm{land})}=\frac{0.3(0.5)}{\Pr(\mathrm{land})}=\frac{0.15}{\Pr(\mathrm{land})}$, After substituting in what we know (on the right above), we still need to calculate $$\Pr(\mathrm{land})$$. Here is the chapter summary from page 45: This chapter introduced the conceptual mechanics of Bayesian data analysis. It can be helpful to create a table: To get the final answer, we divide the number of ways to generate the observed data given the BB card by the total number of ways to generate the observed data (i.e., given any card): Feb. 21, 2019. \], $$\mu_{i} = \alpha + \beta_{R}R_{i} + \beta_{A}A_{i}$$. This site uses Akismet to reduce spam. I'm working through all the examples, both in R and the PyMC3 port to python, but I find the statistics confusing at times and would love to bounce ideas off fellow students. So we can calculate this probability by dividing the number of ways given BB by the total number of ways: $\Pr(+|A) = 0.8$ The Bayesian statistician Bruno de Finetti (1906-1985) began his book on probability theory with the declaration: “PROBABILITY DOES NOT EXIST.” The capitals appeared in the original, so I imagine de Finetti wanted us to shout the statement. save hide report. $\Pr(A | \mathrm{twins}) = \frac{\Pr(\mathrm{twins} | A) \Pr (A)}{\Pr(\mathrm{twins})} = \frac{0.1(0.5)}{0.15} = \frac{1}{3}$. The probability it correctly identifies a species B panda is 0.65. Statistical Rethinking: Chapter 3. This […], This is a tutorial on calculating row-wise means using the dplyr package in R, To show off how R can help you explore interesting and even fun questions using data that is freely available […], Here I work through the practice questions in Chapter 7, “Interactions,” of Statistical Rethinking (McElreath, 2016). $\frac{\Pr(\mathrm{rain},\mathrm{Monday})}{\Pr(\mathrm{Monday})} = \frac{\Pr(\mathrm{rain}|\mathrm{Monday})\Pr(\mathrm{Monday})}{\Pr(\mathrm{Monday})}$ Assume these numbers are known with certainty, from many years of field research. The UNDP Human Development Report 2020 explores how human activity, environmental change, and inequality are changing how we work, live and cooperate. NOTE: Descriptive statistics summarize data to make sense or meaning of a list of numeric values. Finally, there would be no ways for the first card to have been the second side of BW or either side of WW. Option 3 is the probability of it being Monday, given rain. Note that this probability increased from 0.33 to 0.36 when it was observed that the second birth was not twins. Predictor residual plots. Statistical Rethinking is an introduction to applied Bayesian data analysis, aimed at PhD students and researchers in the natural and social sciences. If anyone notices any errors (of which there will inevitably be some), I … As a result, it’s less likely that a card with black sides is pulled from the bag. Imagine that black ink is heavy, and so cards with black sides are heavier than cards with white sides. Each card has two sides, and each side is either black or white. What he meant is that probability is a device for describing uncertainty from the perspective of an observer with limited knowledge; it has no objective reality. Assume again the original card problem, with a single card showing a black side face up. The best intro Bayesian Stats course is beginning its new iteration. $\Pr(A) = 0.5$ Lecture 07 of the Dec 2018 through March 2019 edition of Statistical Rethinking: A Bayesian Course with R and Stan. Option 2 is the probability of rain, given that it is Monday. Statistical Rethinking chapter 5 notes. So suppose now that a veterinarian comes along who has a new genetic test that she claims can identify the species of our mother panda. The probability that it is Monday and that it is raining. If anyone notices any errors (of which there will inevitably be some), I would be … share. So the statement, “the probability of water is 0.7” means that, given our limited knowledge, our estimate of this parameter’s value is 0.7 (but it has some single true value independent of our uncertainty). Of these three ways, only the ways produced by the BB card would allow the other side to also be black. P(test says A | B) = 1 – P (test says B | B) = 1 – 0.65 = 0.35, And for the posterior calculation, you would have to use Again calculate the probability that the other side is black.