/
Exam I review Exam I review

Exam I review - PowerPoint Presentation

celsa-spraggs
celsa-spraggs . @celsa-spraggs
Follow
417 views
Uploaded On 2016-03-07

Exam I review - PPT Presentation

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 ID: 246262

sampling distribution probability distributions distribution sampling distributions probability normal posterior flu predictive variable random sample standard probabilities variance binomial rejection bayesian uncertainty

Share:

Link:

Embed:

Download Presentation from below link

Download Presentation The PPT/PDF document "Exam I review" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.


Presentation Transcript

Slide1

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.Slide2

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.Slide3

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?Slide4

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?Slide5

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?Slide6

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?Slide7

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?Slide8

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

.Slide9

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

?Slide10

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?Slide11

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?Slide12

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.