Estimating probability of binary
events
In many estimation problems we know the distribution of
samples and we only need to estimate the distribution
parameters
In this lecture we will illustrate this with the binomial
distribution, which is the distribut
Derivatives are more difficult than
integrals
How would you measure the speed of the paper
helicopter when it hits the ground?
Comparing simulations to experiments
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Details for previous slide
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Viewgraph norm: position one picture next to
Bioassay example
Example 3.7 from BDA
Bioassay is a procedure applied to new medications to
test toxicity. Here the results are in terms of deaths
caused by different doses of drug.
Binomial distribution where q changes over samples
More realistic example
Outline
Historical note about Bayes rule
Bayesian updating for probability density
functions
Coin trials example
Reading material:
Salary offer estimate
Gelman, Andrew, et al. Bayesian data analysis. CRC
press, 2003, Chapter 1.
Slides based in part on lec
Set theory and Bayes Rule
Sets and set operations
Axioms of probability
Probabilities of sets
Bayes rule
Chapter 2 of Haldar and Mahadevans Probability,
Reliability and Statistical Methods in Engineering Design,
John Wiley, 2000.
Slides based in part on l
Two Parameter normal
We considered estimating the mean alone or the
variance alone
This lecture deals with estimating both together
We will consider various techniques of obtaining
statistics from these distributions.
Posterior expression
objective
Estima
Bayesian inference review
Objective
Bayesian inference
distribution
function
distribution
5
q
Posterior
4
Establish prior of q if any.
Establish likelihood of y conditional on q
Derive posterior distribution
Posterior analysis & prediction
estimate unknow
Calibration with discrepancy
Major references
Calibration lecture is not in the book.
Kennedy, Marc C., and Anthony O'Hagan. "Bayesian calibration of
computer models." Journal of Royal Statistical Society: Series B (2001).
Campbell, Katherine. "Statistica
Binomial probability estimation
Playing chess against a friend you won 3 out of 5
matches and lost 2. Assuming that wins and losses
follow the binomial distribution:
What is the probability that you will win the next match?
What is the probability distrib
Simulation techniques
Summary of the methods we used so far
Other methods
Rejection sampling
Importance sampling
Very good slides from Dr. JooHo Choi of Korea
Aersopace University, so very minor modifications and
kept his format but added his Matlab sc
Basic problem
A random variable X follows the exponential distribution,
p(x)=exp(x) for x=>0. Check how different ways of
sampling will compare in terms of accuracy for
estimating the probability of x>2 with 1,000 samples.
Exact value of probability is e
Importance sampling
Like rejection sampling, we will use a convenient distribution
(proposal distribution) instead of actual one.
Why did we have the condition Mq>p in rejection sampling?
Because we could not accept with probability greater than one.
Im
Fundamental concepts and terminology
of Verification and Validation
Verification is the process that
checks whether mathematical
model was implemented
accurately enough.
Validation is the process that
checks that simulation is
close enough to reality.
Thi
Computer model under uncertainty
In previous lecture on accuracy assessment
We considered mostly deterministic models.
We did not distinguish between epistemic and aleatory uncertainty
In this lecture,
Model under uncertainty is studied.
Validation metric
Method of manufactured solutions
The first stage in code verification is to test for problems for
which we have analytical solutions.
However, often we have codes that are intended for more
complex problems than we can solve analytically.
Codes typically
CCM
T
Validation of Shock Tube
Simulation
Chanyoung Park, Raphael (Rafi) T. Haftka
and
NamHo Kim
Department of Mechanical & Aerospace Engineering,
University of Florida
CCMT
Motivation
Explosive solid particle dispersal

Explosive processes influence d
Summary of Pbox
Probability bound analysis (PBA)
PBA can be implemented by nested Monte Carlo
simulation.
Generate CDF for different instances of the epistemic uncertainty
Pox is given by bounds of probability distribution.
As a result, probability is i