In Chapters 2 **and** 3 , we investigate two different path problems on interval **and** proper interval graphs, as well as we introduce two matrix representations **of** them. First, we investigate in Chapter 2 the complexity status **of** the longest path problem on the class **of** interval graphs. Even if a **graph** is not Hamiltonian, it makes sense in several **applications** **to** search for a longest path, or equivalently, **to** find a maximum induced subgraph **of** the **graph** that is Hamiltonian. However, computing a longest path seems **to** be more difficult than deciding whether or not a **graph** admits a Hamiltonian path. Indeed, it has been proved that even if a **graph** is Hamiltonian, the problem **of** computing a path **of** length n −n ε for any ε < 1 is NP-hard, where n is the number **of** vertices **of** the input **graph** [ 74 ]. Moreover, there is no polynomial-time constant-factor approximation algorithm for the longest path problem unless P=NP [ 74 ]. In contrast **to** the Hamiltonian path problem, there are only few known polynomial algorithms for the longest path problem, **and** these restrict **to** trees **and** some other small **graph** **classes**. In particular, the complexity status **of** the longest path problem on interval graphs was as an open question [ 113 , 114 ], although the Hamiltonian path problem on an interval **graph** G = (V, E) is well known **to** be solved by a greedy approach in linear time O( |V |+|E|) [ 3 ]. We resolve this problem by presenting in Chapter 2 the first polynomial algorithm for the longest path problem on interval graphs **with** running time O(n 4 ), which is based on a dynamic programming

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The focus **of** this thesis is the investigation **of** the facial structure **of** the cardinality constrained matroid, path **and** cycle polytopes. As it might be expected **and** is exemplarily shown for path, cycle, **and** matroid polytopes, an inequality that induces a facet **of** the polytope associated **with** the ordinary problem usually induces a facet **of** the polytope associated **with** the cardinality re- stricted version. However, we are in particular interested in inequalities that cut off the incidence vectors **of** solutions that are feasible for the ordinary problem but infeasible for its cardinality restricted version. In this context, the most important class **of** inequalities for this thesis are the so-called forbidden cardinality inequalities. These inequalities are valid for a polytope associated **with** a cardinality constrained **combinatorial** **optimization** problem independent **of** its specific **combinatorial** structure. Using these inequalities as prototype for inequalities incorporating com- binatorial structures **of** a problem, we derive facet defining inequalities for polytopes associated **with** several cardinality constrained **combinatorial** **optimization** problems, in particular, for the above mentioned polytopes. Moreover, for cardinality constrained path **and** cycle polytopes we derive further **classes** **of** facet defining inequalities **related** **to** cardinality restrictions, also those inequalities specific **to** odd/even path/cycles **and** hop constrained paths.

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We have seen in Chapter 2 how a random tiling **of** a hexagon maps **to** a collection **of** directed non-intersecting lattice paths on a hexagonal lattice. Since these paths only use steps in two lattice directions, we can map such a family **to** a family **of** paths on the square lattice only taking up-steps ((i, j) → (i, j + 1)) **and** right-steps ((i, j) → (i + 1, j)). In the case **of** an (r, s, t)-hexagon we have r paths, the ith path running from (−i + 1, i − 1) **to** (t − i + 1, s + i − 1). Each path can be viewed as a Ferrers diagram (cf. Section 3.4) fitting inside an t × s-rectangle **and** the volume **of** a plane partition is the sum **of** their areas. Thus the volume can be viewed as an area functional on the set **of** families **of** paths. Other, more obvious area functionals are the area between two paths or the area a path encloses **with** a given line if the family is conditioned **to** stay on one side **of** that line (as those associated **with** symmetric tilings, cf. Section 2.1). In this chapter we investigate this problem in the special case **of** only two paths on the square lattice. We compute the area laws for all symmetry subclasses **of** such configurations in the limit **of** large path lengths. This model has been studied in combinatorics [BM96, Ric09b] as well as statistical physics [PB95, Ric06] under the name **of** staircase polygons, parallelogram polygons or polyominoes, where they serve as a simplified model **of** self-avoiding polygons, see also Chapter 5. Explicit expressions for the half-perimeter **and** area generating functions for all symmetry **classes** **of** staircase polygons are given in [LR01], however these are not amenable **to** a first principles approach as applied **to** formula 3.1 in Chapter 3. Instead we analyse functional equations satisfied by the half-perimeter **and** area generating functions **of** the respective **classes** **and** apply the moment method [Bil95, Section 30]. Some **of** the limit laws can also be obtained via bijections **to** **related** **combinatorial** objects (cf. Sections 4.2.3, 4.2.4 **and** 4.2.8), but we give a unified approach applicable **to** all symmetry subclasses.

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Second, for the emerging social network systems over the Internet, simple game **models** also start **to** have potential real **applications**. Consider a **graph** representing friend relationships. Let an advertiser who wants **to** place advertisement on the social network so that each individual has an AD on his own web page or (not exclusive) on one **of** his friends’ pages. The constraints can be written in terms **of** the incidence matrix **of** the **graph** (not the edge-vertex incidence matrix). The optimal solution **to** the covering problem will save advertiser’s cost. Issues on how **to** share the revenue generated among the nodes is a very interesting problem.

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However, as soon as one allows for zero-entries in W , the problem becomes much harder. For the special case **of** a doubly stochastic limit (r = c = 1), Sinkhorn **and** Knopp (1967) show that convergence only holds if the set **of** non-zeros E(W ) satisfies specific **combinatorial** properties. Even in case **of** convergence, the limit W h∞i can show zeros where W has non-zeros, which turns the relative entropy objective into a non-smooth **optimization** problem. As a consequence, the Lagrangian approach can no longer be applied without further considerations. Some publications overlook this pitfall (see Section 3.4.3). They “simply” apply the Lagrangian approach **to** the non-negative setting **and** finally obtain invalid conclusions. Hence, assuming positivity is not just about “simplifying some arguments”, as sloppily stated by Ireland **and** Kullback (1968), but a totally different scenario. Also the proof by Bregman (1967b) claims **to** imply the non-negative case, but it deals only superficially **with** the extension **to** zero-entries. The proof for the RE-optimality **of** IPF was rigorously carried over from the positive **to** the non-negative case by Csiszar (1975), who totally avoids any Lagrangian-type arguments by a measure theoretical approach. Zeros in W h∞i that are not present in W are handled technically sound by considering the Radon-Nikodym derivatives, which are able **to** deal **with** absolute continuous measures, that is **with** E(W h∞i ) ( E(W ). The only drawback **of** this approach is that it does not provide an algorithmic intuition on why optimality holds. What is so special about the IPF-sequence that guarantees that the limit is not just some element from Ω(r, c, W ), but precisely the unique element from Ω(r, c, W ) that is closest **to** W **with** respect **to** relative-entropy-error?

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This paper presents a branch-**and**-price-**and**-cut algorithm for the exact solution **of** the active-passive vehicle-routing problem (APVRP). The APVRP covers a range **of** logistics **applications** where pickup-**and**-delivery requests necessitate a joint op- eration **of** active vehicles (e.g., trucks) **and** passive vehicles (e.g., loading devices such as containers or swap bodies). The objective is **to** minimize a weighted sum **of** the total distance traveled, the total completion time **of** the routes, **and** the num- ber **of** unserved requests. **To** this end, the problem supports a flexible coupling **and** decoupling **of** active **and** passive vehicles at customer locations. Accordingly, the operations **of** the vehicles have **to** be synchronized carefully in the planning. The contribution **of** the paper is twofold: Firstly, we present an exact branch-**and**- price-**and**-cut algorithm for this class **of** routing problems **with** synchronization con- straints. **To** our knowledge, this algorithm is the first such approach that considers explicitly the temporal interdependencies between active **and** passive vehicles. The algorithm is based on a non-trivial network representation that **models** the logical relationships between the different transport tasks necessary **to** fulfill a request as well as the synchronization **of** the movements **of** active **and** passive vehicles. Sec- ondly, we contribute **to** the development **of** branch-**and**-price methods in general, in that we solve, for the first time, an ng-path relaxation **of** a pricing problem **with** linear vertex costs by means **of** a bidirectional labeling algorithm. Computational experiments show that the proposed algorithm delivers improved bounds **and** solu- tions for a number **of** APVRP benchmark instances. It is able **to** solve instances **with** up **to** 76 tasks, 4 active, **and** 8 passive vehicles **to** optimality within two hours **of** CPU time.

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where x denotes the real vector **of** unknown parameters (the point movements) **and** v denotes the real vector **of** unknown residuals, that is, the degree **of** constraint satisfaction. Both, A (referred **to** as the design matrix) **and** l (the vector **of** observations) need **to** be speciﬁed in advance **to** deﬁne the constraints. The constraints are perfectly satisﬁed if v = 0. As this is generally not possible for all constraints, the function v T ·P ·v is minimized, where P deﬁnes the weights between diﬀerent constraints. If there are non-linear constraints, these are usually replaced by their linear approximations. Sarjakoski & Kilpel¨ ainen (1999) **and** Harrie & Sarjakoski (2002) show how **to** solve the problem for large datasets, also considering other generalization operators. Applying the same adjustment technique, Koch & Heipke (2005) **and** Koch (2007) additionally show how **to** cope **with** hard inequality constraints that are needed **to** ensure consistency between DLMs **and** digital terrain **models**. **Related** problems are discussed in the generalization domain, for example, a river must not run uphill (Gaﬀuri, 2007). Least squares adjustment allows diﬀerent generalization operators **to** be handled, yet the existing generalization methods that are based on this technique do not take the discrete nature **of** map generalization into account. Usually, continuous variables are used **to** model a problem. These are not suited, for example, **to** represent whether a vertex **of** an original line is selected for its simpliﬁcation. In their system, Sarjakoski & Kilpel¨ ainen (1999) deﬁne a constraint that attempts **to** pull an unwanted vertex onto the line connecting its predecessor **and** successor. This is a smart workaround **to** also allow for line simpliﬁcation, but **of** course it is not a solution **to** the discrete problem **of** vertex selection, which only allows two stages **and** none in between. Sester (2005) applies adjustment calculus **to** satisfy constraints in building simpliﬁcation, but also points out that it does not solve the whole problem: the elimination **of** details is done in a ﬁrst step, which is not based on **optimization**. The handling **of** hard constraints in **optimization** approaches is seldom addressed in the map generalization literature. Often constraints are relaxed, as they are conﬂicting (Harrie & Weibel, 2007). A few exceptions exist in the context **of** discrete **optimization**, which is addressed in the next section.

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7. Conclusions **and** future perspectives
Most **of** the **combinatorial** **optimization** problems that are found in real-world **applications** have a stochastic nature. Since the vast majority **of** the articles in the **combinatorial** **optimization** litera- ture deal **with** deterministic scenarios, there is a need **to** consider simulation–**optimization** approaches that allow researchers **and** practitioners for solving realistic **models** including uncertainty. Simheuristics contribute **to** fill this gap by extending metaheuristic algorithms in a natural way, so they can also be applied in solving **combinatorial** **optimization** problems **with** stochastic components either in the objective function or in the set **of** constraints. How- ever, the concept **of** simheuristics described in this paper differs from the metaheuristics reported in the SO community [ 9 , 57 ]. In- stead **of** using a pure black box approach, where evaluations are performed only by simulation, simheuristics closely integrate op- timization **and** simulation by incorporating problem-specific infor- mation. Thus, analytical expressions complement the **optimization** process **and** may be used **to** screen poor or infeasible solutions. Since these analytical expressions are problem-specific, they ex- ist prior **to** any simulation run. Therefore, they are not as depen- dent on simulation as the metamodels used in the SO community. Still, they can be enhanced **with** the simulation feedback. Finally, by design they are able **to** provide different alternative solutions **of** similar quality **and** promote the introduction **of** risk or reliability analysis criteria when comparing these solutions, so the decision- maker can choose the solution that best fits his/her utility function according **to** these criteria. In order **to** control the computational time invested in performing simulations, there are some critical is- sues in the design **of** an efficient simheuristic algorithm. One issue is the selection policy **of** promising solutions—the ones that will be sent **to** the simulation component. Another issue is the number **of** replications that must be run for each **of** these promising solu- tions. During the stochastic searching process, simulations **with** a relatively short number **of** replications should be sufficient **to** ob- tain rough estimates **of** the solution value, so that a list **of** elite solutions can be constructed. Once the stochastic searching pro- cess is finished, simulations **with** more replications can be run in order **to** obtain more accurate estimates for each **of** the elite solu- tions. Alternatively, statistical selection methods can be incorpo- rated **to** adjust the simulation length according **to** the difference between solutions. Also, variance reduction techniques can be em- ployed here.

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In the present chapter the gamma model **with** continuous (quantitative) factors is considered. There are wide **applications** where the gamma model **with** its canonical link can be fitted. Nevertheless, there is always a doubt about the suitable link function for outcomes. The common alternative links may come from the power link family that includes the canonical link therefore it is a favorite choice for employment in the thesis. In section 4.1 , we introduce the gamma model highlighting on the **related** assump- tions. Additionally, the notions **of** locally complete **classes** **and** locally essentially com- plete **classes** are presented. In section 4.2 , locally complete **classes** **and** locally essen- tially complete **classes** **of** designs are found leading **to** a considerable reduction **of** the problems **of** locally optimal designs for gamma **models**. From those **classes** locally D- **and** A-optimal designs are derived. Besides, as a gamma model is recognized as a par- ticular generalized linear model the results that are obtained in Chapter 3 for a general setup **of** the generalized linear model will be applied in relevant cases here. The opti- mality conditions will be intuitively characterized by the model parameters **and** hence, those conditions cover relevant subregions **of** the parameter space. So, our results on locally D- or A-optimality are applicable for the majority **of** possible parameter points. In Section 4.3 , we consider a model **with** a single continuous factor. In section

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The thesis is structured into six chapters, representing parts **of** my work in the field **of** parameterized complexity theory. Chapter 1 contains an introduction **to** **and** mo- tivation **of** preprocessing, as well as preliminary explanations **of** **graph**-theoretic **and** (parameterized) complexity-theoretic notation used throughout the thesis. Chapter 2 contains work I did in collaboration **with** my coauthor Johannes Uhlmann on the Two-Layer Planarization problem [ 191 ] which aims at making a given **graph** drawable in two layers without edge crossings by deleting edges. As we shared a room in our o ffices at the Friedrich-Schiller-Universität Jena, he approached me **with** the suggestion **to** look at this problem. Based on earlier work he did **with** Nadja Betzler **and** Jiong Guo, there was a stub manuscript containing some data reduction rules. In joint work, Johannes **and** I developed missing data reduction rules, I came up **with** a uniform way **of** presenting our preprocessing using “tokens”. We proved correctness **of** the whole procedure **and** developed a branching strategy **to** solve Two-Layer Planarization. The paper then was accepted **to** publication at the 7th Annual Conference on Theory **and** **Applications** **of** **Models** **of** Computation (TAMC 2010), yielding an invitation **to** a special issue **of** the journal Theoretical Computer Science. Recently, I picked the paper up again **and** worked on the question whether the branching algorithm presented by Suderman [ 182 ] could be adapted **to** our parameterization. I succeeded **to** some extend, providing a branching algorithm whose asymptotic running time almost matches that **of** Suderman’s algorithm. I implemented the algorithm **and** tested it against published results by Mutzel [ 156 ] **and** Suderman **and** Whitesides [ 183 ]. This work, however, was not published so far.

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Clearly, an alternative approach would be **to** calculate estimators b f 1 **and** b f 2 in the model Z i = f 1 (X i+1 )+
f 2 (X i ) + η i **and** **to** use b f 1 (x) − b f 2 (x) as an estimator **of** f . We will come back **to** **related** **models** below.
The additive model is important for two reasons:
(i) It is the simplest nonparametric regression model **with** several nonparametric components. The theoretical analysis is quite simple because the nonparametric components enter linearly into the model. Furthermore, the mathematical analysis can build on localization arguments from classical smoothing theory. The simple structure allows for completely understanding **of** how the presence **of** additional terms influences estimation **of** each one **of** the nonparametric curves. This question is **related** **to** semiparametric efficiency in **models** **with** a parametric component **and** nonparametric nuissance components. We will come back **to** a short discussion **of** nonparametric efficiency below. (ii) The additive model is also important for practical reasons. It efficiently avoids the curse **of** di- mensionality **of** a full-dimensional nonparametric estimator. Nevertheless, it is a powerful **and** flexible model for high-dimensional data. Higher-dimensional structures can be well approximated by additive functions. As lower-dimensional curves they are also easier **to** visualize **and** hence **to** interpret than a higher-dimensional function.

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The inherent modularity **of** distributed **and** decentralized frameworks can also be beneficial in a multi-agent setting when the number **of** agents in the system is dynamic. For example, when a faulty component needs **to** be isolated **and** shut down for maintenance, the temporarily reduced system can be readily re-optimized within a distributed **optimization** framework. It is worth noting that agents can also be unresponsive or acting atypically, which are known as Byzantine faults. Detection **and** resolution **of** Byzantine faults are not considered within this thesis, but we refer the interested reader **to** an early paper by Lamport et al. ( [LSP82]) **and** the overview by Driscoll ( [DHSZ03]). For distributed continuous convex **optimization**, there are already a number **of** available algorithms such as dual decomposition [Eve63, NS08], Alternating Direction Method **of** Multipliers (ADMM) [BPC + 11, EB92, GM76], or Aug- mented Lagrangian based Alternating Direction Inexact Newton (ALADIN) methods [HFD16], which can all be used **to** solve large-scale strictly con- vex programs **to** global optimality by alternating between solving small-scale convex **optimization** problems **and** sparse linear algebra operations.

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The small-world problem dates back **to** Stanley Milgram, who was the first studying this phenomenon in social networks [16]. He discovered by an experiment, that two ran- domly chosen people are closely **related** **to** each other, despite the fact, that they may be very different 5 . In his experiment, Milgram asked some arbitrarily chosen person liv- ing in Nebraska **to** send a letter **to** a stockbroker in Massachusetts by passing the letter from person **to** person. Furthermore, Milgram made the restriction, that the letter may only be send **to** a person, which is known on a first-name basis. **To** his surprise he found out, that in average the letter was passed only **to** six other people before it reached the stockbroker. The conclusion is, that in the world considered as a social network **of** peo- ple connected through friendship or acquaintanceship the average path length between any two people is rather short [17]. Additionally **to** these short connections between people, most people have a high number **of** friends or acquaintances, which makes the network highly clustered.

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Turning **to** the quadratic standardized forecast error term in Figure 5, it may be deduced that time variation **of** the log predictive likelihood is primarily due **to** the forecast errors. This is not surprising since the covariance matrix **of** the predictive distribution changes slowly **and** smoothly over time while the forecast errors are more volatile. Moreover, the ranking **of** the **models** is **to** some extent reversed, particularly **with** the BVAR having much larger standardized forecast errors than the other **models** over the ﬁrst half **of** the forecast sample. **With** the exception **of** the random walk model, this is broadly consistent **with** the ﬁndings for the point forecasts; see Warne et al. (2013). The reversal in rankings for the forecast error term can also be understood from the behavior **of** second moments, where a given squared forecast error yields a larger value for this term the smaller the uncertainty linked **to** the forecast is. Nevertheless, when compared **with** the forecast uncertainty term in Figure 4 the diﬀerences between the **models** are generally smaller for the forecast error term. This suggests that the model ranking based on the log predictive score is primarily determined by the second moments **of** the predictive distribution in this application.

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upper bound. Young’s modulus **and** density are chosen as the parameters **to** be changed during the **optimization** because they are very much linked **to** the vibration pattern **of** the mechanical structure. The parameters needs change **with** some co-dependency (here a linear dependency) such that the achieved material configuration is kept realistic **and** **to** avoid trivial solutions.

Based on the BIC information criterion **with** heavier penalty for extra parameters, the GTARCH model without spline is preferred, while using the AIC criterion, the Spline-Macro-GTARCH is the superior model. This result holds for both SPX **and** TSX. Note that both selected **models** in- clude the most general GTARCH specification **with** the presence **of** asymmetry in both ARCH **and** GARCH terms. Moreover, the asymmetric term δ goes up **to** 0.24 in the SPX Spline GTARCH model, making the response **to** negative news even more asymmetric compared **to** GTARCH **with**- out spline **with** δ = 0.16. The optimal number **of** knots in the SPX Spline model is 17, while the number **of** knots goes down **to** 8 when we add macroeconomic variables. Macroeconomic vari- ables are useful in modeling the low-frequency component, as their presence reduces the number **of** knots for cycles **and** they have a statistically significant effect on long-run volatility dynamics. Engle **and** Rangel (2008) use macroeconomic variables for panel regressions **of** 48 countries using annual volatility data. In our paper, we model macroeconomic variables for daily volatility fore- casting in the slow-moving component. Thus, statistically significant macroeconomic variables in the low-frequency component could be used for stress testing **of** VaRs, which is a typical regu- latory requirement. The following variables are statistically significant at 10% for predicting the low-frequency volatility component for SPX:

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Considering videos, it becomes clear why we are in particular interested in the average reduction ratio. The diodes are more or less uniformly exposed **to** the stress, **and** the degradation is a long drawn process. Moreover the worst-case reduction ratio for any algorithm is 1 since an image that is represented by a diagonal matrix can not be decomposed into multilines. Note that this also holds for non-consecutive multiline addressing. On the other hand, we advise designers **of** user interfaces for CMLA driven OLED devices **to** bear in mind that they may delay the so-called burn-in effect **of** frequent steady images by smoothing sharp diagonal edges, e.g. by antialiasing. As mentioned before, we claim that an economical hardware implementation becomes possible. The hardware complexity **of** the logics **to** implement mlacompact is only a few thousand gates for QQVGA displays. This yields a sufficiently small area on a silicon wafer that is necessary **to** be in business on the highly competitive display market. However, if we use the same amount **of** RAM as we did on the PC, that is **to** store the complete output, the area will increase significantly. We will address this issue in the following.

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Based on the 3D turn rates **and** accelerations provided by the IMU we analysed characteristic features for each target activity **with** their physical or bio-mechanical explanation, their discriminative power between activities **and** their computation complexity. The features span different window lengths from 32 **to** 512 samples (at 100 Hz), which represent the different natures **of** instantaneous activities (like “jumping”) **to** longer term, repetitive activities like “running”. All features are calculated in real time **with** a frequency **of** 4 Hz **and** discretised into states meaningful **to** distinguish between activities. These have been defined manually in our set up, but this could be automated easily **with** data clustering algorithms. In our implementation, the set **of** features is easily extendable **and** would also cover the integration **of** more sensors into the system seamlessly. For the classification, we decided **to** apply Bayesian techniques. **With** the discretised value ranges **of** all features, we applied a modified learning algorithm for discrete Bayesian Networks (BNs), the Greedy Hill Climber **with** Random Restarts based on the Cooper **and** Herskovits Log score (see [8]) **and** Dirichlet distributions **of** the conditional probability tables, on our 270 minutes activities data set. We limited structure learning **to** a fixed number **of** parents per node **and** imposed causal direction **to** learnt arcs. The learnt structure is shown in Figure 1.

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prior beliefs about the hyperparameter κ are encoded in a prior distribution p(κ). From a conceptual point **of** view, a researcher could introduce another level **of** hierarchy **and** make the prior for κ depend on more hyperparameters as well. Since we are concerned **with** **applications** where the dimensionality **of** κ is already small (such as the time-varying parameter **models** we describe later), we will not pursue this question further in this paper - our approach could be extended in a straightforward manner if a researcher was interested in introducing additional levels **of** hierarchy. We focus here on drawing one vector **of** hy- perparameters, but other vectors **of** hyperparameters could be included in θ (which could be high-dimensional, as in our time-varying parameter VAR later). Draws for those other vectors **of** hyperparameters would then be generated using additional Metropolis steps that have the same structure. If J vectors **of** hyperparameters are present, we denote vector j by κ j (j = 1, . . . , J ) **and** the vector **of** all hyperparameters by ˜ κ = [κ ′ 1 κ ′ 2 . . . κ ′ J ] ′ . When we

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