SLIDE 108 Concentration and approximation of posterior Approximation of posterior
Practical implications of the BvM approximation
Of course, on a realistic practical scale we cannot hope to construct and manipulate p × p matrices such as V. However, the theorem can be used to calculate approximate posterior probabilities using posterior means and covariances estimated by (e.g.) MCMC.
54 Journal of the American Statistical Association, March 1997 Table 2. Brain Phantom Major MCMC Run: Estimates of Posterior Means, Integrated Auto-Correlation Times, and Standard Errors (SE's)
Parameter Mean
T(f)
M ;(f)/N SE High 188.29 12.392 38 .01377 1.6785 Medium 93.524 8.1245 25 .00903 .8502 Low 41.905 12.653 38 .01406 1.4670 3.6189 x 10-3 6.0526 19 .00673 2.2004 x 10-4 3 2.3384 x 10-2 11.003 34 .01223 9.6926 x 10-5
fully Bayesian posterior mean is not as aesthetically pleasing as the posterior mode from the OSL using the fixed "best" prior parameters. The former is slightly noisier and has RMSE value of 6.3670, compared to 5.7003 from the fixed parameter posterior
- mode. The RMSE value of the
fully Bayesian reconstruction is substantially less than the least possible
by ML/EM (8.2179). The MCMC run took 940.3 seconds
10/41 worksta- tion, compared to 30.0 seconds for OSL. Therefore, the comparative computational cheapness
when "good" fixed values
parameters are available discour- ages the use of MCMC to obtain reconstructions. However, in reality the variety
imaged and range
counts recorded may mean that these are not available. Us- ing other parameter values, the OSL algorithm invariably produced grossly oversmoothed
Thus the automation and quality
re- construction is admirable. Furthermore, the fully Bayesian method also allows
to go beyond mere reconstruction, because functionals
the body space can be easily obtained as a by-product
the MCMC. For instance, fully Bayesian posterior pixelwise modes could be obtained. These may be useful for examin- ing a specific region
The OSL fixed-parameter solution would not be capable
as it is a mode for the whole body space. Fully Bayesian posterior medians also
a a
.
CL~~~~~~~~
0.
5 10 15 20 25 30 5 10 15 20 25 30 pixel pixel
(a) (b)
0-
a
LO
a LO
Cu
C
..z.
*.....
0.0
5 10 15 20 25 30 5 10 15 20 25 30 pixel pixel
(c) (d)
Figure
Phantom: (a) Row 25 Truth and Posterior Mean
(
phantom;
(b) Row 25 Truth and Credibility Bounds (---, 90% bounds;
(---, 95%
bounds);
(c) Row 16 Truth
and Posterior Mean
( , Phantom;
(d) Row 16 Truth and Credibility Bounds
(---,
90% bounds; ---, 95% bounds).
can be easily obtained; here the reconstruction composed
was found to be very similar in ap- pearance to the posterior mean. Bayesian interval estimators are also easily obtained. Here I report pixelwise quantiles; however, Besag et al. (1995)
the construction
multaneous bounds for the whole vector
5 graphs two transects
phantom that cross distinct re- gions
The posterior mean and 90% and 95% pix- elwise credibility bounds are superimposed
It can be seen that the posterior mean faithfully reproduces the general truth shape, the major departure being overes- timation
activity region. However, the credibility bounds fully bracket the truth. In general, there may be situations where the credibility bounds do not give full information about the truth. For instance, the ability to successfully reconstruct small area hot spots and edges will be affected by low count/noisy data sets, the sampling bin widths, the reconstruction discretization used, and the
accuracy of the {ats} weight modeling.
We can also use output from the MCMC to get posterior density estimates
parameters. The density es- timates do not have pathological worth but do give insight into appropriate prior parameter values that could, for in- stance, be used to gain fixed-parameter OSL posterior mode reconstructions.
3.2 Real Data
The data relate to a horizontal slice through a brain and comprise 119,405 photon counts recorded by a system
64 detectors at 64 angles. The filtered back-projection re- construction provided by the Bristol Royal Infirmary is on a 64 x 64 grid
Figure 6a displays the cen- tral 48 x 48 pixels. The presence
radial artifacts is a well-documented deficiency
back-projection. Figures 6b and 6c show ML/EM reconstructions after 50 and 100 iterations. These reconstructions define the out- line of the brain better, but they are noisier. Figure 6d dis- plays the OSL solution using the prior parameter values chosen for the simulated data and reveals a clear isotope pattern that comprises several well-defined
ures 6e and 6f present the final realization from the fully Bayesian MCMC run and the resulting estimate
pos- terior mean. The fully Bayesian posterior mean is similar to the OSL fixed-parameter solution but is not as smooth and suggests greater contrast between several regions. Fig- ure 7 displays the posterior mean and credibility bands
a cross-section; the widths
bounds are not constant indicating the varying confidence in parts
solution.
4. DISCUSSION
The work presented here has demonstrated the ability to
representative Bayesian reconstructions
data without the need for prior parameter selections. The process is thus fully automated and can be applied with no supervision. The reconstructions are of comparable qual- ity to OSL ones when "good" fixed prior parameter val- ues are used and better when inappropriate values are used.
Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 68 / 70
