A Network Epidemic Model for Online Community Commissioning Data
A statistical model assuming a preferential attachment network, which is generated by adding nodes sequentially according to a few simple rules, usually describes real-life networks better than a model assuming, for example, a Bernoulli random graph,…
Authors: Clement Lee, Andrew Garbett, Darren J. Wilkinson
A Net w ork Epidemic Mo del for Online Comm unit y Commissioning Data Clemen t Lee 1,2 , Andrew Garb ett 2 , and Darren J Wilkinson 1 1 Sc ho ol of Mathematics and Statistics, Newcastle Univ ersit y 2 Op en Lab, Newcastle Universit y Marc h 28, 2022 Abstract A statistical model assuming a preferential attac hment net w ork, which is generated b y adding no des sequentially according to a few simple rules, usually describes real-life net- w orks b etter than a model assuming, for example, a Bernoulli random graph, in which an y t w o no des hav e the same probability of being connected, do es. Therefore, to study the prop ogation of “infection” across a so cial net work, w e propose a netw ork epidemic mo del b y com bining a sto chastic epidemic mo del and a preferential attachmen t mo del. A simulation study based on the subsequent Marko v Chain Monte Carlo algorithm reveals an iden tifia- bilit y issue with the model parameters. Finally , the netw ork epidemic model is applied to a set of online commissioning data. 1 In tro duction So cial netw ork analysis has b een a p opular research topic o ver the last couple of decades, thanks to the unpreceden tedly large amoun t of internet data a v ailable, and the increasing p ow er of computers to deal with suc h data, whic h details ties b etw een people or ob jects all ov er the w orld. A lot of mo dels ha v e b een developed to c haracterise and/or generate netw orks in v arious w a ys. One w ell-known class of mo dels in the statistical literature is the exp onential random 1 graph mo del (ERGM), in whic h the probability mass function on the graph space is prop ortional to the exp onential of a linear combination of graph statistics; see, for example, Snijders (2002). The Bernoulli random graph (BRG), in which any tw o no des hav e the same probability of b eing connected, independent of any other pair of no des, is a special case of an ERGM. Although the c hoice of graph statistics allows an ER GM to encompass netw orks with differen t characteristics, in general the ERGMs do not describ e real-life net w orks well; see, for example, Snijders (2002) and Hun ter et al. (2008). Instead of characterising a netw ork b y graph statistics, such as the total num b er of degrees, the configuration mo del considers the sequence of the individual degrees; see, for example, Newman (2010), Chapter 13. Eac h no de is assigned a num b er of half-edges according to its degree, and the half-edges are paired at random to connect the no des. Despite its simple rule of net work generation, the configuration mo del may contain multiple edges or self-connecting no des, whic h migh t not occur in real-life net works. Also, the whole net work is not guaran teed to be connected. Moreo v er, even though the individual degrees may b e flexibly mo delled b y a degree distribution, they are not completely indep endent as they hav e to sum to an even integer. One prominent feature of so cial netw orks in real life is that they are scale-free, which means that the degree distribution follows a p o wer law (appro ximately); see, for example, Alb ert, Jeong and Barab´ asi (1999, 2000), and Stumpf, Wiuf and Ma y (2005). The preferen tial attachmen t (P A) mo del by Barab´ asi and Alb ert (1999) is one widely known mo del (Newman, 2010, Chapter 14) that generates such a netw ork with a few parameters and a simple rule. Other models also exist that characterise either the degree distribution, for instance the small-world model by W atts and Strogatz (1998), or other asp ects suc h as ho w clustered the no des in the net work are (V´ azquez et al., 2002). While the ma jority of the net work mo dels fo cus on the top ology of the netw ork, some mo dels are dev elop ed to describe the dynamics within the netw ork, in particular how fast information spreads with resp ect to the structure of the netw ork. As spreading rumours or computer viruses through connections in a so cial net work is similar to spreading a disease through real life con tacts to create an epidemic, most of these models incorp orate certain compartment models in epidemi- ology . F or instance, the Susceptible-Infectious-Reco vered (SIR) mo del splits the p opulation into three compartments according to the stage of the disease of each individual. A susceptible indi- vidual b ecomes infectious up on con tact with an infectious individual, and reco vers after a random 2 p erio d. T raditionally , the infectious p erio d and the contacts made b y an infected individual are assumed to follow an exp onential disribution and a homogeneous P oisson Pro cess, resp ectively . While these assumptions may b e unrealistic for real life data, they are useful as the epidemic pro cess is no w Mark o vian. The dynamics of compartment sizes ov er time can usually b e charac- terised b y a small n umber of parameters in the rate matrix, which is used to obtain the transition probabilities through the Kolmogorov’s equations; see, for example, Wilkinson (2011), Section 5.4. While other kinds of compartment mo dels can be form ulated in a similar wa y , some mo dels depart from the Marko vian assumptions, and will be discussed later. F or more details on the SIR model and its v arian ts, see, for example, Andersson and Britton (2000). Often implicitly assumed in such compartmen t mo dels is that the epidemic is homogeneous mixing, that is, eac h individual can in teract uniformly with all other individuals in the communit y he/she belongs to. How ever, this is not the case when it comes to netw ork epidemics, as one can only infect and b e infected by their neigh b ours in the netw ork, and the collection of neighbours differs from individual to individual. Therefore, modelling an epidemic on a structured p opulation requires relaxing the homogeneous mixing assumption. Instead of assuming the same set of v alues for the parameters gov erning the dynamics, one approach is to apply a separate set of parameter v alues to, for example, each individual or all individuals with the same degree. Such an approac h fo cuses on the mo delling side, and is dominant in the ph ysics literature. A comprehensive review is pro vided by Pastor-Satorras, Castellano, V an Mieghem and V espignani (2015). Our w ork on net w ork epidemic mo delling is motiv ated b y a data set from App Mov ement 1 , whic h is an online platform that enables comm unities to prop ose and design comm unity-commissioned mobile applications (Garbett, Comber, Jenkins and Olivier, 2016). The process of generating the application starts with a comm unity creating a campaign page and sharing it via online so cial netw orks. If we view an individual ha ving seen a campaign and in turn promoting it as b eing “infected” (and “infectious” simultaneously), then the pro cess of sharing a campaign can b e compared to spreading a real-life virus to create an epidemic. The main difference is that suc h an infectious individual cannot p otentially infect an yone in the population but only those connected to them on the social netw orks. F or one campaign, the cumulativ e count of infected and the net w ork of infected users are plotted in Figures 1 and 2, respectively . The former deviates from the typical S shap e of a homogeneous mixing epidemic, while the latter displa ys star-lik e structures and long paths, which typical features in real-life netw orks. It should b e noted that 1 https://app- movement.com 3 this does not represent the complete underlying netw ork G , which is usually unknown. 0 100 200 300 F eb 01 F eb 15 Mar 01 Mar 15 Apr 01 Time Cumulativ e count of infected Figure 1: Cum ulative count of infected for an epidemic of a campaign. The time p oints where the coun t increments are the infection times I . ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1 2 3 4 5 6 7 9 10 11 14 15 16 8 18 19 23 24 25 26 20 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 59 60 29 30 61 66 64 71 74 73 75 78 79 80 81 82 84 83 12 21 69 86 27 72 91 93 98 88 99 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 130 131 101 90 136 134 92 139 138 135 142 132 144 143 89 147 150 151 152 153 148 87 155 94 157 154 159 149 163 164 156 162 85 168 100 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 160 205 206 207 209 211 214 215 216 217 218 219 212 210 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 253 254 204 256 203 260 261 259 264 265 268 267 270 272 273 255 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 298 300 302 303 304 158 307 309 310 311 312 313 314 315 316 318 319 320 321 322 323 324 325 326 327 257 329 169 77 140 274 306 13 17 22 28 58 62 63 65 67 68 70 76 95 96 97 102 103 129 133 137 141 145 146 161 165 166 167 202 208 213 220 221 252 258 262 263 266 269 271 275 297 299 301 305 308 317 328 330 331 332 333 334 335 Figure 2: Netw ork representation of the transmission tree P for the same campaign epidemic as sho wn in Figure 1. 4 Due to the difference in the data b eing applied to, as well as the inclination tow ards inference, epidemic mo dels in the statistics literature pro vide a stark contrast from the classical compart- men t mo del, not only with resp ect to the netw ork issue. First, to accommo date heterogeneities in mixing, Ball et al. (1997) and Britton et al. (2011) prop osed models whic h incorporate t wo lev els and three levels of mixing, resp ectively . Each individual b elongs to b oth the global level and one or more lo cal lev els, such as household, school or workplace, and homogeneous mixing is assumed to tak e place at each level but with a separate rate. Suc h mo dels are prompted by data with detailed information of these lo cal level structures each individual b elongs to, suc h as the 1861 Hagello ch measles outbreak data analysed by Britton et al. (2011). Second, some SIR mo dels and their v ariants relax the assumption that the infectious p erio d follows the exp onential distribution, essentially rendering the epidemic pro cess non-Marko vian. F or instance, Streftaris and Gibson (2002) used the W eibull distribution, while Neal and Rob erts (2005) and Groendyke, W elch and Hun ter (2012) used the Gamma distribution. In general, the compartment dynamics cannot b e represen ted b y a simple differential equation. Third, information is often missing in epidemic datasets, suc h as the infection times and, if a net w ork structure is assumed, the actual netw ork itself. Therefore, models are dev elop ed with a view to inferring these missing data, usually achiev ed by Marko v Chain Monte Carlo (MCMC) algorithms. Examples of mo dels whic h imp ose a netw ork structure include Britton and O’Neill (2002), Neal and Rob erts (2005), Ra y and Marzouk (2008) and Gro endyke, W elc h and Hun ter (2011). In the data considered by these authors, no co v ariates exist to inform if tw o individuals are neigh bours in the net w ork, and the edge inclusion probability parameter is assumed to b e the same for any tw o individuals in the netw ork. Essentially the underlying net work is a BRG, whic h yields a Binomial (or ap- pro ximately Poisson) degree distribution. Such a netw ork mo del seems unrealistic for our App Mo v ement data, compared to a mo del that generates a scale-free netw ork or utilises a p ow er law t yp e degree distribution. In view of the differences in ob jectives and applications shown ab ov e, we prop ose a netw ork epidemic model as an attempt to narrow the gap in the literature. W e fo cus on a Susceptible- Infectious (SI) mo del, in which the epidemic pro cess tak es place on a net work which is assumed to be built from the P A mo del, thus deviating from a BRG. When it comes to inference, the data contains the infection times and p otentially the transmission tree, while the underlying net w ork is unkno wn and therefore treated as laten t v ariables. W e aim at sim ultaneously inferring the infection rate parameter, the parameters gov erning the degree distribution, and the latent structure of the netw ork, in terms of the p osterior edge inclusion probabilities, by using an 5 MCMC algorithm. While the choice of the SI mo del is due to the data in hand, we b elieve the mo del structure and algorithm introduced can b e extended to other compartment mo dels. The rest of the article is divided as follows. The latent netw ork epidemic mo del is introduced in Section 2. Its likelihoo d and its associated MCMC algorithm are derived in Section 3. They are then applied to tw o sets of simulated data in Section 4, and a set of real online commissioning data in Section 5. Section 6 concludes the article. 2 Mo del In this section we in tro duce the latent net work SI epidemic mo del. Describing the formation of the netw ork and the epidemic separately will facilitate the deriv ation of the lik eliho o d in the next section. The notations and definitions are kept to be similar to those in Britton and O’Neill (2002) and Gro endyke et al. (2011). Consider an epidemic in a closed population of size m . Let I = ( I 1 , I 2 , . . . , I m ) denote the ordered v ector of infection times, where I i is the infection time of individual i , and I i ≤ I j for any i < j . W e assume that the first individual is the only initial infected individual. In order to ha v e a temp oral p oint of reference, only the times of m − 1 infections will b e random, and so we define ˜ I = I − I 1 = ( ˜ I 1 = 0 , ˜ I 2 , . . . , ˜ I m ) for conv enience. W e also assume that the observ ation p erio d is long enough to include all infections. Next, consider the undirected random graph G of m no des which represents the social structure of the p opulation, in which the no de i represents the i th individual. Using the adjacency matrix represen tation, if individuals i and j are so cially connected, we write G ij = 1 and call them neigh b ours of eac h other, G ij = 0 otherwise. In this sense G ij can b e interpreted as a potential edge of i and j . W e also assume symmetry in social connections and that each individual is not self-connected, that is, G ij = G j i and G ii = 0, respectively , for 1 ≤ i, j ≤ m . T o characterise G , w e use a mo dified version of the P A mo del b y Barab´ asi and Alb ert (1999), whic h generates a netw ork by sequentially adding no des into it. This requires an order of how the no des enter the netw ork, which is not necessarily the same as the epidemic order. Therefore, w e define a vector random v ariable of the netw ork order, denoted by σ = ( σ 1 , σ 2 , . . . , σ m ), whose 6 supp ort is all m ! p ossible p ermutations of { 1 , 2 , . . . , m } . Node σ i (1 ≤ i ≤ m ), lab elled by the epidemic order, is the i th no de that enters the netw ork. Such order is mainly for the sak e of c haracterisation using the P A mo del, and the net work is assumed to hav e formed b efore the epidemic tak es place, and remain unchanged throughout the course of the epidemic. Such an assumption is reasonable b ecause the timescale of an epidemic is usually muc h smaller than that of net work formation, the pro cess of which is describ ed next. 2.1 Sequence of new edges Initially , there are tw o no des σ 1 and σ 2 whic h are connected i.e. G σ 1 σ 2 = 1. When node σ i (3 ≤ i ≤ m ) enters the netw ork, it connects to X i existing no des, where X i follo ws a censored P oisson distribution with parameter µ and supp ort { 1 , 2 , . . . , i − 1 } , that is, Pr( X i = x ) = e − µ (1 + µ ) , x = 1 , e − µ µ x x ! , x = 2 , 3 . . . , i − 2 , ∞ X z = i − 1 e − µ µ z z ! , x = i − 1 . (1) Indep endence is assumed b etw een X i and X j if i 6 = j . W e mo del the num b er of new edges as a random v ariable because using a constan t n umber of new edges, denoted b y µ 0 , whic h is what the original model b y Barab´ asi and Albert (1999) did, fixes the total n umber of edges to ( m − 2) µ 0 + 1 and mak es the mo del to o restrictiv e. Empirically this censored distribution p erforms b etter than a trunc ate d Poission distribution in terms of iden tifying µ . 2.2 A ttaching edges to no des When node σ i joins the netw ork, according to the original P A rule, an existing node σ j (1 ≤ j < i ) gets connected to no de σ i , that is, G σ i σ j = 1, with probability prop ortional to its current degree P i − 1 k =1 G σ k σ j . T o allow the degree of P A to v ary , w e allow such probability to be a mixture of the current degree and ho w recen tly the no de has joined the net work. T o b e more sp ecific, the pro cess of choosing x i no des is equiv alent to obtaining a weigh ted random sample without 7 replacemen t from { 1 , 2 , . . . , i − 1 } , with the weigh t assigned to node σ j equal to w j , where w j = (1 − γ ) P i − 1 k =1 G σ k σ j P i − 1 l =1 P i − 1 k =1 G σ k σ l + γ j P i − 1 l =1 l , (2) where γ ∈ [0 , 1] and can b e seen as the parameter gov erning the degree of P A. When γ = 0, this reduces to the original P A rule. When γ increases, more w eights are given to latter no des, and the inequalit y in the degrees of the no des is reduced. Such inequality reduction is facilitated by assigning weigh ts according to how recen t the no des join the netw ork, rather than equal weigh ts, in the non-P A comp onen t. Note that, how ever, even in the extreme case where γ = 1, where the degree distribution is unimo dal and closer to symmetry , the mo del do es not reduce to a BR G, where the degree distribution is Binomial with parameters ( m − 1 , p ), where p is the edge inclusion probabilit y , but provides a crude appro ximation to it. 2.3 Constructing the epidemic The Marko vian epidemic pro cess is constructed as follows. A t time 0, the whole p opulation is susceptible except individual 1, who is infected. Once infected at time ˜ I i , individual i makes infectious contacts at points of a homogeneous Poisson pro cess with rate β P m j =1 G ij with its neigh b ours (according to G ), and sta ys infected until the end of the observ ation p erio d. The random transmission tree P , with the same no de as G and whose ro ot is the no de lab elled 1, can b e constructed sim ultaneously . If individual i mak es infectious contact at arbitrary t 0 (go v erned b y the aforemen tioned Poisson process) with susceptible neighbour j , w e write P ij = 1, again using the adjacency matrix represen tation. This implies ˜ I j = t 0 , and P j i = 0 as individual i cannot b e re-infected. Also, P ij = 1 implicit y implies that G ij = ( G j i =)1, as the epidemic can only spread through so cial connections i.e. the edges in G . Also, we assume P ii = 0 as any individual cannot b e infected by themselves. 3 Lik eliho o d and inference W e proceed to compute the likelihoo d, denoted by L , as a function of β , µ , γ and σ . W e assume b oth G and P are given b ecause, as argued b y Britton and O’Neill (2002) and Gro endyke et al. (2011), it is easier to condition on G and P in order to calculate L , and, if they are unobserv ed, 8 include them as latent v ariables in the inference procedure. Two conditional indep endence as- sumptions need to b e noted. Because of the Marko vian nature of the epidemic, P and ˜ I are indep enden t given G . It is also common that the data ( { ˜ I , P } ) and (a subset of ) the parameters ( µ, γ , σ ) are indep endent apriori , giv en G , when mo dels are formulated b y cen tred parameter- isations (Papaspiliopoulos, Rob erts and Sk¨ old, 2003). Therefore, the lik eliho o d can b e brok en do wn into the following comp onents: L := L ( β , µ, γ , σ ) = π ( P , ˜ I , G | β , µ, γ , σ ) = π ( P , ˜ I |G , β , µ, γ , σ ) × π ( G | β , µ, γ , σ ) = π ( P |G ) × π ( ˜ I |G , β ) × π ( G | µ, γ , σ ) . (3) The dropping of any unrelated quantities can b e explained by how the net work and the epi- demic are constructed in Section 2, and is demonstrated in the deriv ations of each comp onent in App endix A, the results of whic h are given b elow: π ( P |G ) = m Y j =2 1 ( j − 1 X i =1 P ij = 1 ) j − 1 X i =1 G ij ! − 1 Y 1 ≤ i 0 } L 1 ( G ; σ , µ ) × ( µ ) a µ − 1 exp( − b µ µ ) 1 { µ > 0 } . Sampling γ : The Metrop olis step for γ is similar to that for µ , as the former is inv olved in 25 L 2 ( G ; σ , γ ) in the likelihoo d. W e propose γ ∗ from a symmetrical prop osal q ( ·| γ ) and accept γ ∗ with probabilit y α γ = 1 ∧ L 2 ( G ; σ , γ ∗ ) × 1 { 0 ≤ γ ∗ ≤ 1 } L 2 ( G ; σ , γ ) × 1 { 0 ≤ γ ≤ 1 } . Sampling σ : T o up date the ordering as a whole, we prop ose σ ∗ , whic h is accepted with probabilit y α σ = 1 ∧ π ( G | µ, γ , σ ∗ ) π ( G | µ, γ , σ ) . This requires a symmetrical proposal on the p ermutation space. Sp ecifically , we use a “random w alk by insertion” metho d used b y Bez´ ak o v´ a, Kalai and San thanam (2006). Two indices i and j are first sampled with replacemen t from { 1 , 2 , . . . , m } uniformly . Without loss of generality , assume that i < j . While the current ordering is σ = ( σ 1 , . . . , σ i − 1 , σ i , σ i +1 , . . . , σ j − 1 , σ j , σ j +1 , . . . , σ m ) , the proposed ordering is σ ∗ = ( σ 1 , . . . , σ i − 1 , σ i +1 , . . . , σ j − 1 , σ j , σ i , σ j +1 , . . . , σ m ) . The intuition is that the i th card of a deck of cards is taken out and inserted in the j th p osition. As ( i, j ) and ( j, i ) hav e the same probability of b eing sampled in their particular orders, the prop osal is symmetrical. This metho d is, according to Bez´ ako v´ a et al. (2006), more efficient than the random sw ap metho d, in which an arbitrary pair of adjacenct indices ( σ i , σ i +1 ) (1 ≤ i < m ) is pic ked, and a swap b et w een them pro duces the prop osed ordering. Theoretical properties are not clear yet to provide guildlines on optimising the n umber of random insertions in eac h MCMC iteration. As it is found out that the ma jority of the computation time p er iteration is taken by updating all p otential edges of G individually , which will b e described b elo w, w e simply prop ose to up date the ordering m times in each iteration, so that each index will on av erage be pick ed and inserted once. It should ho w ev er be noted that an index potentially c hanges its p osition even if it is not selected, as long as its p osition lies b et w een i and j inclusive. Sampling G : W e will use a Gibbs step to up date eac h of the m 2 p oten tial edges in G sequentially , 26 and this requires defining the quantities required first. Unlike a BRG in O’Neill (2002), Neal and Rob erts (2005), Ray and Marzouk (2008) and Groendyke et al. (2011), the p oten tial edges of G are not indep endent anymore, both apriori and ap osteriori . Still, we can update each p oten tial edge G ij (1 ≤ i < j ≤ m ), conditional on all of G except G ij (and G j i b ecause of symmetry), denoted by G − ij . While G − ij is not a prop er adjacency matrix, we also define matrices G 0 − ij := {G − ij , G ij = 0 } and G 1 − ij := {G − ij , G ij = 1 } , so that exactly one of G 0 − ij and G 1 − ij is iden tical to G . Because of the difference in the netw ork ordering and epidemic ordering, for each pair ( i, j ), we pro ceed to sample G σ i σ j instead of G ij . This will not p ose a problem in practice as w e will go through all combinations of ( i, j ) satisfying 1 ≤ i < j ≤ m . Because of the 1-1 relationship b et w een ( i, j ) and ( σ i , σ j ) giv en σ , even tually all the p otential edges will be up dated. F or notational con venience, w e also define s = min( σ i , σ j ) and t = max( σ i , σ j ), whic h implies ˜ I t > ˜ I s . If P st = 1, as mentioned in Section 2.3 and implied b y (4), the four equiv alen t quan tities, namely G σ i σ j , G σ j σ i , G st and G ts , are equal to 1 with p osterior probability 1, regardless of all other parameters and G − st . Therefore, we shall only consider π ( G st |P st = 0 , G − st , · · · ) in detail. Before doing so, w e observe that β can b e in tegrated out in the joint p osterior in (8), whic h is achiev ed b y substituting (5) and the prior of β in (7) into (9), follo wed by integration with resp ect to β : π ( G , β , µ, γ , σ |P , ˜ I ) ∝ π ( P |G ) × β m − 1 exp − β X 1 ≤ i
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