Bayesian Analysis Software User Manual
Contents
1. Figure 1 2 When one of the Bayesian Analysis packages is selected from the Packages pull down menu the appropriate interface is displayed here the interface to the exponential package is dis played A package interface consists of three parts the global pull down menus along the top the package setup widgets just below the global pull down menus and the viewing area the dark blue area at the bottom 24 AN OVERVIEW widget groups are common to all packages However most packages have some variation of the five seen in the Exponential package but some packages have more and some have less For the exponential package here is a brief description of these widget groups Submit Job to Server is a widget group that has three buttons and one text area This widget group is responsible for submitting jobs checking on there status and when necessary canceling jobs e The Run button is used to submit a job to a server If the currently selected server is named Serverl then the Run button will submit the job to Serverl and it will change the Run Status text area to Active or Submitted depending on whether the server uses a queuing facility When the run button is activated most of the widgets on the interface are disabled This is to prevent the user from making changes to the configuration while a job is running e The Get Job button sends a request to the currently selected server requesting the status of the current job I2. Lipper via 13C alpha NMR Biophysical Journal 80 pp 939 951 Jaynes E T 1968 Prior Probabilities IEEE Transactions on Systems Science and Cyber netics SSC 4 pp 227 241 reprinted in 29 Jaynes E T 1978 Where Do We Stand On Maximum Entropy in The Maximum Entropy Formalism R D Levine and M Tribus Eds pp 15 118 Cambridge MIT Press Reprinted in 29 Jaynes E T 1980 Marginalization and Prior Probabilities in Bayesian Analysis in Econometrics and Statistics A Zellner ed North Holland Publishing Company Amsterdam reprinted in 29 Jaynes E T 1983 Papers on Probability Statistics and Statistical Physics a reprint collection D Reidel Dordrecht the Netherlands second edition Kluwer Academic Publishers Dordrecht the Netherlands 1989 Jaynes E T 1957 How Does the Brain do Plausible Reasoning unpublished Stanford University Microwave Laboratory Report No 421 reprinted in Maximum Entropy and Bayesian Methods in Science and Engineering 1 pp 1 24 G J Erickson and C R Smith Eds 1988 Jaynes E T 2003 Probability Theory The Logic of Science edited by G Larry Bret thorst Cambridge University Press Cambridge UK Jeffreys Harold Sir 1939 Theory of Probability Oxford Univ Press London Later editions 1948 1961 Jones John G 2001 Michael A Solomon Suzanne M Cole A Dean Sherry Craig R Malloy An integrated 7H a
3. Nonuniform Sampling Bandwidth and Aliasing in Maximum Entropy and Bayesian Methods in Science and Engineering Joshua Rychert Gary Erickson and C Ray Smith eds pp 1 28 American Institute of Physics USA Bretthorst G Larry Christopher D Kroenke and Jeffrey J Neil 2004 Characterizing Water Diffusion In Fixed Baboon Brain in Bayesian Inference And Maximum Entropy Methods In Science And Engineering Rainer Fischer Roland Preuss and Udo von Toussaint eds AIP conference Proceedings 735 pp 3 15 Bretthorst G Larry William C Hutton Joel R Garbow Joseph J H Ackerman 2005 Exponential parameter estimation in NMR using Bayesian probability theory Concepts in Magnetic Resonance 27A Issue 2 pp 55 63 Bretthorst G Larry William C Hutton Joel R Garbow Joseph J H Ackerman 2005 Ex ponential model selection in NMR using Bayesian probability theory Concepts in Magnetic Resonance 27A Issue 2 pp 64 72 Bretthorst G Larry William C Hutton Joel R Garbow Joseph J H Ackerman 2005 How accurately can parameters from exponential models be estimated A Bayesian view Concepts in Magnetic Resonance 27A Issue 2 pp 73 83 Bretthorst G Larry W C Hutton J R Garbow J J H Ackerman 2008 High Dynamic Range MRS Time Domain Signal Analysis Magn Reson in Med 62 pp 1026 1035 Chandramouli Visvanathan Karin Ekberg William C Schumann Satish C Kalhan John
4. means spectroscopic FID data For more information on this viewer see Section 3 4 2 e The Image Viewer is used to display 4dfp images For more information on this viewer see Section 3 4 2 e The Prior Viewer is used to display and set the prior probabilities used in the Bayesian calculations For more information on this viewer see Section 3 4 4 e The FID Model Viewer is used to display FID models generated by packages that process FID data For more on this viewer see Section 3 4 5 26 AN OVERVIEW e The Plot Results Viewer is used to display the plots associated with an analysis and is the primary method for viewing the results of an analysis For more on this viewer see Section 3 4 6 e The Text Results Viewer is used to display and print the Ascii files that result from an analysis For more on the Text Results viewer see Section 3 4 8 e Finally the File Viewer is used to view the all the files generated by analysis For more on the Text Results viewer see Section 3 4 9 The overview given in this Chapter should give you some indication of what the software can do The Java interface provides a simple user friendly way of setting up a Bayesian Analysis After the analysis is set up the interface will automatically ship the analysis to the selected server The Bayesian Analysis software on that server can run many different types of analysis relevant to NMR in parallel The interface allows the user to leave an analysis while its ru
5. servers change working directories set options etc Each pull down menu has multiple functions and the following Sections explain these menus and how to go about using them 1 2 1 The Global Pull Down Menus The global pull down menus a the top of the interface are always present They allow you to select Bayesian Analysis applications configure servers change WorkDir etc Each item across the top is a pull down menu and each menu has multiple functions These functions are explained in detail in Section 3 Here we give a brief summary of these menus Files is pull down menu that allows you to perform various tasks involving files For example you can load Ascii data FID spectral or image data and images Additionally you can save the current WorkDir and you can restore a previously saved experiment See Section 3 1 1 for more on the files submenu Packages is a pull down menu that allows you to select the Bayesian Analysis package you wish to use Each of the packages is described in more detail in the upcoming Chapters See Section 3 1 2 for a more extensive discussion of the packages pull down menu WorkDir is a pull down menu that allows you to select create or delete a WorkDir Working directories are contained within the Bayes directory in your home account These directories are scratch areas used to contain the loaded data configuration files and the results of running an analysis See Section 3 1 3 for a more extensive dis
6. the Given Polynomial Order Package 17 Unknown Polynomial Order Inl Bayesian Calculations lt e 20 26422 wae ee eee beet ech Ho y eid 171 1 Assigning Priors oe ak a Rand 36308 a Rae he EDAD REC acs 17 1 2 Assigning The Joint Posterior Probability o 17 2 Outputs From the Unknown Polynomial Order Package 18 Errors In Variables 18 1 The Bayesian Calculation oa um 9 ws E S Dew es 18 2 Outputs From The Errors In Variables Package 19 Behrens Fisher 19 3 Bayesian Caleilabiol lt sepais A ee a a amp A DA RA ee ds 19 1 1 The Four Model Selection Probabilities 19 1 1 1 The Means And Variances Are The Same 19 1 1 2 The Mean Are The Same And The Variances Differ 19 1 1 3 The Means Differ And The Variances Are The Same 19 1 1 4 The Means And Variances Differ 19 1 2 The Derived Probabilities 2212 3333 kb 9 o8 o wo A wR ta 19 1 3 Parameter Estimation esos sassa 644444 ses e5GG 45 eee eda s 19 2 Outputs From Behrens Fisher Package eee eee eee 20 Enter Ascii Model 20 1 The Bayesian GCalcilatione 2421 ee ae Se eee AAA RR eS 20 1 1 The Bayesian Calculations Using Eq 20 1 o oo 20 1 2 The Bayesian Calculations Using Eq 20 2 o oo 20 2 Outputs Form The Enter Ascii Model Package o 21 Test You
7. to create a new WorkDir After you create and join a new WorkDir the first thing you must do is to select the package you wish to use The global pull down menus along the top of the startup page are always present on all package interfaces not just the startup page They allow the user to load files select packages configure THE BAYESIAN ANALYSIS SOFTWARE AN OVERVIEW 21 Bayesian Analysis of Common NMR Problems version 4 01 File Package WorkDir Settings Utilities Help To start new analysis select the package you wish to run under Package menu 3 Washington University in St Louis To restore analysis BayesEnterAscii saved in AbscissaTesting press Restore Analysis SCHOOL OF MEDICINE button RESTORE ANALYSIS Bayesian Analysis of Common NMR Problems Developed at Washington University in St Louis Mallinckrodt Institute of Radiology Bayesian Analysis Software developed by Java Interface developed by Larry Bretthorst Ph D Karen Marutyan Ph D Research Associate Professor of Radiology Post doctoral Researcher Washington University St Louis MO Washington University St Louis MO gbretthorst wustl edu marutyan wustl edu Lely Dci Figure 1 1 The Bayesian Analysis Startup Page allows you to select what functions you wish to perform For example you might restore an old analysis change a setting run one of the utility programs or select a new WorkDir or a new Bayesian Analysis package 22 AN OVERVIEW
8. 0 Bretthorst G Larry 1990 Bayesian Analysis III Examples Relevant to NMR J Magn Reson 88 pp 571 595 Bretthorst G Larry 1991 Bayesian Analysis IV Noise and Computing Time Considera tions J Magn Reson 93 pp 369 394 Bretthorst G Larry 1992 Bayesian Analysis V Amplitude Estimation for Multiple Well Separated Sinusoids J Magn Reson 98 pp 501 523 Bretthorst G Larry 1992 Estimating The Ratio Of Two Amplitudes In Nuclear Magnetic Resonance Data in Mazimum Entropy and Bayesian Methods C R Smith et al eds pp 67 77 Kluwer Academic Publishers the Netherlands Bretthorst G Larry 1993 On The Difference In Means in Physics amp Probability Essays in honor of Edwin T Jaynes W T Grandy and P W Milonni eds pp 177 194 Cambridge University Press England Bretthorst G Larry 1996 An Introduction To Model Selection Using Bayesian Probability Theory in Maximum Entropy and Bayesian Methods G R Heidbreder ed pp 1 42 Kluwer Academic Publishers Printed in the Netherlands 415 416 12 13 14 15 16 17 18 19 20 21 22 23 no i BIBLIOGRAPHY Bretthorst G Larry 1999 The Near Irrelevance of Sampling Frequency Distributions in Maximum Entropy and Bayesian Methods W von der Linden et al eds pp 21 46 Kluwer Academic Publishers the Netherlands Bretthorst G Larry 2001
9. 151 8 2 The Bayes Analyze Model Equation 2 e 153 8 3 The Bayesian Calculations cea eR Y EG EE ONS a 159 8 4 Levenberg Marquardt And Newton Raphson s 163 8 5 Outputs From The Bayes Analyze Package o e 8 5 1 The bayes params nnnn and bayes model nnnn Files 8 5 1 1 The Bayes Analyze File Header lcs 85 1 2 The Global Parameters coso ri 8 5 1 3 The Model Components aa occu u llle 802 The bayesoubput nbne Piles 2 4 42 66844 aa a bee SE ee 8 5 3 The bayes probabilitiesnnnn File o 8 5 4 The bayes log nnnn File o 8 5 5 The bayes status nnnn and bayes accepted nnnn Files 8 5 5 1 The bayes model nnnr File o 8 5 6 The bayesssummaryl nnnn File o 8 5 7 The bayes ssummary2 nnnn File a accs csta t esasta o 8 5 8 The bayesisummearys innnn Files 22 2 t t x R3 Ey 8 6 Bayes Analyze Error Messages eh 9 Big Peak Little Peak 9 1 The Bayesian Calculation lacio Oe OE Rm a UR ORE a e RC RCRUS 9 2 Outputs From The Big Peak Little Peak Package leen 10 Metabolic Analysis 10 1 The Metabole Model 2 222299 mom mo RK KARE eGR MEA ww REESE 10 2 The Bayesian Calculation oa cir see a See a AA eR ee Y s 10 3 The Metabolite Models s s 605s e ea boo L
10. 22 24 REXETI ES The Subrodtie Body ex Wack ee ee mo 48 9 box s E 6 Model Subroutines With Marginalization F the Bayes Directory Organization G 4dfp Overview H Outlier Detection Bibliography 343 343 345 347 361 363 365 367 367 368 369 371 372 373 374 374 375 376 381 385 391 391 394 396 398 399 400 405 407 411 415 List of Figures 3 21 3 22 3 23 3 24 3 25 3 26 3 27 3 28 3 29 3 30 3 31 The Start Up Window i sosa RA RSA 4 DO 9 3 3 9 4 RE eS 21 Example Package Interface sosa e ee y a EA OE RU XU Y E 23 Whe Start Up WIDdONW es e Seka ee eee a X PPP E A ACE SOR SI see G 30 Tbe Piles Wend 2 uo eu eee deemed des o Roh oho RR mene 31 The Load Image Selection Menu o o a Aa A ee eee 33 The Packages Menu 24 4 2 9 9 9999 a a A 37 The Working Directory Pull Down Menu o 42 The Working Directory Po pup ceo 444 604 aaa daa taaa ateo m RES 43 The Settings Pull Down Menu ee 44 The McMC Parameters Po pup 2 02 42 LES ee See RBAUS os 44 The Edi Server POPUD cs rama 4 YA Wank e e ce udo box xk OES ow 45 The Submit Job Widget Group o Roto he pee atta eui 9 Ee me n 48 The Server Widget Group cnc Behe eae a RR AU EA 49 the Ascii Data viewer s sc cora osoa ls ee 50 the tid Data viewer uu c a aralara ee eee ee Eee eee ee es 52 The Pid Data Viewer Display Type o ca sra sunud 044444 24 Ry 53 The Fid Data
11. Bayesian Analysis Users Guide Release 4 00 Manual Version 1 G Larry Bretthorst Biomedical MR Laboratory Washington University School Of Medicine Campus Box 8227 Room 2313 East Bldg 4525 Scott Ave St Louis MO 63110 http bayes wustl edu Email larry bayes wustl edu August 21 2013 Contents Manual Status 1 An Overview Of The Bayesian Analysis Software Ll The Server Software duco mec eee eR Gea an CP Pee ee 1 4 463 12 The Client Interiace lt i sone m Romo m HRA ORG RO ROW A OX xoxo xs 1 2 1 The Global Pull Down Menus ccr m do EE la 1 2 2 The Package Interface 266 voe Ro sema m x 3 E Re s 1 2 9 The Viewers 6 4 60 ob ok owonchom mox bode dee nee ee ada OX R44 4 4 43 2 Installing the Software 3 the Client Interface 2I The Global Pul Down Men s Rom T oy eee eee SEA s l Ahe Piles mend o e ARA OX a RA 351 2 the Packages Me o 2o v wv AAA WX S oos A9 the WorkDir MENU saxo REDE S 9 ow SHER RRS OSS Row mew d Sled the Settings Mend oso noria 3o odo rot ok RR XX Ee pops 21 9 the D bbiesdBeHU uon Rom SA C 8 OR 8 9 X Aw Bl MR Dee Alb wire Help mel ulna a a a SOUS mde a qe AUR ee e RRE 3 2 The Submit Job To Server ar a 449 Las oe ey ow 4 3 3303 9 3 VR es mu The Server ares fa seek ke GG Rue oe dem m LAUR E RM AU ORE EE ue 24 Interface Viewers i qe go eo oe e OA EEE ee RAO Ao Y OC 34 1 the Asci Data Viewer o 9 442 24444455565 8 Be mom eg RO dede ae the Bd Data Viewer s a R
12. Binned Histogram package estimates a binned density function with error bars In the near future we will be enhancing this package to perform model selection That is to say the binned histogram package will automatically determine both the number of bins and smoothing need to describe the density function e The Linear Phasing package produces linearly phased images In NMR the complex image data have phases that vary across the image in a linear fashion These linear phases are present because of the gradients that are used to generate an MR image The linear phasing package estimates the value of the zero and first order phases in the phase encode and readout domains and then unwraps this phase so that the image can be displayed in absorption mode e The Non Linear phasing package phases images that have phases that are varying in a Non Linear fashion In this package the phases are estimated on a pixel by pixel basis and the estimated phase is used to generate an absorption mode image e The Image Pixels package loads a one of the predefined Ascii models and then uses that model to analyze images on a pixel by pixel basis The loaded models can be generated by the users or they can be loaded from a system library that we provide 20 AN OVERVIEW e The Image Pixels Model Selection package extends the concepts in Analyze Image Pixels to model selection In this package the user can load a number of different models that describe the signal in a p
13. Viewer the Options Menu a 54 The Image Viewer 2c won x Sog x xe heme OX OE emm OE PEOR ERR cR dU EOS 57 The Image Viewer Right Mouse Menu o e eee 58 he Prior Viewer I ie eae OEGE O e c9 wd 3 box ERS ss 63 The Fid Model Viewer 2 2 o 242 ses RR Es ERE Ge ee EEE SS 66 The Data Model and Resiuals 3s 24 GSE m9 e m aaa ee es 69 The Plot Information popup cc soa 44 5 Ae m mcm memor om ek ee Re 70 The Posterior Probabilities aos s so areca tie oo 71 The Posterior Probabilities Vs Parameter Value noana o 73 The Posterior Probabilities Vs Parameter Value a Skewed Example 74 The Expected Log Likelihood o o e eee ee eee 76 The Seatter Plols 22x33 ope oum a vom a ald 77 The Los Probability Plot 4 oO Wo SS sie eee Ga OY ORR Ged 79 The Text Results Viewer cocinera 644 04 2 oe eee ee eee bees 81 The Bayes Condensed File Room m mme A 84 Bortran C Model VIeWeE osse en ex E PIONEER Ey Ye es 87 Portia C Model Viewer vs oss aes RE A es 88 10 4 1 4 2 4 3 4 4 4 5 4 6 4 7 5 1 6 1 6 2 6 3 Fe 8 15 8 16 8 17 8 18 Oo 9 2 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 Frequency Estimation Using The DFT llle 104 JAMASeS TAR 105 Nonuniformly Nonsimultaneously Sampled Sinusoid 4 119 AMAS SPACE e ee c 120 Which ls The Critical Time o sa iie s iio ee 09 3 Ge 9 ed ea eS 122 Example Frequency Estimation o l
14. Wahren and Bernard R Landau 1997 Quantifying gluconeogenesis during fasting Amer ican Journal of Physiology 273 pp H1209 H1215 Cox R T 1961 The Algebra of Probable Inference Johns Hopkins Univ Press Baltimore d Avignon Andr G Larry Bretthorst Marlyn Emerson Holtzer and Alfred Holtzer 1998 Site Specific Thermodynamics and Kinetics of a Coiled Coil Transiton by Spin Inversion Trans fer NMR Biophysical Journal 74 pp 3190 3197 d Avignon Andr G Larry Bretthorst Marlyn Emerson Holtzer and Alfred Holtzer 1999 Thermodynamics and Kinetics of a Folded Folded Transition at Valine 9 of a GCN4 Like Leucine Zipper Biophysical Journal 76 pp 2752 2759 Gilks W R S Richardson and D J Spiegelhalter 1996 Markov Chain Monte Carlo in Practice Chapman amp Hall London Goggans Paul M and Ying Chi 2004 Using Thermodynamic Integration to Calculate the Posterior Probability in Bayesian Model Selection Problems in Bayesian Inference and Mazi mum Entropy Methods in Science and Engineering 23rd International Workshop Volume 707 pp 59 66 BIBLIOGRAPHY 417 25 26 27 28 29 30 31 32 33 34 35 36 37 38 Holtzer Marlyn Emerson G Larry Bretthorst D Andr d Avignon Ruth Hogue Angelette Lisa Mints and Alfred Holtzer 2001 Temperature Dependence of the Folding and Unfolding Kinetics of the GCN4 Leucine
15. ck to their default values Save is will bring up a popup that allows you to navigate to the location where you want to save the current WorkDir and then to Save the current WorkDir The Set button will save a WorkDir 1 2 3 The Viewers After a job has been run and retrieved by the interface the interface unpacks the result of the analysis After unpacking the run status is set to Run and the various viewers located at the bottom of the interface can be used to look at the results of an analysis These viewers are act to display various kinds of data The buttons along the center of the interface activate the various viewers These Viewers are Ascii Data Viewer FID Data Viewer Fid Model Viewer Plot Results Viewer Text Results Viewer File Viewer used by the interface to display different kinds of data Because the display requirements for different types of data are very different there are many different viewers Not all viewers show up on all packages On the Exponential package the viewers shown above there are seven of these viewers and this is pretty typical of all packages For more information on these viewers see Chapter 3 4 Here we are just going to briefly list the viewers and note there primary function e The Ascii Data Viewer is used to display Ascii data For more information on this viewer see Section 3 4 1 e The FID Data Viewer allows you to look at both the time and frequency domain FID data Here FID data
16. cussion of working directories Settings is a pull down menu that allows you to configure the Bayesian Analysis packages The various menu items allow you to configure the Markov chain Monte Carlo simulations see Section 3 1 4 add delete and modify server settings See Section 3 1 4 and it allows you to configure some optional features of the software Utilities is a pull down menu that allows you to start a memory monitor get information on the system you are running and allows you to determine if there is an updated version of the Bayesian Analysis software See Section 3 1 5 for more on the utilities Help is a pull down menu that allows you to view information about the current installation of the Bayesian Analysis software and it allows you to visit the Bayesian Analysis Software home page 1 2 2 The Package Interface When one of the packages is selected the interface displays that package interface For example if the Exponential package is selected the interface shown in Fig 1 2 is displayed This interface is very similar to the interface of many other packages and we will use it to illustrate some of the general features of the Interface First note that the global menus that were present on the Bayesian Analysis Home Page are present on all package interfaces Second below the global menus is an area that is used to configure a package Each set of widgets are enclosed in a highlighted box We are going to call these enclose
17. d widgets widget groups and we will name them based on the name above each group So on the Exponential interface there are five widget groups The first two Submit Job to Server and Server THE BAYESIAN ANALYSIS SOFTWARE AN OVERVIEW 23 Given and Unknown Number of Exponentials test2 Host bayes File Package WorkDir Settings Utilities Help Submit Job to Server Server Model E Analysis Option SaveReset fr f f gt r l RUN J Cancel Set Status Set Order 1 v Find Outliers G Save Get Job Not Run bmrw75 Include Constant O Reset Ascii Data Viewer FID Data Viewer Image Viewer Prior Viewer Fid Model Viewer PlotResults Viewer TextResults Viewer File Viewer Print copy Save Save as Enable Editing jScroll Up V settings Instructions Probability model MCMC values Bayes params To use the Exponential package Console log Load an accii file Bayes accepted Image Abscissa Specify the number of exponentials or specify unknown to enable Fortran Ist automatic model determination When the number of exponentials is given specify whether or not a constant is present Select the server to run the analysis Probability model Bayes params J E Run the analysis using the Run button Console log Log Output Use Get Job to get the results from the server Model Status Summary Best Model Summary2 Best Summary Summary3 Best Regions
18. del Interface 222224 330 Ascii Model Selection Interface eooo ss zo o m kk RR RR R4 4n 332 Absorption Model Images iue no o9 ons Ox WO 9o mUR RR ROX RO E E E E 334 Bayes Phase Interface osos mon ox a 335 Bayes Phase Listing 2 44 4 4 xk OX x X X e AEE MR OX E a a a 341 12 24 1 24 2 28 1 A 1 p D 2 D3 E 1 E 2 Ea E 5 E 6 G 1 Hl H2 Nonlmear Phasing Example a SG ee wx hok buo cod ox AY X 9 5 344 Nonlinear Phasing Interface og essu mona ces kA RR e 348 Image Pixels Example 3444 6 ee ee Rufo ded orc AR S AUS Sex SOR 362 Ascii Data File Format Se a bx RR RUBENS ee eS SEE 368 The McMC Values Report Header lees 386 MeMC Values Report The Middle ooo a R a S a 387 The McMC Values Report The End o o 388 Writing Models A Fortran Example e eee eee 392 Writing Models A C Example aaa 393 Writing Models The Parameter File o 395 Writing Models Fortran Declarations o o naci a a ei aa a a a a eee eens 399 Writing Models Fortran Example 402 Writing Models The Parameter File so s sa cci audda enana iadaaa 403 The FD File Header 225492299 RR ee A XR Pee RRA 409 the Posterior Probability for the Number of Outliers a a ouaaa aaa 412 The Data Model and Residual Plot With Outliers lll 414 List of Tables 8 1 Multiplet Relative Amplitudes eee eee e
19. e Peak Interface 2 2 0 maa eee ee ee ee 198 The Time Dependent Parameters o a a e e 208 The Bayes Metabolite Interface o e e ee ee 210 Bayes Metabolite Viewer ao arad iape kaa a aa aa a E A 212 Bayes Metabolite Probabilities List o oea cna saraa e a 217 The IPGD D20 Metabolite ecos 3399 Romo Be ee dee R E eee eee eee 219 Bayes Metabolite IPGD_D20 Spectrum eee 220 Bayes Metabolite The Fraction of Glucose 500000005 221 Glutamate Example Spectrum 22 2444 20444 45 Eee eee ds 223 Estimating The Foo y and Fag Parameters eee eee eee 226 Bayes Metabolite The Ethyl Ether Example 227 12 1 12 2 12 3 13 1 13 2 13 3 13 4 14 1 14 2 14 3 14 4 15 1 15 2 15 3 16 1 16 2 17 1 172 17 3 18 1 18 2 19 1 19 2 19 3 19 4 19 5 19 6 19 7 20 1 22 1 23 1 23 2 23 3 the Find Resonances interface 2A 230 Difusion Tensor Interfate u lt lt aa eee Roe e SR XU ATE Res 238 Diffusion Tensor Parameter Estimates ea eens 246 Diffusion Tensor Posterior Probability For The Model 246 The Big Magnetization Package Interface o 250 Big Magnetization Transfer Example Fid cesses 252 Big Magnetization Transfer Expansion 0 00 0 ee ee eens 253 Big Magnetization Transfer Peak Pick llle 254 Magnetization Tran
20. ee 8 2 Bayes Analyze Models ou o eR Rv a E d SES s 8 3 Bayes Analyze Short Descriptions aoa ews oec soa oem a a aa a e e e a 13 16 Chapter 1 An Overview Of The Bayesian Analysis Software The Bayesian Analysis Software developed at Washington University is a client server based software package that analyzes common problems in the sciences using Bayesian Probability theory The Software is a client server software package consisting of three distinct sets of software The Server software the Client software and the Installation software The Server software actually runs the Bayesian analysis The Client software is an interface that functions as a buffer between the user and the server software Finally there is an Installation procedure that downloads and installs software The software is loosely divided into a series of programs which we refer to as packages Each package addresses a specific kind of problem For example the exponential package estimates the parameters associated with exponential models All of the calculations presented in this manual use Bayesian probability 1 35 theory to estimate the parameters or to perform model selection For those unfamiliar with Bayesian Probability theory Chapter 4 contains a tutorial and there are a number of excellent tutorials 30 39 3 11 and books 32 58 60 55 31 in the literature Most but not all of the packages described in this manual use Markov chain Monte Carlo t
21. erver software contains the programs that run the Bayesian analysis packages while the Client Interface allows one too easily access these programs Here is a list of the packages with a brief description of each The Client Interface Chapter Chapter 3 contains a more extensive description of the packages and the later Chapters in this manual contain detail information about each package The Exponential package estimates the decay rate constants and amplitudes of signals known to be decaying exponentially The Unknown Exponential package estimates the decay rate constants and amplitudes of signals known to be decaying exponentially when the number of exponential components are unknown The Inversion Recovery package is a special type of exponential analysis that is very common in NMR In this problem the NMR signal starts at a negative value and decays to a positive value The Diffusion Tensor package analyzes NMR diffusion measurements using one two or three diffusion tensor models with or without a constant The Enter Ascii Model package allows the user to define a model and then use Bayesian Probability theory to analyze data using that model The Enter Ascii Model Selection package utilizes the models generated for Enter Ascii to do model selection The Test Ascii Model model package supports the other packages that use Ascii Models by giving the user a means of testing models The Magnetization Transfer two sites package solve
22. esume Ed UR b Rus 3 59 Image Viewer 22592299 WA RO x ox o Ro x A Roy x wo 34 31 the Image List Area 2x 46g roe t k o a ee EE S432 the Set Imag area 222 kG S m Ru ook S 34 9 9 the mage Viewing BPO pg 666 64 a ye ed UE E REX ew we S 3 4 3 4 the Grayscale area on the bottom odo he Pizel Mioarei ong 9 xRoxoeox AAA A EEE eS 3 4 3 6 the Image Statistics area sore RA 29 14 Prior Viewer 22 Such a RA A dem EORR dns 24 5 tid Model Viewer coord ED deir e RR GO o ee is 24 5 The tid Model Formato c cox koc ReROGR kr Rb RR memos 14 17 17 22 22 25 34 5 2 The bid Model Reports gt o oo ba ew ee ee es 67 9 45 Plot Results Viewer 222 4 448 44 d hse 244 ee eee 3 68 3 4 6 1 the Data Model and Residuals Plots 70 34 62 the Posterior Probabilities Plots sa eera 22224284 ee 94 71 3 4 7 the Posterior Probability Vs Parameter Samples plot 72 3 4 7 1 the Expected Log Likelihood Plot 75 34 12 the Scatter Plots 2 420 606 ee RR b omo 9o e eee ee 75 34 5 9 the Log Probability Plot 2 24404 44 4 464404 sad ow I x d s 78 3 4 8 Text Results Viewer coace s macaa 06006 08084445 ao Lees 80 3419 Res Viene pois Oe X ux uec e ch ou ORES ees 86 3 410 Fortran C Gode Viewer 444466 66 oe RR REESE SP 86 3 4 10 1 Fortran C Model Viewer Popup Editor 88 An Introduction to Bayesian Probability Theory 91 4l The Rules of Probabilit
23. f the status is other than Run the Run Status text area is updated with the current status and nothing else happens If the current status is Run the job is fetched from the server and the appropriate files are updated Finally the Run status text area is set to Run If for some reason the job failed the Run Status text area is set to Error e The Cancel button will send a request to the server to cancel a job When the server receives this request it will determine if the job is running and if so the job is killed and the temporary work directories containing the job are removed If the job has already finished the temporary work directories are removed e The Run Status text area on the bottom right of the Submit Job widget group is used to display the current status of a job Server is a global widget group that has two buttons and one text area In general terms this widget group allows you to set the current server e The server Set button allows you to set the current server When you click on this button a pull down menu appears containing a list of all of the servers that you have configured on the interface Note there may be other servers but if you have not told the interface about them they will not appear in this pull down menu Clicking on a server will cause it to be set as the current server The current server is displayed in the server name text are under this button At the bottom of pull down menu is an item Edit Ser
24. hoes in the Presence of a Time Dependent Field Gradient Journal of Chemical Physics 42 1 pp 288 292 Taylor D G Bushell M C 1985 The spatial mapping of translational diffusion coefficients by the NMR imaging technique Physics in Medicine and Biology 30 4 pp 345 349 Tribus M 1969 Rational Descriptions Decisions and Designs Pergamon Press Oxford Woodward P M 1953 Probability and Information Theory with Applications to Radar McGraw Hill N Y Second edition 1987 R E Krieger Pub Co Malabar Florida 1990 Zellner A 1971 An Introduction to Bayesian Inference in Econometrics John Wiley and Sons New York
25. ion through Alternate Pathways Involving the Citric Acid Cycle of the Heart by 13C NMR Spectroscopy Journal of Biological Chemistry Vol 263 No 15 pp 6964 6971 Malloy Craig R A Dean Sherry F Mark H Jeffrey 1990 Analysis of tricarboxylic acid cycle of the heart using C isotope isomers American Journal of Physiology 259 pp H987 H995 Merboldt K Hanicke W Frahm J 1969 Self diffusion NMR imaging using stimulated echoes Journal of Magnetic Resonance 64 3 pp 479 486 Metropolis Nicholas Arianna W Rosenbluth Marshall N Rosenbluth Augusta H Teller and Edward Teller 1953 Equation of State Calculations by Fast Computing Machines Journal of Chemical Physics The previous link is to the Americain Institute of Physics and if you do not have access to Science Sitations you many not be able to retrieve this paper Neal Radford M 1993 Probabilistic Inference Using Markov Chain Monte Carlo Methods technical report CRG TR 93 1 Dept of Computer Science University of Toronto Neil Jeffrey J and G Larry Bretthorst 1993 On the Use of Bayesian Probability Theory for Analysis of Exponential Decay Data An Example Taken from Intravoxel Incoherent Motion Experiments Magn Reson in Med 29 pp 642 647 Nyquist H 1924 Certain Factors Affecting Telegraph Speed Bell System Technical Jour nal 3 pp 324 346 Nyquist H 1928 Certain Topics in Telegraph Trans
26. ixel and then the program will compute the posterior probability for the model Outputs include the posterior probability for the model indicator as well as parameter maps of the parameters 1 2 The Client Interface The interface to the Bayesian Analysis software is a Java interface that runs on any machine having Java 6 or higher Assuming the Bayesian Analysis software has been installed on a server at your site for arguments sake lets call this machine your server net then you can bring up the interface the client software by issuing javaws http your server net 8080 Bayes launch jnlp where javaws is the Java web start utility and comes with most Java installations your server net should be replaced by your server name or IP address and you should replace 8080 by the port number used by your installation see Chapter 2 for a description of how to install the software If you do not have the software installed on your local machines you can download the interface directly from Washington University javaws http bayes wustl edu Bayes launch jnlp This version of the interface will allow you to view the packages and to determine what is available However because the software has not been installed on one of your machines you will not be able to run an analysis Assuming you use one of these to methods to start the interface it will displays the default startup page shown in Fig 1 1 The purpose of the
27. mission Theory Transactions AIEE 3 p 617 644 Press W H S A Teukolsky W T Vetterling and B P Flannary 1992 Numerical Recipes The Art of Scientific Computing Second Edition Cambridge University Press Cambridge UK Scargle J D 1982 Studies in Astronomical Time Series Analysis II Statistical Aspects of Spectral Analysis of Unevenly Sampled Data Astrophysical Journal 263 pp 835 853 Scargle J D 1989 Studies in Astronomical Time Series Analysis III Fourier Transforms Autocorrelation and Cross correlation Functions of Unevenly Spaced Data Astrophysical Jour nal 343 pp 874 887 Schuster A 1905 The Periodogram and its Optical Analogy Proceedings of the Royal Society of London 77 p 136 140 Shannon C E 1948 A Mathematical Theory of Communication Bell Syst Tech J 27 pp 379 423 Shore J E R W Johnson 1981 Properties of cross entropy minimization IEEE Trans on Information Theory IT 27 No 4 pp 472 482 BIBLIOGRAPHY 419 54 Shore J E R W Johnson 1980 Axiomatic derivation of the principle of maximum entropy 55 56 57 58 59 60 and the principle of minimum cross entropy IEEE Trans on Information Theory IT 26 No 1 pp 26 37 Sivia D S and J Skilling 2006 Data Analysis A Bayesian Tutorial Oxford University Press USA Stejskal E O Tanner J E 1965 Spin Diffusion Measurements Spin Ec
28. nd C NMR study of gluconeogenesis and TCA cycle flux in humans American Journal of Physiology Endocrinology and Metabolism 281 pp H848 H856 Kotyk John N G Hoffman W C Hutton G Larry Bretthorst and J J H Ackerman 1992 Comparison of Fourier and Bayesian Analysis of NMR Signals I Well Separated Resonances The Single Frequency Case J Magn Reson 98 pp 483 500 Laplace Pierre Simon 1814 A Philosophical Essay on Probabilities John Wiley amp Sons London Chapman amp Hall Limited 1902 Translated from the 6th edition by F W Truscott and F L Emory Lartillot N and H Philippe 2006 Computing Bayes Factors Using Thermodynamic Inte gration Systematic Biology 55 2 pp 195 207 Le Bihan D 1985 E Breton Imagerie de diffusion in vivo par rsonance C R Acad Sci Paris 301 15 pp 1109 1112 Lomb N R 1976 Least Squares Frequency Analysis of Unevenly Spaced Data Astrophys ical and Space Science 39 pp 447 462 418 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 BIBLIOGRAPHY Loredo T J 1990 From Laplace To SN 1987A Bayesian Inference In Astrophysics in Maximum Entropy and Bayesian Methods P F Fougere ed Kluwer Academic Publishers Dordrecht The Netherlands Malloy Craig R A Dean Sherry F Mark H Jeffrey 1988 Evaluation of Carbon Flux and Substrate Select
29. nning and then come back to that analysis at a later time and simply pick up the analysis from the point they left off The user can determine the status of a job while its running and then fetch the job when its completed The interface provides a convenient way of displaying the results of the analysis in graphical form and finally allows the user to view and print the outputs from an analysis Bibliography 1 Bayes Rev T 1763 An Essay Toward Solving a Problem in the Doctrine of Chances Philos Trans R Soc London 53 pp 370 418 reprinted in Biometrika 45 pp 293 315 1958 and Facsimiles of Two Papers by Bayes with commentary by W Edwards Deming New York Hafner 1963 Bretthorst G Larry 1988 Bayesian Spectrum Analysis and Parameter Estimation in Lecture Notes in Statistics 48 J Berger S Fienberg J Gani K Krickenberg and B Singer eds Springer Verlag New York New York Bretthorst G Larry 1990 An Introduction to Parameter Estimation Using Bayesian Prob ability Theory in Maximum Entropy and Bayesian Methods Dartmouth College 1989 P Foug re ed Kluwer Academic Publishers Dordrecht the Netherlands pp 53 79 Bretthorst G Larry 1990 Bayesian Analysis I Parameter Estimation Using Quadrature NMR Models J Magn Reson 88 pp 533 551 Bretthorst G Larry 1990 Bayesian Analysis II Signal Detection And Model Selection J Magn Reson 88 pp 552 57
30. o approximate the Bayesian posterior probabilities For those unfamiliar with Markov chain see 23 44 and Sec tion B gives a description of how the various packages implement the Markov chain Monte Carlo calculations 1 1 The Server Software Before we describe the interface we briefly describe the server software and how the client software interfaces to it The server the machine that actually runs the Bayesian Analysis can be any multi core LinuxPC either 32 or 64 bit running GNU Linux CintOS 4 7 or higher or a Sun system running Solaris 9 or 10 When the software is installed on the server the installation procedure downloads the latest version of the software from Washington University and installs it on the server see Chapter 2 for instructions on how to install the software The server software consists of three parts a web server a set of scripts that are used by the web server and the programs the implement the Bayesian probability theory calculations The web server handles the communications between the client and the server applications The clients send requests to the servers and the servers use 17 18 AN OVERVIEW a set of scripts to handle these requests These scripts do things as simple as listing the process currently running on the server to things as complicated as unpacking an analysis and then running the appropriate software In the following Chapters we will describe each of these software packages The s
31. r Own ASCII Model 22 Ascii Model Selection 23 Phasing An Image 23 1 The Bayesian Calculation visera Re eee ee ea be a 23 2 Using The Package 267 269 273 277 279 279 280 282 285 287 288 289 291 295 297 300 303 303 306 307 309 310 311 312 313 314 321 323 323 324 327 329 331 24 Phasing An Image Using Non Linear Phases 241 The Model Eguation ss 2 24321 A 24 2 The Bayesian Calculations 22s m RR 24 9 The VnmrJ and Vamo Interfaces 2 222226 RA 28 Analyze Image Pixel 28 1 Modification History uu 264 4 44 800 om SUE dee oo de A 29 Image Pixel Model Selection A Ascii Data File Formats A Ascii Input Data Files os o icicu a a S RR RR A A 2 Ascii Image File Formats llle A 3 The Abscissa File Format 00000 B Markov chain Monte Carlo With Simulated Annealing B 1 Metropolis Hastings Algorithm o o B 2 Moaltiple Simulations cn ee ee a Bo eaumulated Annealine oo cs 2619 9c r a B 4 The Annealing Schedule lt lt o eee Bb Killine Simulations a menm 9o UE G44 GRE SS B6 the Proposal s soca kk wees ee ee eA a Cee ds C Thermodynamic Integration D McMC Values Report E Writing Fortran C Models E 1 Model Subroutines No Marginalization E 2 The Parameter File coccion 204660404 b44h 4044 E 3 The Subroutine Interface E 4 The Subroutine Declarations lt 2 22
32. s can be added to the library of models but there are no facilities for building these models within the interface e The Behrens Fisher package solves the classical medical testing problem given two experi ments that consist of repeated measurements of the same quantity where in the second mea surement one has change some experiential parameter determine if the experiments are the same or if they differ e The Errors in Variables package solves the problem of straight line fitting when there are errors in both the measured data and in the measured time or abscissa value The implementation in this package allows the user to set the order of the polynomial to be fit so its a little more general that just straight line fitting e The Polynomial Models package fits polynomials of either a given or an unknown order to the input data When the unknown model is selected the programs that implement the calculation compute the posterior probability for the order of the polynomial needed to fit the data down to the noise e The Maximum Entropy Histograms density estimation package is a ASCII package that takes as its input a sample drawn from an unknown density function It then computes the posterior probability for the number of nontrivial moments in the data i e the number of Lagrange multipliers need by the Maximum Entropy density function Its output is the estimated density function with error bars on the estimated density function e The
33. s the Block McConnell equations to obtain the exchange rate constants for two site magnetization exchange The Magnetization Transfer Kinetics package is a magnetization transfer package that solves the Block McConnell equations at multiple temperatures and concentrations to derive the entropy and enthalpies of the the exchange process The Big Magnetization Transfer package solves the magnetization transfer problem when one of the sites can be considered infinite compared to the other The Bayes Analyze package is a time domain frequency estimation package that is fully capable of determining the number of resonances in an FID and estimating the resonance parameters The Big Peak Little Peak package analyzes time domain FID data in which there is a single big peak that may be many orders of magnitude larger in intensity the big peak than the metabolic peaks the little peaks of interest THE BAYESIAN ANALYSIS SOFTWARE AN OVERVIEW 19 e The Find Resonances package analyzes NMR FID data looking for resonances The program is a model selection program that is attempting to determine the number of resonances in the data and estimate the parameters associated with those resonances e The Metabolite package analyzes FID data from a number of known samples for example a C13 FID of Glutamate The intensity of the Glutamate resonances are related to each other through a metabolic model This model can be very simple or very complex Metabolic model
34. sfer Interface bia a a E a a a 256 Magnetization Transfer Peak Pick o oo a a a eee 262 Magnetization Transfer Example Data oea cac errau aa ee eene 263 Magnetization Transfer Example Spectrum aaaea a 264 Magnetization Transfer Kinetics Interface auaa 268 Magnetization Transfer Kinetics Arrhernius Plot llle 274 Magnetization Transfer Kinetics Water Viscosity Table 275 Given Polynomial Order Package Interface o 278 Given Polynomial Order Scatter Plot o aa naaa 284 Unknown Polynomial Order Interface o a a a a a a a a a a lea 286 The Distribution of Models lt s s ecos aa sacas E datert SR diit 290 Unknown Polynomial Order Package Posterior Probability 292 Errors In Variables Interface coo 22cm oO Rn 296 Errors Im Variables McMC Values File 0040544464444 m 4 4444 Seo 302 the Behrens Fisher interface o 304 Behrens Fisher Hypotheses Tested o eee 305 Behrens Fisher Console Log hasara 315 Behrens Fisher Status Listing eee 316 Behrens Fisher McMC Values File The Preamble 317 Behrens Fisher McMC Values File The Middle llle 318 Behrens Fisher McMC Values File The End llle 319 Enter Ascii Model Interface o een 322 Test Your Own Ascii Mo
35. startup page is to allow you to restart an analysis When you exit the interface or changes working directories the interface saves the current settings in a special Java properties file When the interface start it consults this file and determines what your last WorkDir was and how to restart that analysis If an analysis was saved the interface displays the messages shown in Fig 1 1 the lines starting To restore analysis This line contains the name of the package that was being processed in this case the package name was AnalyzelmagePixels and the analysis was saved in a WorkDir named Given If the Restore Analysis button is activated then the Given AnalyzelmagePixels analysis will be restored to its previous status When the interface finishes restoring the analysis it will function exactly like you never exited the WorkDir or interface If you do not want to restore an analysis then changing the package will delete the contents of the current WorkDir and configure the WorkDir for the new package If you do not want to change packages but want to check on another analysis then changing the current working directory using the WorkDir menu will cause the interface to switch to the new WorkDir and assuming that WorkDir contains a previous analysis that analysis will be restored to its previous status Finally if you wish to start a completely new analysis then selecting WorkDir Edit will bring up a popup that will allow you
36. t o o oca aap cewa g kaaa a E E EE e a 123 Estimating The Sinusoids Parameters o e oo oor birra daa 125 the Exponential interface llamar a 130 the Unknown Exponential interface 0 0020000 ee eee 136 Tbe Distribution of Models 2 21 4 2424240608480 44b444 ene 141 Exponential Probability for the Model o a 142 the Inversion Recovery interface e e e e 144 Bayes Analyze Interface escasa Aaa 148 Bayes Analyze Fid Model Viewer een 152 The Bayes Analyze File Header 22e 170 The bayes mbise Wiles fon Lu onus ee o eH REA eee ee ee 172 Bayes Analyze Global Parameters 2 een 175 Bayes Analyze Model File 2 2 ee 176 Bayes Analyze Initial Model ee 178 Base qD Locarthm Or her QUOS ubere aee ep eh ee A Oe A A an 178 The bayes outp t nbnnn Report 2 222222 93 he eS 179 Bayes Analyze Uncorrelated Output ee ee 180 The bayes probabilitiesmnnn File oo se e sass 224444 ERR VR 182 The bayeslos n nn Pie 25 a See ke ee ye AS aE PRE Rove eo 185 The bayes status onon File ccoo c E EEE rss EEE SS 187 The b ayes modelonan File 2222 99 cs kG EDS ERE RE OR Pes 188 The bayes model nnnn File Uncorrelated Resonances 189 Bayes Analyze Summary Header e 189 The Summary2 Report 2444 2 264 Rx m ee don ae A ek m 190 The Summary2 Report s s cls sma 9 dede CO ORE EGE RR Eee eRe 191 The Big Peak Littl
37. vers that can be used to modify your list of servers Activating this widget will bring up a popup Chapter 3 1 4 that allows you to modify your current servers and to add new ones if desired This Server Edit popup is also available under the Settings Server Setup menu e The server Status button will send a request for a list of jobs currently running on the server On Linux and Sun systems this request is a simple ps The results of this request are displayed in the Text Viewer at the bottom of the interface e The current Server is displayed in the Server Name text area under the two button in the Server widget group THE BAYESIAN ANALYSIS SOFTWARE AN OVERVIEW 25 Model is a widget group that is specific to the Exponential package In the exponential package the Model widget group servers three purposes to set the order of the exponential model to be processed to indicate if a constant offset is present and indicate if the number of exponentials is unknown For a more detailed description of these widgets see the chapters on the exponential packages Chapters 5 and 6 Analysis Options is a widget group that shows up on many packages The exact content of this widget group is specific to each package Here there is a single widget that indicates whether or not the program is to attempt outlier detection For more on the outlier model and how it is handled in the calculations see Chapter H Reset will resets all optional settings ba
38. xx ee hee Oe ed eS 10 3 1 The IPGD D20 Metabolite 22222 kk RRR RR RRA A RE a de 10 3 2 The Glutamate 2 0 Metabolite ra 6 2066 62244408 28 ee eee es 10 3 3 The Glutamate 3 0 Metabolite 10 4 The Example Metabolite llle 10 5 Outputs From The Bayes Metabolite Package llle 11 Find Resonances 11 1 The Bayesian Calculations 21e RR REX ee Ree a a 11 2 Outputs From The Bayes Find Resonances Package 12 Diffusion Tensor Analysis 12 1 The Bayesian Calculation seine RO moo e de SOR RUE oe ee eS 12 2 Using The Package 13 Big Magnetization Transfer 13 1 The Bayesian Calculation sics x09 x o s Rak 4 x XO memo ee ROR aS 13 2 Outputs From The Big Magnetization Transfer Package 14 Magnetization Transfer 14 1 The Bayesian Calculation y isis RR mm a De ee ee we ee 14 2 Using The Package 167 169 169 174 175 177 181 184 187 188 189 190 191 192 197 199 206 209 213 215 218 218 222 225 226 228 229 231 236 237 244 15 Magnetization Transfer Kinetics 151 The Bayesian Calculation amp 2442044448 044 ooo poe mo 9 Rok RR eee eee y 3 15 2 Using The Package 16 Given Polynomial Order 16 1 Theo Bayesian Calculatiu uuu wa mme eR oh cy ORG GE A DR a S TO Ll Gram Schmidt g coa dor aa RR ede ei S ooh eR Rum m moe e ge dd du 16 1 2 Th Bayesian Calculation oca 24 roa sco ck m kon tmr RR RUE RR n RO AU 16 2 Outputs From
39. y Theory 2o k ko xo dee eee x 4 4 a X m RO Ee 91 42 Assigning Probabilities 2 666444 cee coo a br ee a 94 4 3 Example Parameter Estimation e a 101 43 1 Define The Problem 2 0 ia mpm ede a na ea XR 102 4 3 1 1 The Discrete Fourier Transform lt ss sec o o 102 43012 JAMBES Dc a o e E dh 105 4 3 2 State The Model Single Frequency Estimation 106 430 Apply Probability Theory 20399 39 9 3 oy a 107 ASA Assign The Probabilities 65 323293 ee oko ko E Ro EEG XXE be ees 110 4 3 5 Evaluate The Sums and Integrals elles 112 4 3 6 How Probability Generalizes The Discrete Fourier Transform 115 LOL PORN 50x 0x Eee S lh ADAC Stee e QE e ep ag ul ie Se eem 118 4 9 8 Parameter Estimates 29 00885024 koc x 93 3 99 9 d ege qn 124 44 Summary and Conclusions aaa da a o eoe HA RR ee aa 127 Given Exponential Model 129 DL The Bayesian Calculation oes wo m 9 x ee RE A EE 131 5 2 Outputs From The Given Exponential Package 133 Unknown Number of Exponentials 135 6 1 The Bayesian CAIGDBUIORS oscar 4 4 6 Od Roe Rome ee ETE RC RU 137 6 2 Outputs From The Unknown Number of Exponentials Package 140 Inversion Recovery 143 Til The Bayesian Calculation 6 a mox c x RA X X ED OR O3 RF RR A 145 7 2 Outputs From The Inversion Recovery Package 146 Bayes Analyze 147 Ol Bayes Model 03 6 s eet tPA ee EUR bee beet REESE OOS
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