1. The Nathan Kline Institute, NY
Applied Bayesian Inference: A Short Course Lecture 3 – Computational Techniques
Kevin H Knuth, Ph.D.
Center for Advanced Brain Imaging and Cognitive Neuroscience and Schizophrenia,
The Nathan Kline Institute, NY

2. Outline Exploring the Posterior Probability
Obtaining Answers to Questions
Cost Functions
MAP estimates
Markov Chain Monte Carlo
More Examples

3. Bayesian Inference Bayes' Theorem
describes how our prior knowledge about a model,
based on our prior information I, is modified by the
acquisition of new information or data:
Rev. Thomas Bayes
Likelihood
Posterior Probability
Prior Probability
Evidence

4. Exploring the Posterior
We have looked at
one-dimensional problems
The data is
The signal model is
Our estimate is
Probability
Phase Angle

5. Exploring the Posterior
And two-dimensional problems... a b
Log P

6. Looking at the Mode The mode of the probability density
is the parameter value at which the
peak occurs. This is often called the
Maximum A Posteriori (MAP)
estimate.
It is often most easily found by looking for the parameter value
at which the derivative of the probability (or Log p) is zero.
The variance can be estimated from the local curvature at the peak
One Dimensional Case
Multi-Dimensional Case

7. Looking at the Mean The mean is also commonly reported.
For a symmetric density it is equal to
the mode.
The variance can be computed by
calculating the expected squared
deviation from the mean, or sometimes
confidence intervals are defined to
delineate regions of high probability.

8. An Example with Difficulties

9. Stellar Distances from Parallax Hajian, Knuth, Armstrong
Here we consider a “simple” one-dimensional problem.
Given a measurement of stellar parallax, infer the distance to the star. D 

10. What is the Answer??
If we are given a mean and standard deviation for the parallax,
we can derive a Gaussian prior probability.
The probability for the distance can readily be found by the transform
where

11. What is the Answer?? Resulting in Do we report the mode? The mean?
The most probable distance estimate
does not correspond with the best
parallax estimate!
The result could make a big difference for someone planning a trip
Distance - D

12. Warning!
Be careful, the mode of the posterior of a parameter value will not
in general correspond to the mode of a transform of that parameter!
Often researchers will compute one optimal parameter and perform a
transform assuming that the result is optimal.
A physics example of this difficulty is a comparison of the
frequency spectrum and wavelength spectrum of blackbody radiation.

13. Cost Functions
To obtain a certain answer we must employ additional optimality
criteria.
By minimizing the expected squared error with respect to our
optimized distance we obtain as a “best” estimate of the distance
the mean.

14. Cost Functions
Other optimality criteria lead to different “best” estimates.
Minimization of the expected absolute value of the error
results in the median
Whereas, maximization of leads to the mode.
The mean minimizes the expected squared deviation of our chosen
answer from the correct solution, whereas the mode in the least-
squares approximation minimizes the expected squared deviation of
our predicted results and the data.

15. Marginalizing Problems Away
When one has a multi-dimensional posterior where many of the
parameters are uninteresting, we can marginalize over those
parameters to reduce the dimensionality of the problem.
However, rarely can more than 3 or 4 parameters be analytically
marginalized over.
Also conventional numerical integration techniques often fail due
to accumulation of round-off error

16. Other Difficulties How does one handle Multiply-peaked Distributions?
P(x|data,I) x

17. Other Difficulties
Or other multi-dimensional difficulties

18. Sampling from the Posterior
There is a way to sample from the posterior using a dynamical
system that evolves according to the probability density
Markov chain Monte Carlo
Allow exploration of high-dimensional parameter spaces
These techniques can actually increase in accuracy with higher
dimensional problems.
Converge slowly - O(n-1/2)

19. Markov chain Monte Carlo
Start with a position in parameter space X
Make a transition to a new point Y
with the transition probability T(Y|X)
We want the relative occurrences
of X and Y to be proportional to
the ratio of their probabilities.
We can control this by either accepting or rejecting the transition
with an acceptance probability . Y X
Model Parameter Space - M

20. Strolling along Hypothesis Space
Metropolis-Hastings Algorithm
The transition is performed by incrementing X with a random vector.
We employ a symmetric transition probability
with an acceptance probability of

21. Markov chain Monte Carlo
If the step size for the transition is too small, almost every step is
accepted, but it takes many steps for the samples to be independent.
If the step size is too large then the transitions are rarely accepted.
It is best to keep the acceptance rate between 30% and 70%

22. Running the Simulation
We evolve many (50) simulations
simultaneously.
In my MCMC I change one parameter
at a time. The numerous simulations
allow one to adjust the step size for
each parameter to make sure that the acceptance rate is high enough.
Bad runs (low probability simulations) can be abandoned and
good ones duplicated. Duplicated runs will diverge in time.

23. Simulated Annealing
Bring in the data slowly by writing the posterior as
By varying  from 0 to 1 we can slowly turn on the effect of the data.
= 0
= 0.5
= 1

24. Benefits of Annealing
Now with
We define a function
so that

25. Estimating the Evidence
We find
so that
we have
so we can calculate the evidence using simulated annealing.
This is sometimes called Thermodynamic Integration.

26. A MCMC Example

27. Estimating the BOLD Response Knuth, Ardekani, Helpern ISMRM 2001
Estimate the shape of the hemodynamic response function or
Blood Oxygenation Level Dependent (BOLD) response in an
event-related functional Magnetic Resonance Imaging (er-fMRI)
experiment.
Time (sec)
Amplitude (arb units)

28. Modeling the Experiment
We assume a parameterized form of the response (a unit amplitude normalized Gamma function)
Amplitude A, time dilation ,
shape parameter . The peak occurs
at time  
Time (sec)
Amplitude (arb units)
Given a series of stimuli at times {1, 2, ...i} we expect the total
hemodynamic response to be

29. Modeling the Experiment
We expect that there will be a baseline voxel intensity,
as well as a possible linear drift of the intensity due to the
magnetic field drifting during the experiment.
The intensity of a voxel is modeled as
where a is the linear drift, b is the baseline intensity and n(t) is
an unpredictable noise component.

30. The Data
The experiment was a visual oddball experiment where the subject pressed a button in response to the oddball. Data, v(t), was taken from a single voxel in the motor strip.
The presentation times of S=36 oddball stimuli are used in modeling the expected response.
A total of 256 gradient echo single shot EPI images were obtained using a Siemens 1.5T Magneton Vision system with: TR=2s, TE=60ms, image size of 64x64, 20 axial slices, 5mm slice thickness, and FOV=250mm.
Voxel Intensity
Time (seconds)

31. Assigning Probabilities
We have using Bayes’ Theorem
Assign a Gaussian for the Likelihood
The prior probabilities are
Cutoffs are employed as A, ,  are nonnegative,  positive, and
 reflects time scales shorter than 30sec

32. Solving the Problem The posterior is then
we marginalize over  to obtain
We still have a five-dimensional space to work in and must use MCMC.

33. Looking at the Posterior
We can take some slices of the logarithm of the five-dimensional posterior through some likely parameter values...
Log Probability
Log Probability
Amplitude A
Baseline Intensity b
Slice taken through
 = 1
 = 5
a = 0
b = 400
Slice taken through
A = 1
 = 1
 = 5
a = 0

34. Monte Carlo Results MCMC ran for 225,00 iterations, last 25,000 kept.
A = 3.38±1.28 (arb. intensity units)
 = 4.84±4.10 (unitless)
 = 1.28±1.73s
a = ± (arb int/sec)
b = ±0.517 (arb int)
peak time =   = 2.980.27s
(but notice the OTHER histogram peak!)

35. Monte Carlo Results
A density plot of the 25,000 sampled waveforms gives
a feel for the response and our uncertainty.
A = 3.38±1.28 (arb. intensity units)
peak time =   = 2.980.27s