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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
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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
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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
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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.