Markov chain Monte Carlo and stochastic origin ensembles methods Comparison of a simple application for a Compton imager detector

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1 Markov chain Monte Carlo and stochastic origin ensembles methods Comparison of a simple application for a Compton imager detector Pierre-Luc Drouin DRDC Ottawa Research Centre Defence Research and Development Canada Scientific Report DRDC-RDDC-2016-R124 September 2016

2 c Her Majesty the Queen in Right of Canada, as represented by the Minister of National Defence, 2016 c Sa Majesté la Reine (en droit du Canada), telle que réprésentée par le ministre de la Défense nationale, 2016

3 Abstract This document aims to summarise Markov Chain Monte Carlo (MCMC) methods, in particular, the Metropolis-Hastings algorithm and the Stochastic Origin Ensembles (SOE) method, in a concise and notation-consistent manner. These methods are commonly used to perform model parameter estimation for a population, based on a measured sample, through the sampling of the probability distribution for these parameters. A simple application of SOE is then demonstrated using simulation data from a Compton imager detector. Significance for defence and security In Radiation and Nuclear (RN) defence, many detection technologies rely on model parameter estimation to perform threat detection, identification and/or localisation. Such algorithms are often required when the direct observation of the parameters of concern is not possible. Some detection systems, such as Compton imagers and muon tomography systems, need to make extensive use of such algorithms compared to traditional technologies. This document summarises a set of algorithms which represent good candidates for the estimation of the parameters for these new technologies. Hardware and software development of such systems is currently ongoing at DRDC Ottawa Research Centre. DRDC-RDDC-2016-R124 i

4 Résumé Ce document vise à résumer les méthodes de Monte-Carlo par Chaînes de Markov, et en particulier, l algorithme de Metropolis-Hastings et la méthode des Ensembles d Origine Stochastique (EOS), d une manière concise et tout en utilisant une notation consistante. Ces méthodes sont fréquemment utilisées afin d estimer les paramètres d un modèle pour une population, basé sur la mesure d un échantillon, par le biais de l échantillonnage de la distribution probabilistique pour ces paramètres. Une application simple de la métode EOS est ensuite démontrée en utilisant des donnés simulées provenant d un imageur Compton. Importance pour la défense et la sécurité En défence Radiologique et Nucléaire (RN), plusieurs technologies de détection dépendent sur l estimation de paramètres d un modèle afin d accomplir la détection de menaces, leur identification et/ou leur localisation. De tels algorithmes sont souvent requis lorsque l observation directe des paramètres concernés n est pas possible. Certains systèmes de détection, tels que les imageurs Compton et les systèmes de tomographie muonique, requièrent un usage intensif de tels algorithmes comparativement aux technologies traditionnelles. Ce document résume un ensemble d algorithmes qui représentent de bons candidats afin d estimer les paramètres pour ces nouvelles technologies. Du développement d équipement ainsi que logiciel est en cours au RDDC Centre de recherches d Ottawa. ii DRDC-RDDC-2016-R124

5 Table of contents Abstract Significance fordefence andsecurity i i Résumé ii Importance pour la défense et la sécurité ii Tableofcontents iii Listoffigures Listoftables iv iv 1 Introduction Metropolis Hastings algorithm Sampling from the probability density function of an evaluated model Stochasticoriginensemblesmethod ApplicationofSOE foracomptonimager ComparisonoftheSOE algorithm LimitationsofthesimpleSOE algorithm Conclusion References DRDC-RDDC-2016-R124 iii

6 List of figures Figure 1: Figure 2: Images of a simulated 137 Cs point source located 10 off-axis as produced using simple back projection (top), smoothed MLEM (left) andsoe (right) Images of a simulated 137 Cs C-shaped source as produced using simple back projection, MLEM and SOE with 2 ( Coarse ) and 1 ( Fine ) priorpdf binning List of tables Table 1: Measured mean width for a 2D Gaussian fitted on the distributions resultingfrom thethreedifferentalgorithms iv DRDC-RDDC-2016-R124

7 1 Introduction In statistical data analysis, parameter estimation techniques are commonly used to determine model parameters for a population, based on a measured sample. Maximum-likelihood estimation is a well known and widely used approach, which aims at determining the model parameters that maximise the probability of the measured sample. Although this technique is well suited for a large number of applications, it can be difficult to apply for some situations in practice, for example when the number of unknown model parameters is very large and/or when the model can be defined more easily by introducing latent (hidden) variables. This document first presents a summary of Markov Chain Monte Carlo (MCMC) methods, in particular, the Metropolis Hastings algorithm, which can be used to perform parameter estimation in such situations. The Stochastic Origin Ensembles (SOE) method, which represents a particular application of the Metropolis-Hastings algorithm, is then described, and results from a simple application of this method are then compared to alternative algorithms. As shown in this document, advanced radiation imaging systems, such as Compton imagers, greatly benefit from using models which use a large number of latent variables. These imagers allow for a quick localisation of point or extended radioactive sources that can be partially shielded by the material of a cluttered environment. Such a task can represent a real challenge when attempted using traditional detectors. 2 Metropolis Hastings algorithm The Metropolis Hastings algorithm [1, 2] is a random walk Monte Carlo method, which is a subclass of the MCMC methods, and that samples from a Probability Density Function (PDF) f ( θ), through the usage of a state vector θ which is randomly moving in the parameter space of the PDF. A Markov process is uniquely defined by the expression of a transition PDF f ( θ θ ) which provides the probability density of transiting from state θ to state θ. A Markov chain, defined as a sequence of such state vectors, is thus memoryless, since the probability of transition to the following state depends only on the current state. A Markov process reaches a unique stationary distribution π( θ)= f ( θ) asymptotically when the two following conditions are met: the condition of detailed balance, π( θ) f ( θ θ )=π( θ ) f ( θ θ), (1) and the Markov process must be ergodic, meaning that the process must be aperiodic and must be able to return to any given state θ in a finite number of steps (irreducible process). DRDC-RDDC-2016-R124 1

8 The Metropolis-Hastings algorithm ensures that Condition (1) is met by enforcing it directly through the relationship between f ( θ) and f ( θ θ ): f ( θ) f ( θ θ )=f( θ ) f ( θ θ) (2) f ( θ θ ) f ( θ θ) = f ( θ ) f ( θ). (3) The transition PDF f ( θ θ ) is then expressed as the product between a proposal PDF g( θ θ ) and an acceptance PDF h( θ θ ): f ( θ (trans.) θ )=f( θ prop. θ, θ accept. θ ) = f ( θ prop. θ ) f ( θ accept. θ θ prop. θ ) =g( θ θ )h( θ θ ). (4) In order to fulfill the ergodicity condition, g( θ θ ) must be aperiodic and both g( θ θ ) and h( θ θ ) must allow f ( θ θ ) to be irreducible. From Equations (3) and (4), the condition on h( θ θ ) is h( θ θ ) h( θ θ) = g( θ θ) f ( θ ) g( θ θ ) f ( θ). (5) The Metropolis-Hastings algorithm uses the following expression for h( θ θ ) to satisfy the above constraint: { h( θ θ )=min 1, g( θ θ) f ( } θ ) g( θ θ ) f (. (6) θ) Equation (6) satisfies the irreducibility condition when g( θ θ ) is irreducible. The Metropolis-Hastings algorithm thus relies uniquely on g( θ θ ) to satisfy the ergodicity condition, while the expression for h( θ θ ) ensures that the condition of detailed balanced can be reached. Note also that due to the form of Equation (6), the Metropolis-Hastings expression for the transition PDF f ( θ θ ) is insensitive to the normalisation or scaling of f ( θ), meaning that the expression for f ( θ) needs only to be known up to a scaling factor that does not depend on θ, which can greatly simplify computation in practice. Using the above results and an initial state θ s with s = 0, the Metropolis-Hastings algorithm can be executed as follows: 1. Draw a random proposed state θ according to the chosen ergodic proposal PDF g( θ s θ ). 2 DRDC-RDDC-2016-R124

9 2. Compute g( θ θ s ) f ( θ ). If the resulting value is greater or equal to either 1 or otherwise, to a random number drawn uniformly in the interval ]0,1], the proposed state g( θ s θ ) f ( θ s ) is accepted, θ s+1 = θ, s is incremented by 1 and the algorithm continues to Step θ s+1 = θ s, s is incremented by 1 and the algorithm continues to Step 1. This procedure allows the distribution of states to converge to f ( θ) once the equilibrium is reached. However, the choice for the initial state θ 0 as well as the proposal PDF g( θ θ ) can greatly affect the required number of steps required to reach this regime. Also, unless g( θ θ ) has a negligible dependency on θ and the acceptance of θ is very likely, there can be a significant autocorrelation within the MCMC chain, such that consecutive states cannot be considered to be statistically independent. The usage of correlated states for estimator computation can lead to biased results. However, choosing a function g( θ θ ) having a weak dependency on θ often results in a very small probability of acceptance for θ. It can thus be advantageous to accept a higher dependency within g( θ θ ) to increase the probability of acceptance. A reduced chain of states with low autocorrelation can then be obtained by sampling the original chain at a fixed interval which is sufficiently large for the desired intent. A Metropolis-Hastings algorithm thus often involves the rejection of a number of burn-in steps required to reach the equilibrium, followed by a sampling of the remaining steps to insure a sufficiently low autocorrelation. The determination of the number of burn-in steps and the sampling rate is discussed in detail in [3]. In the case of multi-dimensional state vectors, a common method to easily increase the level of acceptance consists in a proposal function that randomly updates a single component of θ at each step. This implies a sampling rate of the chain which is greater than the dimension of the state vector to achieve statistical independence between the states of the reduced chain. 3 Sampling from the probability density function of an evaluated model MCMC methods can be used to sample from the PDF of a model which is evaluated using a data sample. Let the conditional probability density of a single measured event e, as computed by a model characterised by a set of parameters θ,bedefinedas f ( x e θ), where x e represents the coordinate of the event in the measurement parameter space. Assuming the statistical independence of the events within a measured sample, the PDF of a sample, its likelihood function, can be expressed as L(sample θ)= f (sample θ)= n events e=1 f ( x e θ), (7) DRDC-RDDC-2016-R124 3

10 where n events is the number of measured events. In the common case of a measurement where the number of measured events is Poisson distributed, an extended likelihood function is defined as L e (sample θ)=f e (sample θ)=p(n events ν) f (sample θ) = e ν ν n events n events! n events e=1 f ( x e θ), (8) where ν is a parameter, a subcomponent of θ, for the Poisson Probability Mass Function (PMF) P(n ν)= e ν ν n n!, which corresponds to the average number of measured events. Estimation methods, such as Maximum Likelihood (ML) and Maximum Likelihood Expectation Maximisation (MLEM), provide estimators ˆ θ that maximise the likelihood L(e) of the measured sample. ML methods can estimate the variance of the estimators through different methods such as contour lines in the likelihood parameter space. In contrast, MCMC methods can be used to sample from the f (e) ( θ sample) PDF directly, such that the resulting posterior distribution can be used to evaluate any metric. These posterior distributions contain more information than provided by ML methods and are thus suited to characterise estimator uncertainties in the case of non-gaussian statistics. f ( θ) is often expressed as a function of the likelihood function, through the usage of Bayes theorem: f (e) ( θ sample)=l (e) (sample f p ( θ) θ) f p (sample), (9) where f (e) ( θ sample) represents the MCMC PDF f ( θ). In the above expression, f p ( θ) is the prior PDF for the model parameters, while the PDF f p (sample) is the prior for the measured sample. Sampling simplifications with the Metropolis-Hastings algorithm As previously mentioned, for the Metropolis-Hastings algorithm, sampling of a PDF can be performed as long as the PDF is known up to a scaling factor which does not depend on the sampled parameters. This can simplify the task, notably when expressing the PDF as a function of the likelihood function. From Equation (9), it becomes no longer relevant to evaluate the f p (sample) expression and the PDF is now more simply interpreted as f (e) ( θ sample) L (e) (sample θ) f p ( θ). (10) 4 Stochastic origin ensembles method The StochasticOriginEnsemblesmethod [4] represents a particular applicationof themetropolis-hastings algorithm for a scenario where the parameter vector θ can be subdivided 4 DRDC-RDDC-2016-R124

11 in a set of φ subcomponents θ ( φ 1, φ 2,..., φ nevents), (11) each one respectively associated to a corresponding event in the measured sample. The φ components can represent hidden variables of the process and are considered to be independent from each other since they are associated to independently measured events. A priori, these components nonetheless follow an unknown distribution, and this distribution is approximated using the distribution of the φ components within θ. The f p ( θ) prior PDF is then computed by evaluating the probability density of each subcomponent in this distribution. Usually, the posterior sample of θ parameter values, as provided by the resulting Markov chain, is then used to generate a distribution in the φ parameter space. Due to the statistical independence of the φ subcomponents, f p ( θ) can be expressed as f p ( θ)= n events e=1 f ( φ e ), (12) where f ( φ) is the unknown distribution which is approximated using the distribution of the φ components within θ: f ( φ) f b ( φ θ). (13) In the above expression, f b ( φ θ) is a binned PDF in the φ parameter space, which is populated using the θ sample. The approximation of f ( φ) using f b ( φ θ) does not affect the convergence of the Markov chain to the stationary distribution; it is guarantied by the Metropolis-Hastings algorithm. We define a linearisedbin indexk for the binned φ parameter space and let n(k θ) represent the number of components within the θ sample that fall in bin k. If the function b( φ) provides the bin index k associated to the φ coordinate, then f b ( φ θ) is proportional to Using Equations (12) to (14), we then have f p ( θ) n events e=1 f b ( φ e θ) f b ( φ θ) n(b( φ) θ). (14) n events n(b( φ e ) n bins θ)= e=1 k=1 n(k θ) n(k θ), (15) where n bins is the total number of bins within f b ( φ θ). With the SOE algorithm, the ratio f ( θ ) f ( θ) from the Metropolis-Hastings algorithm is thus given by f ( θ ) f ( θ) L(sample θ n ) bins n(k θ ) n(k θ ) L(sample θ) k=1 n(k θ) n(k θ), (16) DRDC-RDDC-2016-R124 5

12 and similar to the case of an extended likelihood. If the implementation of the algorithm is such that a single component of θ is updated by the chosen proposal function g( θ θ ), then the above expression can be simplified even further. We define o and d as the origin and destination bin indices for the proposed transition, respectively. Thus, we have, when o d : such that n bins n(k θ ) n(k θ ) k=1 n(k θ) n(k θ) n(o θ )=n(o θ) 1 (17) n(d θ )=n(d θ)+1, (18) = [n(o θ) 1] n(o θ) 1 [n(d θ)+1] n(d θ)+1 n(o θ) n(o θ) n(d θ) n(d. (19) θ) This finally gives f ( θ ) f ( θ) = L(sample θ ) 1, if o = d L(sample [n(o θ) θ) 1] n(o θ) 1 [n(d θ)+1] n(d θ)+1, otherwise. (20) n(o θ) n(o θ) n(d θ) n(d θ) The above ratio can thus be used within the Metropolis-Hastings algorithm as described in the previous section to sample from f ( θ) when the model parameters are associated to the events in the measured sample. 5 Application of SOE for a Compton imager A simple application of the SOE method can be performed for a Compton imager [5], where one wants to determine the distribution of φ, defined as the angular origin of the detected radiation. For a simplified physical model where the PDF for the angular origin of a given detected event consists in a ring resulting from back projection of the Compton cone [6], the likelihood of the angular origin for the event is constant along that ring and otherwise null. This model assumes that the energy of the analysed photons is perfectly known. If one chooses a function g( θ θ ) which only proposes origins along these rings, the expression for h( θ θ ) can be easily evaluated, since the ratio of likelihood values in Equation (20) is equal to 1 and the proposal function is symmetrical. The resulting algorithm to generate the Markov chain for this simple Compton imager SOE is thus given by: 1. Pick an initial origin for θ, using random positions along the back projected rings of the measured events. 2. Compute the values for n(k θ) by filling an histogram, and record the bin index for each event. 6 DRDC-RDDC-2016-R124

13 3. Draw a random event index e uniformly out of the n events events in the measured sample. 4. Pick a random position along the back projected ring for the selected event and determine the proposed destination bin d. 5. Compute f ( θ ) as given by Equation (20) (where the ratio of likelihood values is equal f ( θ s ) to 1), using the destination bin d and the recorded current bin index for the event e as the o bin. If the resulting value is greater or equal to either 1 or otherwise, to a random number drawn uniformly in the interval ]0,1], the proposed state is accepted, n(o θ ) and n(d θ ) and the current bin index for event e are updated, θ s+1 = θ, s is incremented by 1 and the algorithm continues to Step θ s+1 = θ s, s is incremented by 1 and the algorithm continues to Step The resulting chain must be sampled at least every n events steps. 6 Comparison of the SOE algorithm In this section, results from the application of the SOE algorithm to Compton imager simulation data are presented and compared to alternate algorithms. The algorithm was tested using 1453 detected gamma rays from a 137 Cs point source located 10 off-axis. The prior SOE histogram was binned using bins from 90 to 90 in each direction. A total of steps were generated, including burn-in steps, and one step every 100 steps was then used to generated the posterior distribution. Figure 1 shows the resulting distribution, along with corresponding results which were obtained in [6] using simple back projection and smoothed MLEM. Corresponding angular resolution results for this simulation are shown in Table 1, where a 2D Gaussian distribution was fitted using a range of ±10 around the peak. When comparing MLEM and SOR results to back projection results, there is an obvious improvement for both the source image and the angular resolution values, as these two algorithms allow the elimination of the circular patterns produced by the back projection of the Compton cones. The mean Gaussian width is reduced by more than 50%, which is a very significant improvement considering that the three algorithms are not using assumptions regarding the source s spatial distribution. When comparing MLEM and SOE results, Figure 1 shows slight clustering of events outside the source location with the MLEM technique, which is not present with SOE. The computed angular resolution is slightly better with SOE, and the fit range for the 2D Gaussian excluded most of the visible clusters in MLEM s results. An infinitesimally thin C-shaped 137 Cs source was also imaged, using the same simulated dataset that was processed in [6]. The SOE algorithm was run twice to observe the effect of different prior PDF binning. The results are presented in Figure 2. Similarly to DRDC-RDDC-2016-R124 7

14 A Figure 1: Images of a simulated 137 Cs point source located 10 off-axis as produced using simple back projection (top), smoothed MLEM (left) and SOE (right). Table 1: Measured mean width for a 2D Gaussian fitted on the distributions resulting from the three different algorithms. Algorithm Resolution [ ] Simple back projection 7.1±0.1 MLEM 3.2±0.1 SOE 3.1±0.1 8 DRDC-RDDC-2016-R124

15 B!"#$ LL!"#$% A S S! "#$ A S S! "#$%& A A 2 Figure 2: Images of a simulated 137 Cs C-shaped source as produced using simple back projection, MLEM and SOE with 2 ( Coarse ) and 1 ( Fine ) prior PDF binning. DRDC-RDDC-2016-R124 9

16 the results obtained with the point source, the figure shows a drastic improvement of the reconstructed image when comparing SOE with the back projection method, which translates into a higher contrast. When qualitatively comparing SOE results with MLEM, we observe a dependence of the outcome on the binning that is chosen for the prior PDF, as one would expect since this effectively approximates the probability density to be uniform within a given bin. Correlations between image bins are thus reduced as the prior PDF binning becomes finer, but this is done at the expense of increased statistical noise, since the prior PDF is approximated using the measured sample itself. When comparing image blur, the MLEM result appears to fall between the two SOE results. The SOE result with finer binning could benefit from a postsmoothing filter in order to reduce the statistical noise effects. Similar structures have been observed with the MLEM method when the number of iterations becomes too large. 7 Limitations of the simple SOE algorithm The simple SOE method based on a back projection model which was described in Sections 5 and 6 has the advantage of being analytical and easy to compute. However, it has many of the flaws of the other methods based on this model: it assumes that the detector measures energies exactly, such that back projected events lie along an infinitesimal ring rather than a broad uncertainty band; the detector angular response is assumed to be uniform; energy deposition is assumed to occur in the centroid of the detector pixels; the analysed measured sample is assumed to be pure, without any contamination such as multiple scattering within the scatter plane, backscattering off the absorber plane or undetected energy escaping either plane; the source is assumed to be at an infinite distance from the detector; and the lateral position of the scatter pixel with respect to the origin of the back projection angular space is neglected. This list of approximations can contribute to various unwanted effects, such as image blur, distortion, artificial image structures, etc. A more realistic model which would avoid so many approximations is thus desirable. Except for the last two items, the above approximations will be addressed in the future. 10 DRDC-RDDC-2016-R124

17 8 Conclusion In this document, the Metropolis-Hastings algorithm was presented, following a summary of MCMC methods. The SOE method was then explained, as a special case of the Metropolis-Hastings algorithm, and a simple implementation for a Compton imager was demonstrated and compared to alternate algorithms. When compared to the back projection method, the SOE method showed a drastic improvement of the reconstructed image. The latter method produced results which were comparable to the results obtained with MLEM. Compared to MLEM, SOE showed less clustering of events outside the simulated point source location. DRDC-RDDC-2016-R124 11

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19 References [1] Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E. (1953), Equation of State Calculations by Fast Computing Machines, The Journal of Chemical Physics, 21(6), [2] Hastings, W. K. (1970), Monte Carlo Sampling Methods Using Markov Chains and Their Applications, Biometrika, 57(1), [3] Raftery, A. E. and Lewis, S. M. (1995), The Number of Iterations, Convergence Diagnostics and Generic Metropolis Algorithms, In Practical Markov Chain Monte Carlo (W.R. Gilks, D.J. Spiegelhalter and S. Richardson, eds.), pp , Chapman and Hall. [4] Sitek, A. (2008), Representation of photon limited data in emission tomography using origin ensembles, Physics in medicine and biology, 53(12), [5] Andreyev, A. (2009), Stochastic image reconstruction method for Compton camera, In Nuclear Science Symposium Conference Record (NSS/MIC), 2009 IEEE, pp , IEEE. [6] Ueno, R. (2016), Development of the GEANT4 Simulation for the Compton Gamma-Ray Camera, (DRDC-RDDC-2016-C138) Defence Research and Development Canada Ottawa Research Centre. DRDC-RDDC-2016-R124 13

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21 DOCUMENT CONTROL DATA (Security markings for the title, abstract and indexing annotation must be entered when the document is Classified or Protected.) 1. ORIGINATOR (The name and address of the organization preparing the document. Organizations for whom the document was prepared, e.g. Centre sponsoring a contractor s report, or tasking agency, are entered in section 8.) 2a. SECURITY MARKING (Overall security marking of the document, including supplemental markings if applicable.) UNCLASSIFIED 3701 Carling Avenue, Ottawa ON K1A 0Z4, Canada 2b. CONTROLLED GOODS (NON-CONTROLLED GOODS) DMC A REVIEW: GCEC DECEMBER TITLE (The complete document title as indicated on the title page. Its classification should be indicated by the appropriate abbreviation (S, C or U) in parentheses after the title.) Markov chain Monte Carlo and stochastic origin ensembles methods 4. AUTHORS (Last name, followed by initials ranks, titles, etc. not to be used.) Drouin, P.-L. 5. DATE OF PUBLICATION (Month and year of publication of document.) September a. NO. OF PAGES (Total containing information. Include Annexes, Appendices, etc.) 22 6b. NO. OF REFS (Total cited in document.) 6 7. DESCRIPTIVE NOTES (The category of the document, e.g. technical report, technical note or memorandum. If appropriate, enter the type of report, e.g. interim, progress, summary, annual or final. Give the inclusive dates when a specific reporting period is covered.) Scientific Report 8. SPONSORING ACTIVITY (The name of the department project office or laboratory sponsoring the research and development include address.) DRDC Ottawa Research Centre 3701 Carling Avenue, Ottawa ON K1A 0Z4, Canada 9a. PROJECT OR GRANT NO. (If appropriate, the applicable research and development project or grant number under which the document was written. Please specify whether project or grant.) 9b. CONTRACT NO. (If appropriate, the applicable number under which the document was written.) 10a. ORIGINATOR S DOCUMENT NUMBER (The official document number by which the document is identified by the originating activity. This number must be unique to this document.) DRDC-RDDC-2016-R124 10b. OTHER DOCUMENT NO(s). (Any other numbers which may be assigned this document either by the originator or by the sponsor.) 11. DOCUMENT AVAILABILITY (Any limitations on further dissemination of the document, other than those imposed by security classification.) Unlimited 12. DOCUMENT ANNOUNCEMENT (Any limitation to the bibliographic announcement of this document. This will normally correspond to the Document Availability (11). However, where further distribution (beyond the audience specifiedin (11)) is possible, a wider announcement audience may be selected.) Unlimited

22 13. ABSTRACT (A brief and factual summary of the document. It may also appear elsewhere in the body of the document itself. It is highly desirable that the abstract of classified documents be unclassified. Each paragraph of the abstract shall begin with an indication of the security classification of the information in the paragraph (unless the document itself is unclassified) represented as (S), (C), or (U). It is not necessary to include here abstracts in both official languages unless the text is bilingual.) This document aims to summarise Markov Chain Monte Carlo (MCMC) methods, in particular, the Metropolis-Hastings algorithm and the Stochastic Origin Ensembles (SOE) method, in a concise and notation-consistent manner. These methods are commonly used to perform model parameter estimation for a population, based on a measured sample, through the sampling of the probability distribution for these parameters. A simple application of SOE is then demonstrated using simulation data from a Compton imager detector. 14. KEYWORDS, DESCRIPTORS or IDENTIFIERS (Technically meaningful terms or short phrases that characterize a document and could be helpful in cataloguing the document. They should be selected so that no security classification is required. Identifiers, such as equipment model designation, trade name, military project code name, geographic location may also be included. If possible keywords should be selected from a published thesaurus. e.g. Thesaurus of Engineering and Scientific Terms (TEST) and that thesaurus identified. If it is not possible to select indexing terms which are Unclassified, the classification of each should be indicated as with the title.) Markov chain Monte Carlo Metropolis-Hastings Stochastic Origin Ensembles Parameter estimation Maximum likelihood Maximum Likelihood Expectation Maximisation