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Exam I review • Understanding the meaning of the terminology we use. • Quick calculations that indicate understanding of the basis of methods. • Many of the possible questions are already sprinkled in the lecture slides. Introduction to Uncertainty • • • • Aleatory and Epistemic uncertainty Uncertainty reduction measures Histograms, pdfs and cdfs Example problem: A farmer has a model for predicting the yield of his crop based on the amount of rain measured over his field, soil conditions, number of days of sunshine, and average temperatures during the growing season. List the epistemic and aleatory uncertainties that render this model less than perfect. Random Variable Distributions • Properties of the normal distributions • Mean, median, mode, standard deviation, variance, coefficient of variance, probability plots. • Light and heavy tails, extreme distributions. • Example: Indicate two different plots that you can use to get a quick estimate of the mode of a distribution from a sample. • Example: What is the probability that a standard lognormal variable is larger than 2? Set theory and Bayes’ law • • • • Sets terminology, notation, operations, and axioms Venn diagrams Conditional probabilities and Bayes’ rule Example: A patient who had received a flu shot shows up at a doctor’s office complaining of flu symptoms. You know that for his age group the vaccine is 70% effective, and that the symptoms indicate the flu 80% of the time when one has the flu, and 20% of the time when one does not have it. What is the probability that the patient does not have the flu? Bayesian posteriors • Difference between classical and Bayesian probabilities. • Bayes’ rule for pdfs. Prior, likelihood and posteriors. • Example: You are testing a coin for bias to show heads. The first five tosses were all heads. What is the likelihood that it is unbiased? • Example: A random variable follows the random distribution p(x)=2x in [0,1]. A friend tells you that he observed an occurrence of this variable and it was larger than 0.5. What is the probability distribution of the occurrence your friend observed? Bayesian estimation with the binomial distribution • The binomial and beta distributions. • Posterior and predictive distributions after n observations. • Conjugate prior. Conjugacy. • Example: A family has two sons and is expecting another. By how much did the probability of having three sons changed from before the first son to now? • Example: The formulas for the binomial and beta distributions appear almost the same. However, there is a fundamental difference. What is it? Single parameter normal • Posterior and predictive mean and standard deviations. • Chi square distribution. • Example: How does the posterior distribution of the variance depends on the known mean? • Example: You are sampling from a normal distribution with the variance known to be 4. A single sample was at x=0. What Matlab command would you use to sample the predictive distribution? Two parameter normal • • • • Posterior and predictive distribution. Marginal and conditional distributions. Methods of sampling from posterior distribution. Example: the pdf of the random variables x and y is proportional to xy in the unit square. What are the marginal distributions? Describe how you would obtain samples of x,y. Bioassay problem • Terminology, description of method, logit transformation. • Inverse CDF method. • Grid method. • Predictive distribution and LD50 • Example: How would you sample the triangular distribution p=2x, using the inverse CDF with Matlab? Simulation techniques • Summary and questions • Rejection sampling. • Example: Write the Matlab code for sampling the triangular distribution p=2x, using the normal distribution as proposal distribution and rejection sampling. • Example: How would you use the normal to minimize the number of rejected samples? Importance sampling • Comparison with rejection sampling. • Calculation of moments and probabilities. Indicator functions. • Accuracy of probability calculation. • Example: x and y are standard normal, and you want to use sampling to estimate the probability of xy>10. What would be a good proposal distribution for the sampling? Markov Chain Monte Carlo • Transition matrix, Markov chain, Metropolis algorithm for discrete probabilities. • Example: Indicate two different ways of finding the long range transition matrix.