3731:
3284:
3726:{\displaystyle {\begin{aligned}{\widetilde {\pi }}({\boldsymbol {\theta }}_{k}|{\boldsymbol {y}})&\propto \left.{\frac {\pi \left({\boldsymbol {x}},{\boldsymbol {\theta }}_{k},{\boldsymbol {y}}\right)}{{\widetilde {\pi }}_{G}\left({\boldsymbol {x}}|{\boldsymbol {\theta }}_{k},{\boldsymbol {y}}\right)}}\right\vert _{{\boldsymbol {x}}={\boldsymbol {x}}^{*}({\boldsymbol {\theta }}_{k})},\\&\propto \left.{\frac {\pi ({\boldsymbol {y}}|{\boldsymbol {x}},{\boldsymbol {\theta }}_{k})\pi ({\boldsymbol {x}}|{\boldsymbol {\theta }}_{k})\pi ({\boldsymbol {\theta }}_{k})}{{\widetilde {\pi }}_{G}\left({\boldsymbol {x}}|{\boldsymbol {\theta }}_{k},{\boldsymbol {y}}\right)}}\right\vert _{{\boldsymbol {x}}={\boldsymbol {x}}^{*}({\boldsymbol {\theta }}_{k})},\end{aligned}}}
1801:
2409:
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268:
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2022:
1796:{\displaystyle {\begin{aligned}\pi ({\boldsymbol {x}},{\boldsymbol {\theta }}|{\boldsymbol {y}})&\propto \pi ({\boldsymbol {\theta }})\pi ({\boldsymbol {x}}|{\boldsymbol {\theta }})\prod _{i}\pi (y_{i}|\eta _{i},{\boldsymbol {\theta }})\\&\propto \pi ({\boldsymbol {\theta }})\left|{\boldsymbol {Q_{\theta }}}\right|^{1/2}\exp \left(-{\frac {1}{2}}{\boldsymbol {x}}^{T}{\boldsymbol {Q_{\theta }}}{\boldsymbol {x}}+\sum _{i}\log \left\right).\end{aligned}}}
1806:
3189:
2404:{\displaystyle {\begin{array}{rcl}{\widetilde {\pi }}(x_{i}|{\boldsymbol {y}})&=&\int {\widetilde {\pi }}(x_{i}|{\boldsymbol {\theta }},{\boldsymbol {y}}){\widetilde {\pi }}({\boldsymbol {\theta }}|{\boldsymbol {y}})d{\boldsymbol {\theta }}\\{\widetilde {\pi }}(\theta _{j}|{\boldsymbol {y}})&=&\int {\widetilde {\pi }}({\boldsymbol {\theta }}|{\boldsymbol {y}})d{\boldsymbol {\theta }}_{-j},\end{array}}}
2810:
1412:
3045:
2623:
36:
1038:
2017:{\displaystyle {\begin{array}{rcl}\pi (x_{i}|{\boldsymbol {y}})&=&\int \pi (x_{i}|{\boldsymbol {\theta }},{\boldsymbol {y}})\pi ({\boldsymbol {\theta }}|{\boldsymbol {y}})d{\boldsymbol {\theta }}\\\pi (\theta _{j}|{\boldsymbol {y}})&=&\int \pi ({\boldsymbol {\theta }}|{\boldsymbol {y}})d{\boldsymbol {\theta }}_{-j},\end{array}}}
1270:
903:
773:
3184:{\displaystyle {\begin{aligned}{\pi }({\boldsymbol {\theta }}|{\boldsymbol {y}})={\frac {\pi \left({\boldsymbol {x}},{\boldsymbol {\theta }},{\boldsymbol {y}}\right)}{\pi \left({\boldsymbol {x}}|{\boldsymbol {\theta }},{\boldsymbol {y}}\right)\pi ({\boldsymbol {y}})}},\end{aligned}}}
2805:{\displaystyle {\begin{aligned}{\widetilde {\pi }}(x_{i}|{\boldsymbol {y}})=\sum _{k}{\widetilde {\pi }}\left(x_{i}|{\boldsymbol {\theta }}_{k},{\boldsymbol {y}}\right)\times {\widetilde {\pi }}({\boldsymbol {\theta }}_{k}|{\boldsymbol {y}})\times \Delta _{k},\end{aligned}}}
2140:
1407:{\displaystyle \pi ({\boldsymbol {x}},{\boldsymbol {\theta }}|{\boldsymbol {y}})={\frac {\pi ({\boldsymbol {y}}|{\boldsymbol {x}},{\boldsymbol {\theta }})\pi ({\boldsymbol {x}}|{\boldsymbol {\theta }})\pi ({\boldsymbol {\theta }})}{\pi ({\boldsymbol {y}})}},}
3807:
4293:
661:
3873:
4185:
3239:
3040:
2987:
519:. The linear predictor can take the form of a (Bayesian) additive model. All latent effects (the linear predictor, the intercept, coefficients of possible covariates, and so on) are collectively denoted by the vector
1033:{\displaystyle \pi ({\boldsymbol {x}}|{\boldsymbol {\theta }})\propto \left|{\boldsymbol {Q_{\theta }}}\right|^{1/2}\exp \left(-{\frac {1}{2}}{\boldsymbol {x}}^{T}{\boldsymbol {Q_{\theta }}}{\boldsymbol {x}}\right),}
330:
methods to compute posterior marginal distributions. Due to its relative speed even with large data sets for certain problems and models, INLA has been a popular inference method in applied statistics, in particular
2596:
3276:
3953:
4090:
4047:
4128:
2542:
4348:
1206:
898:
4295:
is more involved, and the INLA method provides three options for this: Gaussian approximation, Laplace approximation, or the simplified
Laplace approximation. For the numerical integration to obtain
1130:
425:
3289:
3050:
2905:
2628:
1466:
2455:
2027:
4216:
2501:
1161:
4611:
Lindgren, Finn; Rue, Håvard; Lindström, Johan (2011). "An explicit link between
Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach".
2937:
3906:
1067:
3736:
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2618:
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1262:
1089:
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513:
456:
483:
4378:
Rue, HĂĄvard; Martino, Sara; Chopin, Nicolas (2009). "Approximate
Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations".
298:
768:{\displaystyle \pi ({\boldsymbol {y}}|{\boldsymbol {x}},{\boldsymbol {\theta }})=\prod _{i\in {\mathcal {I}}}\pi (y_{i}|\eta _{i},{\boldsymbol {\theta }}),}
89:
4413:
Taylor, Benjamin M.; Diggle, Peter J. (2014). "INLA or MCMC? A tutorial and comparative evaluation for spatial prediction in log-Gaussian Cox processes".
900:
is a
Gaussian Markov Random Field (GMRF) (that is, a multivariate Gaussian with additional conditional independence properties) with probability density
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348:
4747:
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3911:
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2506:
1803:
Obtaining the exact posterior is generally a very difficult problem. In INLA, the main aim is to approximate the posterior marginals
326:. It is designed for a class of models called latent Gaussian models (LGMs), for which it can be a fast and accurate alternative for
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4541:
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291:
254:
821:(some elements may be unobserved, and for these INLA computes a posterior predictive distribution). Note that the linear predictor
4298:
181:
84:
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The trick in the
Laplace approximation above is the fact that the Gaussian approximation is applied on the full conditional of
1166:
871:
207:
1098:
352:
145:
370:
2135:{\displaystyle {\boldsymbol {\theta }}_{-j}=\left(\theta _{1},\dots ,\theta _{j-1},\theta _{j+1},\dots ,\theta _{m}\right)}
2864:
4658:"Using a Bayesian modelling approach (INLA-SPDE) to predict the occurrence of the Spinetail Devil Ray (Mobular mobular)"
2414:
284:
176:
114:
4771:
4190:
3802:{\displaystyle {\widetilde {\pi }}_{G}\left({\boldsymbol {x}}|{\boldsymbol {\theta }}_{k},{\boldsymbol {y}}\right)}
3279:
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1135:
166:
135:
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1043:
228:
109:
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249:
161:
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1072:
639:
592:
548:
824:
140:
1217:
344:
43:
4557:
Opitz, T. (2017). "Latent
Gaussian modeling and INLA: A review with focus on space-time applications".
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1417:
1223:
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802:
617:
570:
522:
267:
1132:
is its determinant. The precision matrix is sparse due to the GMRF assumption. The prior distribution
4669:
4049:
itself need not be close to a
Gaussian, and so the Gaussian approximation is not directly applied on
223:
104:
74:
778:
4288:{\displaystyle {\widetilde {\pi }}\left(x_{i}|{\boldsymbol {\theta }}_{k},{\boldsymbol {y}}\right)}
3810:
323:
55:
47:
27:
3868:{\displaystyle {\pi }\left({\boldsymbol {x}}|{\boldsymbol {\theta }}_{k},{\boldsymbol {y}}\right)}
4638:
4566:
4440:
4422:
4395:
2837:
428:
319:
272:
197:
69:
3956:
99:
4350:, also three options are available: grid search, central composite design, or empirical Bayes.
491:
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79:
51:
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1092:
332:
94:
4006:. Applying the approximation here improves the accuracy of the method, since the posterior
461:
130:
4673:
4460:"Bayesian computation for Log-Gaussian Cox processes: a comparative analysis of methods"
4690:
4657:
4484:
4459:
3984:
in the denominator since it is usually close to a
Gaussian due to the GMRF property of
542:
2153:
1811:
1163:
for the hyperparameters need not be
Gaussian. However, the number of hyperparameters,
4760:
4642:
4624:
4391:
4444:
4399:
340:
244:
4475:
4180:{\displaystyle {\widetilde {\pi }}({\boldsymbol {\theta }}_{k}|{\boldsymbol {y}})}
4436:
4656:
Lezama-Ochoa, N.; Grazia
Pennino, M.; Hall, M. A.; Lopez, J.; Murua, H. (2020).
4092:. The second important property of a GMRF, the sparsity of the precision matrix
2457:
is an approximated posterior density. The approximation to the marginal density
4681:
3234:{\displaystyle {\widetilde {\pi }}({\boldsymbol {\theta }}|{\boldsymbol {y}})}
3035:{\displaystyle {\widetilde {\pi }}({\boldsymbol {\theta }}|{\boldsymbol {y}})}
2982:{\displaystyle {\widetilde {\pi }}({\boldsymbol {\theta }}|{\boldsymbol {y}})}
427:
denote the response variable (that is, the observations) which belongs to an
4699:
4493:
4588:
Geospatial Health Data: Modeling and Visualization with R-INLA and Shiny
336:
35:
2591:{\displaystyle \pi (x_{i}|{\boldsymbol {\theta }},{\boldsymbol {y}})}
4571:
3271:{\displaystyle {\boldsymbol {\theta }}={\boldsymbol {\theta }}_{k}}
614:
The observations are assumed to be conditionally independent given
4427:
3948:{\displaystyle {\boldsymbol {x}}^{*}({\boldsymbol {\theta }}_{k})}
2145:
A key idea of INLA is to construct nested approximations given by
4085:{\displaystyle {\pi }({\boldsymbol {\theta }}|{\boldsymbol {y}})}
4042:{\displaystyle {\pi }({\boldsymbol {\theta }}|{\boldsymbol {y}})}
868:
For the model to be a latent Gaussian model, it is assumed that
4123:{\displaystyle {\boldsymbol {Q}}_{{\boldsymbol {\theta }}_{k}}}
2537:{\displaystyle \pi ({\boldsymbol {\theta }}|{\boldsymbol {y}})}
4343:{\displaystyle {\widetilde {\pi }}(x_{i}|{\boldsymbol {y}})}
784:
713:
3511:
3341:
4713:
4507:
Wang, Xiaofeng; Yue, Yu Ryan; Faraway, Julian J. (2018).
3955:. The mode can be found numerically for example with the
1201:{\displaystyle m=\mathrm {dim} ({\boldsymbol {\theta }})}
893:{\displaystyle {\boldsymbol {x}}|{\boldsymbol {\theta }}}
1125:{\displaystyle \left|{\boldsymbol {Q_{\theta }}}\right|}
4534:
Spatial and Spatio-temporal Bayesian Models with R-INLA
3241:
is obtained at a specific value of the hyperparameters
2503:
is obtained in a nested fashion by first approximating
420:{\displaystyle {\boldsymbol {y}}=(y_{1},\dots ,y_{n})}
4301:
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2900:{\displaystyle \pi (\theta _{j}|{\boldsymbol {y}})}
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2450:{\displaystyle {\widetilde {\pi }}(\cdot |\cdot )}
2449:
2403:
2134:
2016:
1795:
1450:
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1406:
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1234:
1216:In Bayesian inference, one wants to solve for the
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4634:20.500.11820/1084d335-e5b4-4867-9245-ec9c4f6f4645
4464:Journal of Statistical Computation and Simulation
4415:Journal of Statistical Computation and Simulation
799:is the set of indices for observed elements of
611:are random variables with prior distributions.
355:. The INLA method is implemented in the R-INLA
4559:Journal de la Société Française de Statistique
4532:Blangiardo, Marta; Cameletti, Michela (2015).
1208:, is assumed to be small (say, less than 15).
4458:Teng, M.; Nathoo, F.; Johnson, T. D. (2017).
4211:{\displaystyle {{\boldsymbol {\theta }}_{k}}}
2496:{\displaystyle \pi (x_{i}|{\boldsymbol {y}})}
1156:{\displaystyle \pi ({\boldsymbol {\theta }})}
343:. It is also possible to combine INLA with a
292:
8:
2932:{\displaystyle {\boldsymbol {\theta }}_{-j}}
4130:, is required for efficient computation of
3901:{\displaystyle {\boldsymbol {\theta }}_{k}}
1062:{\displaystyle {\boldsymbol {Q_{\theta }}}}
2812:where the summation is over the values of
351:to study e.g. spatial point processes and
299:
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18:
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1212:Approximate Bayesian inference with INLA
349:stochastic partial differential equation
312:Integrated nested Laplace approximations
172:Integrated nested Laplace approximations
4373:
4371:
4369:
4367:
4365:
4363:
4359:
4333:
4276:
4262:
4221:Obtaining the approximate distribution
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3130:
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3025:
3015:
2972:
2962:
2916:
2907:is computed by numerically integrating
2890:
2827:{\displaystyle {\boldsymbol {\theta }}}
2820:
2775:
2759:
2730:
2716:
2662:
2613:{\displaystyle {\boldsymbol {\theta }}}
2606:
2598:, and then numerically integrating out
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1451:{\displaystyle {\boldsymbol {\theta }}}
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1422:
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1257:{\displaystyle {\boldsymbol {\theta }}}
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1112:
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1084:{\displaystyle {\boldsymbol {\theta }}}
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651:{\displaystyle {\boldsymbol {\theta }}}
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604:{\displaystyle {\boldsymbol {\theta }}}
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560:{\displaystyle {\boldsymbol {\theta }}}
553:
527:
375:
236:
215:
189:
153:
122:
61:
26:
4509:Bayesian Regression Modeling with INLA
836:{\displaystyle {\boldsymbol {\eta }}}
7:
2992:To get the approximate distribution
2834:, with integration weights given by
1414:the joint posterior distribution of
2842:
2786:
1183:
1180:
1177:
14:
3999:{\displaystyle {\boldsymbol {x}}}
3977:{\displaystyle {\boldsymbol {x}}}
1429:{\displaystyle {\boldsymbol {x}}}
1235:{\displaystyle {\boldsymbol {x}}}
858:{\displaystyle {\boldsymbol {x}}}
814:{\displaystyle {\boldsymbol {y}}}
629:{\displaystyle {\boldsymbol {x}}}
582:{\displaystyle {\boldsymbol {x}}}
534:{\displaystyle {\boldsymbol {x}}}
4625:10.1111/j.1467-9868.2011.00777.x
4392:10.1111/j.1467-9868.2008.00700.x
266:
182:Approximate Bayesian computation
34:
208:Maximum a posteriori estimation
4738:Gomez-Rubio, Virgilio (2021).
4536:. John Wiley & Sons, Ltd.
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1543:
1534:
1525:
1519:
1511:
1498:
1489:
1472:
1395:
1387:
1379:
1371:
1365:
1356:
1347:
1341:
1324:
1315:
1303:
1294:
1277:
1195:
1187:
1150:
1142:
928:
919:
910:
881:
792:{\displaystyle {\mathcal {I}}}
759:
737:
723:
694:
677:
668:
567:. As per Bayesian statistics,
414:
382:
318:) is a method for approximate
1:
4476:10.1080/00949655.2017.1326117
4740:Bayesian inference with INLA
4437:10.1080/00949655.2013.788653
3191:as the starting point. Then
545:of the model are denoted by
115:Principle of maximum entropy
3042:, one can use the relation
2854:{\displaystyle \Delta _{k}}
353:species distribution models
85:Bernstein–von Mises theorem
4788:
4682:10.1038/s41598-020-73879-3
508:{\displaystyle \eta _{i}}
110:Principle of indifference
16:Bayesian inference method
4767:Computational statistics
4742:. Chapman and Hall/CRC.
4590:. Chapman and Hall/CRC.
4511:. Chapman and Hall/CRC.
1220:of the latent variables
451:{\displaystyle \mu _{i}}
328:Markov chain Monte Carlo
162:Markov chain Monte Carlo
3280:Laplace's approximation
2861:. The approximation of
167:Laplace's approximation
154:Posterior approximation
4586:Moraga, Paula (2019).
4344:
4289:
4212:
4181:
4124:
4086:
4043:
4000:
3978:
3949:
3902:
3869:
3811:Gaussian approximation
3803:
3727:
3272:
3235:
3185:
3036:
2983:
2933:
2901:
2855:
2828:
2806:
2614:
2592:
2538:
2497:
2451:
2405:
2136:
2018:
1797:
1452:
1430:
1408:
1258:
1236:
1218:posterior distribution
1202:
1157:
1126:
1085:
1063:
1034:
894:
859:
837:
815:
793:
769:
652:
630:
605:
583:
561:
535:
509:
479:
452:
421:
363:Latent Gaussian models
273:Mathematics portal
216:Evidence approximation
4613:J. R. Statist. Soc. B
4380:J. R. Statist. Soc. B
4345:
4290:
4213:
4182:
4125:
4087:
4044:
4001:
3979:
3957:Newton-Raphson method
3950:
3903:
3870:
3804:
3728:
3273:
3236:
3186:
3037:
2984:
2934:
2902:
2856:
2829:
2807:
2615:
2593:
2539:
2498:
2452:
2406:
2137:
2019:
1798:
1453:
1431:
1409:
1259:
1237:
1203:
1158:
1127:
1086:
1064:
1035:
895:
860:
838:
816:
794:
770:
653:
631:
606:
584:
562:
536:
510:
480:
478:{\displaystyle y_{i}}
453:
422:
345:finite element method
177:Variational inference
4299:
4225:
4191:
4134:
4096:
4053:
4010:
3988:
3966:
3912:
3883:
3817:
3737:
3285:
3245:
3195:
3046:
2996:
2943:
2911:
2865:
2838:
2816:
2624:
2602:
2548:
2507:
2461:
2415:
2149:
2028:
1807:
1462:
1440:
1418:
1271:
1246:
1224:
1167:
1136:
1099:
1073:
1044:
904:
872:
847:
825:
803:
779:
662:
640:
618:
593:
571:
549:
523:
492:
485:) being linked to a
462:
435:
371:
255:Posterior predictive
224:Evidence lower bound
105:Likelihood principle
75:Bayesian probability
4674:2020NatSR..1018822L
515:via an appropriate
28:Bayesian statistics
22:Part of a series on
4772:Bayesian inference
4662:Scientific Reports
4340:
4285:
4208:
4177:
4120:
4082:
4039:
3996:
3974:
3945:
3898:
3865:
3799:
3723:
3721:
3268:
3231:
3181:
3179:
3032:
2979:
2929:
2897:
2851:
2824:
2802:
2800:
2681:
2610:
2588:
2534:
2493:
2447:
2401:
2399:
2132:
2014:
2012:
1793:
1791:
1722:
1555:
1448:
1426:
1404:
1254:
1232:
1198:
1153:
1122:
1091:-dependent sparse
1081:
1059:
1030:
890:
855:
833:
811:
789:
765:
719:
648:
626:
601:
579:
557:
531:
505:
475:
448:
429:exponential family
417:
333:spatial statistics
320:Bayesian inference
198:Bayesian estimator
146:Hierarchical model
70:Bayesian inference
4749:978-1-03-217453-2
4470:(11): 2227–2252.
4421:(10): 2266–2284.
4311:
4237:
4146:
3750:
3669:
3619:
3449:
3399:
3301:
3207:
3172:
3008:
2955:
2751:
2691:
2672:
2640:
2427:
2352:
2304:
2259:
2213:
2165:
1713:
1679:
1546:
1399:
991:
700:
309:
308:
203:Credible interval
136:Linear regression
4779:
4753:
4725:
4724:
4722:
4720:
4714:"R-INLA Project"
4710:
4704:
4703:
4693:
4653:
4647:
4646:
4636:
4608:
4602:
4601:
4583:
4577:
4576:
4574:
4554:
4548:
4547:
4529:
4523:
4522:
4504:
4498:
4497:
4487:
4455:
4449:
4448:
4430:
4410:
4404:
4403:
4375:
4349:
4347:
4346:
4341:
4336:
4331:
4326:
4325:
4313:
4312:
4304:
4294:
4292:
4291:
4286:
4284:
4280:
4279:
4271:
4270:
4265:
4259:
4254:
4253:
4239:
4238:
4230:
4217:
4215:
4214:
4209:
4207:
4206:
4205:
4200:
4186:
4184:
4183:
4178:
4173:
4168:
4163:
4162:
4157:
4148:
4147:
4139:
4129:
4127:
4126:
4121:
4119:
4118:
4117:
4116:
4111:
4104:
4091:
4089:
4088:
4083:
4078:
4073:
4068:
4060:
4048:
4046:
4045:
4040:
4035:
4030:
4025:
4017:
4005:
4003:
4002:
3997:
3995:
3983:
3981:
3980:
3975:
3973:
3954:
3952:
3951:
3946:
3941:
3940:
3935:
3926:
3925:
3920:
3907:
3905:
3904:
3899:
3897:
3896:
3891:
3874:
3872:
3871:
3866:
3864:
3860:
3859:
3851:
3850:
3845:
3839:
3834:
3824:
3808:
3806:
3805:
3800:
3798:
3794:
3793:
3785:
3784:
3779:
3773:
3768:
3758:
3757:
3752:
3751:
3743:
3732:
3730:
3729:
3724:
3722:
3715:
3714:
3710:
3709:
3704:
3695:
3694:
3689:
3680:
3674:
3670:
3668:
3667:
3663:
3662:
3654:
3653:
3648:
3642:
3637:
3627:
3626:
3621:
3620:
3612:
3607:
3603:
3602:
3597:
3582:
3581:
3576:
3570:
3565:
3551:
3550:
3545:
3536:
3531:
3526:
3514:
3502:
3495:
3494:
3490:
3489:
3484:
3475:
3474:
3469:
3460:
3454:
3450:
3448:
3447:
3443:
3442:
3434:
3433:
3428:
3422:
3417:
3407:
3406:
3401:
3400:
3392:
3387:
3386:
3382:
3381:
3373:
3372:
3367:
3358:
3344:
3328:
3323:
3318:
3317:
3312:
3303:
3302:
3294:
3277:
3275:
3274:
3269:
3267:
3266:
3261:
3252:
3240:
3238:
3237:
3232:
3227:
3222:
3217:
3209:
3208:
3200:
3190:
3188:
3187:
3182:
3180:
3173:
3171:
3167:
3156:
3152:
3151:
3143:
3138:
3133:
3119:
3118:
3114:
3113:
3105:
3097:
3083:
3075:
3070:
3065:
3057:
3041:
3039:
3038:
3033:
3028:
3023:
3018:
3010:
3009:
3001:
2988:
2986:
2985:
2980:
2975:
2970:
2965:
2957:
2956:
2948:
2938:
2936:
2935:
2930:
2928:
2927:
2919:
2906:
2904:
2903:
2898:
2893:
2888:
2883:
2882:
2860:
2858:
2857:
2852:
2850:
2849:
2833:
2831:
2830:
2825:
2823:
2811:
2809:
2808:
2803:
2801:
2794:
2793:
2778:
2773:
2768:
2767:
2762:
2753:
2752:
2744:
2738:
2734:
2733:
2725:
2724:
2719:
2713:
2708:
2707:
2693:
2692:
2684:
2680:
2665:
2660:
2655:
2654:
2642:
2641:
2633:
2619:
2617:
2616:
2611:
2609:
2597:
2595:
2594:
2589:
2584:
2576:
2571:
2566:
2565:
2543:
2541:
2540:
2535:
2530:
2525:
2520:
2502:
2500:
2499:
2494:
2489:
2484:
2479:
2478:
2456:
2454:
2453:
2448:
2440:
2429:
2428:
2420:
2410:
2408:
2407:
2402:
2400:
2393:
2392:
2384:
2372:
2367:
2362:
2354:
2353:
2345:
2329:
2324:
2319:
2318:
2306:
2305:
2297:
2290:
2279:
2274:
2269:
2261:
2260:
2252:
2246:
2238:
2233:
2228:
2227:
2215:
2214:
2206:
2190:
2185:
2180:
2179:
2167:
2166:
2158:
2141:
2139:
2138:
2133:
2131:
2127:
2126:
2125:
2107:
2106:
2088:
2087:
2063:
2062:
2045:
2044:
2036:
2023:
2021:
2020:
2015:
2013:
2006:
2005:
1997:
1985:
1980:
1975:
1951:
1946:
1941:
1940:
1921:
1910:
1905:
1900:
1886:
1878:
1873:
1868:
1867:
1839:
1834:
1829:
1828:
1802:
1800:
1799:
1794:
1792:
1785:
1781:
1780:
1776:
1772:
1764:
1763:
1754:
1749:
1748:
1721:
1709:
1704:
1703:
1702:
1692:
1691:
1686:
1680:
1672:
1656:
1655:
1651:
1642:
1638:
1637:
1636:
1618:
1601:
1594:
1586:
1585:
1576:
1571:
1570:
1554:
1542:
1537:
1532:
1518:
1497:
1492:
1487:
1479:
1457:
1455:
1454:
1449:
1447:
1435:
1433:
1432:
1427:
1425:
1413:
1411:
1410:
1405:
1400:
1398:
1394:
1382:
1378:
1364:
1359:
1354:
1340:
1332:
1327:
1322:
1310:
1302:
1297:
1292:
1284:
1263:
1261:
1260:
1255:
1253:
1241:
1239:
1238:
1233:
1231:
1207:
1205:
1204:
1199:
1194:
1186:
1162:
1160:
1159:
1154:
1149:
1131:
1129:
1128:
1123:
1121:
1117:
1116:
1115:
1093:precision matrix
1090:
1088:
1087:
1082:
1080:
1068:
1066:
1065:
1060:
1058:
1057:
1056:
1039:
1037:
1036:
1031:
1026:
1022:
1021:
1016:
1015:
1014:
1004:
1003:
998:
992:
984:
968:
967:
963:
954:
950:
949:
948:
927:
922:
917:
899:
897:
896:
891:
889:
884:
879:
864:
862:
861:
856:
854:
842:
840:
839:
834:
832:
820:
818:
817:
812:
810:
798:
796:
795:
790:
788:
787:
774:
772:
771:
766:
758:
750:
749:
740:
735:
734:
718:
717:
716:
693:
685:
680:
675:
657:
655:
654:
649:
647:
635:
633:
632:
627:
625:
610:
608:
607:
602:
600:
588:
586:
585:
580:
578:
566:
564:
563:
558:
556:
540:
538:
537:
532:
530:
514:
512:
511:
506:
504:
503:
487:linear predictor
484:
482:
481:
476:
474:
473:
457:
455:
454:
449:
447:
446:
431:, with the mean
426:
424:
423:
418:
413:
412:
394:
393:
378:
324:Laplace's method
301:
294:
287:
271:
270:
237:Model evaluation
38:
19:
4787:
4786:
4782:
4781:
4780:
4778:
4777:
4776:
4757:
4756:
4750:
4737:
4734:
4732:Further reading
4729:
4728:
4718:
4716:
4712:
4711:
4707:
4655:
4654:
4650:
4610:
4609:
4605:
4598:
4585:
4584:
4580:
4556:
4555:
4551:
4544:
4531:
4530:
4526:
4519:
4506:
4505:
4501:
4457:
4456:
4452:
4412:
4411:
4407:
4377:
4376:
4361:
4356:
4317:
4297:
4296:
4260:
4245:
4244:
4240:
4223:
4222:
4195:
4189:
4188:
4187:for each value
4152:
4132:
4131:
4106:
4099:
4094:
4093:
4051:
4050:
4008:
4007:
3986:
3985:
3964:
3963:
3930:
3915:
3910:
3909:
3886:
3881:
3880:
3840:
3829:
3825:
3815:
3814:
3774:
3763:
3759:
3740:
3735:
3734:
3720:
3719:
3699:
3684:
3643:
3632:
3628:
3609:
3608:
3592:
3571:
3540:
3515:
3510:
3509:
3500:
3499:
3479:
3464:
3423:
3412:
3408:
3389:
3388:
3362:
3353:
3349:
3345:
3340:
3339:
3332:
3307:
3283:
3282:
3256:
3243:
3242:
3193:
3192:
3178:
3177:
3128:
3124:
3120:
3092:
3088:
3084:
3044:
3043:
2994:
2993:
2941:
2940:
2914:
2909:
2908:
2874:
2863:
2862:
2841:
2836:
2835:
2814:
2813:
2799:
2798:
2785:
2757:
2714:
2699:
2698:
2694:
2646:
2622:
2621:
2600:
2599:
2557:
2546:
2545:
2505:
2504:
2470:
2459:
2458:
2413:
2412:
2398:
2397:
2379:
2338:
2333:
2310:
2292:
2291:
2219:
2199:
2194:
2171:
2147:
2146:
2117:
2092:
2073:
2054:
2053:
2049:
2031:
2026:
2025:
2011:
2010:
1992:
1960:
1955:
1932:
1923:
1922:
1859:
1848:
1843:
1820:
1805:
1804:
1790:
1789:
1755:
1740:
1733:
1729:
1694:
1681:
1667:
1663:
1628:
1623:
1622:
1599:
1598:
1577:
1562:
1501:
1460:
1459:
1438:
1437:
1416:
1415:
1383:
1311:
1269:
1268:
1244:
1243:
1222:
1221:
1214:
1165:
1164:
1134:
1133:
1107:
1102:
1097:
1096:
1071:
1070:
1048:
1042:
1041:
1006:
993:
979:
975:
940:
935:
934:
902:
901:
870:
869:
845:
844:
823:
822:
801:
800:
777:
776:
741:
726:
660:
659:
638:
637:
616:
615:
591:
590:
569:
568:
547:
546:
543:hyperparameters
521:
520:
495:
490:
489:
465:
460:
459:
438:
433:
432:
404:
385:
369:
368:
365:
305:
265:
250:Model averaging
229:Nested sampling
141:Empirical Bayes
131:Conjugate prior
100:Cromwell's rule
17:
12:
11:
5:
4785:
4783:
4775:
4774:
4769:
4759:
4758:
4755:
4754:
4748:
4733:
4730:
4727:
4726:
4705:
4648:
4619:(4): 423–498.
4603:
4596:
4578:
4549:
4542:
4524:
4517:
4499:
4450:
4405:
4386:(2): 319–392.
4358:
4357:
4355:
4352:
4339:
4335:
4330:
4324:
4320:
4316:
4310:
4307:
4283:
4278:
4274:
4269:
4264:
4258:
4252:
4248:
4243:
4236:
4233:
4204:
4199:
4176:
4172:
4167:
4161:
4156:
4151:
4145:
4142:
4115:
4110:
4103:
4081:
4077:
4072:
4067:
4063:
4059:
4038:
4034:
4029:
4024:
4020:
4016:
3994:
3972:
3944:
3939:
3934:
3929:
3924:
3919:
3895:
3890:
3863:
3858:
3854:
3849:
3844:
3838:
3833:
3828:
3823:
3797:
3792:
3788:
3783:
3778:
3772:
3767:
3762:
3756:
3749:
3746:
3718:
3713:
3708:
3703:
3698:
3693:
3688:
3683:
3679:
3673:
3666:
3661:
3657:
3652:
3647:
3641:
3636:
3631:
3625:
3618:
3615:
3606:
3601:
3596:
3591:
3588:
3585:
3580:
3575:
3569:
3564:
3560:
3557:
3554:
3549:
3544:
3539:
3535:
3530:
3525:
3521:
3518:
3512:
3508:
3505:
3503:
3501:
3498:
3493:
3488:
3483:
3478:
3473:
3468:
3463:
3459:
3453:
3446:
3441:
3437:
3432:
3427:
3421:
3416:
3411:
3405:
3398:
3395:
3385:
3380:
3376:
3371:
3366:
3361:
3357:
3352:
3348:
3342:
3338:
3335:
3333:
3331:
3327:
3322:
3316:
3311:
3306:
3300:
3297:
3291:
3290:
3265:
3260:
3255:
3251:
3230:
3226:
3221:
3216:
3212:
3206:
3203:
3176:
3170:
3166:
3162:
3159:
3155:
3150:
3146:
3142:
3137:
3132:
3127:
3123:
3117:
3112:
3108:
3104:
3100:
3096:
3091:
3087:
3081:
3078:
3074:
3069:
3064:
3060:
3056:
3052:
3051:
3031:
3027:
3022:
3017:
3013:
3007:
3004:
2978:
2974:
2969:
2964:
2960:
2954:
2951:
2926:
2923:
2918:
2896:
2892:
2887:
2881:
2877:
2873:
2870:
2848:
2844:
2822:
2797:
2792:
2788:
2784:
2781:
2777:
2772:
2766:
2761:
2756:
2750:
2747:
2741:
2737:
2732:
2728:
2723:
2718:
2712:
2706:
2702:
2697:
2690:
2687:
2679:
2675:
2671:
2668:
2664:
2659:
2653:
2649:
2645:
2639:
2636:
2630:
2629:
2608:
2587:
2583:
2579:
2575:
2570:
2564:
2560:
2556:
2553:
2533:
2529:
2524:
2519:
2515:
2512:
2492:
2488:
2483:
2477:
2473:
2469:
2466:
2446:
2443:
2439:
2435:
2432:
2426:
2423:
2396:
2391:
2388:
2383:
2378:
2375:
2371:
2366:
2361:
2357:
2351:
2348:
2342:
2339:
2337:
2334:
2332:
2328:
2323:
2317:
2313:
2309:
2303:
2300:
2294:
2293:
2289:
2285:
2282:
2278:
2273:
2268:
2264:
2258:
2255:
2249:
2245:
2241:
2237:
2232:
2226:
2222:
2218:
2212:
2209:
2203:
2200:
2198:
2195:
2193:
2189:
2184:
2178:
2174:
2170:
2164:
2161:
2155:
2154:
2130:
2124:
2120:
2116:
2113:
2110:
2105:
2102:
2099:
2095:
2091:
2086:
2083:
2080:
2076:
2072:
2069:
2066:
2061:
2057:
2052:
2048:
2043:
2040:
2035:
2009:
2004:
2001:
1996:
1991:
1988:
1984:
1979:
1974:
1970:
1967:
1964:
1961:
1959:
1956:
1954:
1950:
1945:
1939:
1935:
1931:
1928:
1925:
1924:
1920:
1916:
1913:
1909:
1904:
1899:
1895:
1892:
1889:
1885:
1881:
1877:
1872:
1866:
1862:
1858:
1855:
1852:
1849:
1847:
1844:
1842:
1838:
1833:
1827:
1823:
1819:
1816:
1813:
1812:
1788:
1784:
1779:
1775:
1771:
1767:
1762:
1758:
1753:
1747:
1743:
1739:
1736:
1732:
1728:
1725:
1720:
1716:
1712:
1708:
1701:
1697:
1690:
1685:
1678:
1675:
1670:
1666:
1662:
1659:
1654:
1650:
1646:
1641:
1635:
1631:
1626:
1621:
1617:
1613:
1610:
1607:
1604:
1602:
1600:
1597:
1593:
1589:
1584:
1580:
1575:
1569:
1565:
1561:
1558:
1553:
1549:
1545:
1541:
1536:
1531:
1527:
1524:
1521:
1517:
1513:
1510:
1507:
1504:
1502:
1500:
1496:
1491:
1486:
1482:
1478:
1474:
1471:
1468:
1467:
1446:
1424:
1403:
1397:
1393:
1389:
1386:
1381:
1377:
1373:
1370:
1367:
1363:
1358:
1353:
1349:
1346:
1343:
1339:
1335:
1331:
1326:
1321:
1317:
1314:
1308:
1305:
1301:
1296:
1291:
1287:
1283:
1279:
1276:
1266:Bayes' theorem
1252:
1230:
1213:
1210:
1197:
1193:
1189:
1185:
1182:
1179:
1175:
1172:
1152:
1148:
1144:
1141:
1120:
1114:
1110:
1105:
1079:
1055:
1051:
1029:
1025:
1020:
1013:
1009:
1002:
997:
990:
987:
982:
978:
974:
971:
966:
962:
958:
953:
947:
943:
938:
933:
930:
926:
921:
916:
912:
909:
888:
883:
878:
853:
831:
809:
786:
764:
761:
757:
753:
748:
744:
739:
733:
729:
725:
722:
715:
710:
707:
703:
699:
696:
692:
688:
684:
679:
674:
670:
667:
646:
624:
599:
577:
555:
529:
502:
498:
472:
468:
445:
441:
416:
411:
407:
403:
400:
397:
392:
388:
384:
381:
377:
364:
361:
347:solution of a
307:
306:
304:
303:
296:
289:
281:
278:
277:
276:
275:
260:
259:
258:
257:
252:
247:
239:
238:
234:
233:
232:
231:
226:
218:
217:
213:
212:
211:
210:
205:
200:
192:
191:
187:
186:
185:
184:
179:
174:
169:
164:
156:
155:
151:
150:
149:
148:
143:
138:
133:
125:
124:
123:Model building
120:
119:
118:
117:
112:
107:
102:
97:
92:
87:
82:
80:Bayes' theorem
77:
72:
64:
63:
59:
58:
40:
39:
31:
30:
24:
23:
15:
13:
10:
9:
6:
4:
3:
2:
4784:
4773:
4770:
4768:
4765:
4764:
4762:
4751:
4745:
4741:
4736:
4735:
4731:
4715:
4709:
4706:
4701:
4697:
4692:
4687:
4683:
4679:
4675:
4671:
4667:
4663:
4659:
4652:
4649:
4644:
4640:
4635:
4630:
4626:
4622:
4618:
4614:
4607:
4604:
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4597:9780367357955
4593:
4589:
4582:
4579:
4573:
4568:
4564:
4560:
4553:
4550:
4545:
4543:9781118326558
4539:
4535:
4528:
4525:
4520:
4518:9781498727259
4514:
4510:
4503:
4500:
4495:
4491:
4486:
4481:
4477:
4473:
4469:
4465:
4461:
4454:
4451:
4446:
4442:
4438:
4434:
4429:
4424:
4420:
4416:
4409:
4406:
4401:
4397:
4393:
4389:
4385:
4381:
4374:
4372:
4370:
4368:
4366:
4364:
4360:
4353:
4351:
4322:
4318:
4308:
4305:
4281:
4272:
4267:
4250:
4246:
4241:
4234:
4231:
4219:
4202:
4159:
4143:
4140:
4113:
4057:
4014:
3960:
3958:
3937:
3922:
3893:
3878:
3861:
3852:
3847:
3826:
3821:
3812:
3795:
3786:
3781:
3760:
3754:
3747:
3744:
3716:
3706:
3691:
3681:
3671:
3664:
3655:
3650:
3629:
3623:
3616:
3613:
3599:
3586:
3578:
3555:
3547:
3537:
3516:
3506:
3504:
3496:
3486:
3471:
3461:
3451:
3444:
3435:
3430:
3409:
3403:
3396:
3393:
3383:
3374:
3369:
3359:
3350:
3346:
3336:
3334:
3314:
3298:
3295:
3281:
3263:
3253:
3204:
3201:
3174:
3157:
3153:
3144:
3125:
3121:
3115:
3106:
3098:
3089:
3085:
3079:
3054:
3005:
3002:
2990:
2952:
2949:
2924:
2921:
2879:
2875:
2868:
2846:
2795:
2790:
2782:
2764:
2748:
2745:
2739:
2735:
2726:
2721:
2704:
2700:
2695:
2688:
2685:
2677:
2673:
2669:
2651:
2647:
2637:
2634:
2577:
2562:
2558:
2551:
2510:
2475:
2471:
2464:
2441:
2433:
2424:
2421:
2394:
2389:
2386:
2376:
2349:
2346:
2340:
2335:
2315:
2311:
2301:
2298:
2283:
2256:
2253:
2239:
2224:
2220:
2210:
2207:
2201:
2196:
2176:
2172:
2162:
2159:
2143:
2128:
2122:
2118:
2114:
2111:
2108:
2103:
2100:
2097:
2093:
2089:
2084:
2081:
2078:
2074:
2070:
2067:
2064:
2059:
2055:
2050:
2046:
2041:
2038:
2007:
2002:
1999:
1989:
1965:
1962:
1957:
1937:
1933:
1926:
1914:
1890:
1879:
1864:
1860:
1853:
1850:
1845:
1825:
1821:
1814:
1786:
1782:
1777:
1765:
1760:
1756:
1745:
1741:
1734:
1730:
1726:
1723:
1718:
1714:
1710:
1688:
1676:
1673:
1668:
1664:
1660:
1657:
1652:
1648:
1644:
1639:
1624:
1608:
1605:
1603:
1587:
1582:
1578:
1567:
1563:
1556:
1551:
1547:
1522:
1508:
1505:
1503:
1480:
1469:
1401:
1384:
1368:
1344:
1333:
1312:
1306:
1285:
1274:
1267:
1219:
1211:
1209:
1173:
1170:
1139:
1118:
1103:
1094:
1027:
1023:
1000:
988:
985:
980:
976:
972:
969:
964:
960:
956:
951:
936:
931:
907:
866:
762:
751:
746:
742:
731:
727:
720:
708:
705:
701:
697:
686:
665:
612:
544:
518:
517:link function
500:
496:
488:
470:
466:
443:
439:
430:
409:
405:
401:
398:
395:
390:
386:
379:
362:
360:
358:
354:
350:
346:
342:
338:
334:
329:
325:
321:
317:
313:
302:
297:
295:
290:
288:
283:
282:
280:
279:
274:
269:
264:
263:
262:
261:
256:
253:
251:
248:
246:
243:
242:
241:
240:
235:
230:
227:
225:
222:
221:
220:
219:
214:
209:
206:
204:
201:
199:
196:
195:
194:
193:
188:
183:
180:
178:
175:
173:
170:
168:
165:
163:
160:
159:
158:
157:
152:
147:
144:
142:
139:
137:
134:
132:
129:
128:
127:
126:
121:
116:
113:
111:
108:
106:
103:
101:
98:
96:
95:Cox's theorem
93:
91:
88:
86:
83:
81:
78:
76:
73:
71:
68:
67:
66:
65:
60:
57:
53:
49:
45:
42:
41:
37:
33:
32:
29:
25:
21:
20:
4739:
4717:. Retrieved
4708:
4668:(1): 18822.
4665:
4661:
4651:
4616:
4612:
4606:
4587:
4581:
4562:
4558:
4552:
4533:
4527:
4508:
4502:
4467:
4463:
4453:
4418:
4414:
4408:
4383:
4379:
4220:
3961:
2991:
2144:
1458:is given by
1215:
867:
613:
366:
341:epidemiology
315:
311:
310:
245:Bayes factor
171:
3879:at a given
1264:. Applying
843:is part of
4761:Categories
4572:1708.02723
4354:References
190:Estimators
62:Background
48:Likelihood
4643:120949984
4565:: 62–85.
4428:1202.1738
4309:~
4306:π
4263:θ
4235:~
4232:π
4198:θ
4155:θ
4144:~
4141:π
4109:θ
4066:θ
4058:π
4023:θ
4015:π
3933:θ
3923:∗
3889:θ
3843:θ
3822:π
3777:θ
3748:~
3745:π
3702:θ
3692:∗
3646:θ
3617:~
3614:π
3595:θ
3587:π
3574:θ
3556:π
3543:θ
3517:π
3507:∝
3482:θ
3472:∗
3426:θ
3397:~
3394:π
3365:θ
3347:π
3337:∝
3310:θ
3299:~
3296:π
3259:θ
3250:θ
3215:θ
3205:~
3202:π
3158:π
3141:θ
3122:π
3103:θ
3086:π
3063:θ
3055:π
3016:θ
3006:~
3003:π
2963:θ
2953:~
2950:π
2939:out from
2922:−
2917:θ
2876:θ
2869:π
2843:Δ
2821:θ
2787:Δ
2783:×
2760:θ
2749:~
2746:π
2740:×
2717:θ
2689:~
2686:π
2674:∑
2638:~
2635:π
2607:θ
2574:θ
2552:π
2518:θ
2511:π
2465:π
2442:⋅
2434:⋅
2425:~
2422:π
2387:−
2382:θ
2360:θ
2350:~
2347:π
2341:∫
2312:θ
2302:~
2299:π
2288:θ
2267:θ
2257:~
2254:π
2236:θ
2211:~
2208:π
2202:∫
2163:~
2160:π
2119:θ
2112:…
2094:θ
2082:−
2075:θ
2068:…
2056:θ
2039:−
2034:θ
2000:−
1995:θ
1973:θ
1966:π
1963:∫
1934:θ
1927:π
1919:θ
1898:θ
1891:π
1876:θ
1854:π
1851:∫
1815:π
1770:θ
1757:η
1735:π
1727:
1715:∑
1700:θ
1669:−
1661:
1634:θ
1616:θ
1609:π
1606:∝
1592:θ
1579:η
1557:π
1548:∏
1540:θ
1523:π
1516:θ
1509:π
1506:∝
1485:θ
1470:π
1445:θ
1385:π
1376:θ
1369:π
1362:θ
1345:π
1338:θ
1313:π
1290:θ
1275:π
1251:θ
1192:θ
1147:θ
1140:π
1113:θ
1078:θ
1054:θ
1012:θ
981:−
973:
946:θ
932:∝
925:θ
908:π
887:θ
830:η
756:θ
743:η
721:π
709:∈
702:∏
691:θ
666:π
645:θ
598:θ
554:θ
497:η
440:μ
399:…
359:package.
322:based on
90:Coherence
44:Posterior
4719:21 April
4700:33139744
4494:29200537
4445:88511801
56:Evidence
4691:7606447
4670:Bibcode
4485:5708893
4400:1657669
3809:is the
337:ecology
4746:
4698:
4688:
4641:
4594:
4540:
4515:
4492:
4482:
4443:
4398:
3875:whose
3733:where
2411:where
2024:where
1040:where
775:where
541:. The
339:, and
4639:S2CID
4567:arXiv
4441:S2CID
4423:arXiv
4396:S2CID
3278:with
1069:is a
52:Prior
4744:ISBN
4721:2022
4696:PMID
4592:ISBN
4538:ISBN
4513:ISBN
4490:PMID
3877:mode
2544:and
1436:and
1242:and
1095:and
636:and
589:and
458:(of
367:Let
316:INLA
4686:PMC
4678:doi
4629:hdl
4621:doi
4563:158
4480:PMC
4472:doi
4433:doi
4388:doi
3959:.
3908:is
3813:to
2620:as
1724:log
1658:exp
970:exp
4763::
4694:.
4684:.
4676:.
4666:10
4664:.
4660:.
4637:.
4627:.
4617:73
4615:.
4561:.
4488:.
4478:.
4468:87
4466:.
4462:.
4439:.
4431:.
4419:84
4417:.
4394:.
4384:71
4382:.
4362:^
4218:.
2989:.
2142:.
865:.
658::
335:,
54:Ă·
50:Ă—
46:=
4752:.
4723:.
4702:.
4680::
4672::
4645:.
4631::
4623::
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4575:.
4569::
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4447:.
4435::
4425::
4402:.
4390::
4338:)
4334:y
4329:|
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3697:(
3687:x
3682:=
3678:x
3672:|
3665:)
3660:y
3656:,
3651:k
3640:|
3635:x
3630:(
3624:G
3605:)
3600:k
3590:(
3584:)
3579:k
3568:|
3563:x
3559:(
3553:)
3548:k
3538:,
3534:x
3529:|
3524:y
3520:(
3497:,
3492:)
3487:k
3477:(
3467:x
3462:=
3458:x
3452:|
3445:)
3440:y
3436:,
3431:k
3420:|
3415:x
3410:(
3404:G
3384:)
3379:y
3375:,
3370:k
3360:,
3356:x
3351:(
3330:)
3326:y
3321:|
3315:k
3305:(
3264:k
3254:=
3229:)
3225:y
3220:|
3211:(
3175:,
3169:)
3165:y
3161:(
3154:)
3149:y
3145:,
3136:|
3131:x
3126:(
3116:)
3111:y
3107:,
3099:,
3095:x
3090:(
3080:=
3077:)
3073:y
3068:|
3059:(
3030:)
3026:y
3021:|
3012:(
2977:)
2973:y
2968:|
2959:(
2925:j
2895:)
2891:y
2886:|
2880:j
2872:(
2847:k
2796:,
2791:k
2780:)
2776:y
2771:|
2765:k
2755:(
2736:)
2731:y
2727:,
2722:k
2711:|
2705:i
2701:x
2696:(
2678:k
2670:=
2667:)
2663:y
2658:|
2652:i
2648:x
2644:(
2586:)
2582:y
2578:,
2569:|
2563:i
2559:x
2555:(
2532:)
2528:y
2523:|
2514:(
2491:)
2487:y
2482:|
2476:i
2472:x
2468:(
2445:)
2438:|
2431:(
2395:,
2390:j
2377:d
2374:)
2370:y
2365:|
2356:(
2336:=
2331:)
2327:y
2322:|
2316:j
2308:(
2284:d
2281:)
2277:y
2272:|
2263:(
2248:)
2244:y
2240:,
2231:|
2225:i
2221:x
2217:(
2197:=
2192:)
2188:y
2183:|
2177:i
2173:x
2169:(
2129:)
2123:m
2115:,
2109:,
2104:1
2101:+
2098:j
2090:,
2085:1
2079:j
2071:,
2065:,
2060:1
2051:(
2047:=
2042:j
2008:,
2003:j
1990:d
1987:)
1983:y
1978:|
1969:(
1958:=
1953:)
1949:y
1944:|
1938:j
1930:(
1915:d
1912:)
1908:y
1903:|
1894:(
1888:)
1884:y
1880:,
1871:|
1865:i
1861:x
1857:(
1846:=
1841:)
1837:y
1832:|
1826:i
1822:x
1818:(
1787:.
1783:)
1778:]
1774:)
1766:,
1761:i
1752:|
1746:i
1742:y
1738:(
1731:[
1719:i
1711:+
1707:x
1696:Q
1689:T
1684:x
1677:2
1674:1
1665:(
1653:2
1649:/
1645:1
1640:|
1630:Q
1625:|
1620:)
1612:(
1596:)
1588:,
1583:i
1574:|
1568:i
1564:y
1560:(
1552:i
1544:)
1535:|
1530:x
1526:(
1520:)
1512:(
1499:)
1495:y
1490:|
1481:,
1477:x
1473:(
1423:x
1402:,
1396:)
1392:y
1388:(
1380:)
1372:(
1366:)
1357:|
1352:x
1348:(
1342:)
1334:,
1330:x
1325:|
1320:y
1316:(
1307:=
1304:)
1300:y
1295:|
1286:,
1282:x
1278:(
1229:x
1196:)
1188:(
1184:m
1181:i
1178:d
1174:=
1171:m
1151:)
1143:(
1119:|
1109:Q
1104:|
1050:Q
1028:,
1024:)
1019:x
1008:Q
1001:T
996:x
989:2
986:1
977:(
965:2
961:/
957:1
952:|
942:Q
937:|
929:)
920:|
915:x
911:(
882:|
877:x
852:x
808:y
785:I
763:,
760:)
752:,
747:i
738:|
732:i
728:y
724:(
714:I
706:i
698:=
695:)
687:,
683:x
678:|
673:y
669:(
623:x
576:x
528:x
501:i
471:i
467:y
444:i
415:)
410:n
406:y
402:,
396:,
391:1
387:y
383:(
380:=
376:y
357:R
314:(
300:e
293:t
286:v
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