HNPE: Leveraging Global Parameters for Neural Posterior Estimation

1 Introduction . Simulation-based inference ( SBI ) has the potential to revolutionize experimental science as it opens the door to the inversion of arbitrary complex non-linear computer models , such as those found in physics , biology , or neuroscience ( Cranmer et al. , 2020 ) . The only requirement is to have access to a mathematical model implemented as a simulator . When applied to biophysical models and simulators in neuroscience ( e.g . Leon et al . 2013 ) , it could estimate properties of the brain closer to the cellular level , thus closing the gap between the neuroimaging and computational neuroscience communities . Grounded in Bayesian statistics , recent SBI techniques profit from recent advances in deep generative modeling to approximate the posterior distributions over the full simulator parameters . Their intrinsic quantification of uncertainties reveals whether certain parameters are worth ( or not ) scientific interpretation given some experimental observation . SBI is concerned with the estimation of a conditional distribution over parameters of interest ✓ . Given some observation x0 , the goal is to compute the posterior p ( ✓|x0 ) . It generally happens that some of these parameters are strongly coupled , leading to very structured posteriors with low dimensional sets of equally likely parameters values . For example , this happens when the data generative process depends only on the products of some parameters : multiplying one of such parameters by a constant and another by its inverse will not affect the output . Performing Bayesian inference on such models naturally leads to a “ ridge ” or “ banana shape ” in the posterior landscape , as seen e.g . in Figure 4 of Gonçalves et al . ( 2020 ) . More formally the present challenge is posed as soon as the model likelihood 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . function is non-injective w.r.t . ✓ , and is not strictly due to the presence of noise on the output observations . In statistics and econometrics literature , such models are called partially identified models ( Gustafson , 2014 ) . To alleviate the ill-posedness of the estimation problem , one may consider a hierarchical Bayesian model ( Gelman and Hill , 2007 ) where certain parameters are shared among different observations . In other words , the model ’ s parameters ✓i for an observation xi are partitioned into ✓i = { ↵i , } , where ↵i is a set of sample specific ( or local ) parameters , and corresponds to shared ( or global ) parameters . For this broad class of hierarchical models , the posterior distribution for a set X = { x1 , . . . , xN } of N observations can be written as ( Tran et al. , 2017 ) : p ( ↵1 , . . . , ↵N , |X ) / p ( ) NY i=1 p ( xi|↵i , ) p ( ↵i| ) . ( 1 ) Hierarchical models share statistical strength across observations , hence resulting in sharper posteriors and more reliable estimates of the ( global and local ) parameters and their uncertainty . Examples of applications of hierarchical models are topic models ( Blei et al. , 2003 ) , matrix factorization algorithms ( Salakhutdinov et al. , 2013 ) , including Bayesian non-parametrics strategies ( Teh and Jordan , 2010 ) , and population genetics ( Bazin et al. , 2010 ) . In this work , we further assume that the likelihood function p ( xi|↵i , ) is implicit and intractable , which implies that traditional MCMC methods can not be used to estimate the posterior distribution . This setup leads to so-called likelihood-free inference ( LFI ) problems and many algorithms ( Papamakarios and Murray , 2016 ; Greenberg et al. , 2019 ; Hermans et al. , 2020 ; Durkan et al. , 2020b ) have recently been developed to carry out inference under this scenario . These methods all operate by learning parts of the Bayes ’ rule , such as the likelihood function , the likelihood-to-evidence ratio , or the posterior itself . Approaches for LFI in hierarchical models exist , but are limited . Bazin et al . ( 2010 ) extend approximate Bayesian computation ( ABC ) into a two-step procedure in which local and global variables are estimated . Tran et al . ( 2017 ) adapt variational inference to hierarchical implicit models using a GAN-like training approach , while Brehmer et al . ( 2019 ) and Hermans et al . ( 2020 ) extend amortized likelihood ratios to deal with global parameters , but can not do inference on local parameters . Motivated by the posterior estimates of individual samples , we consider a sequential neural posterior estimation approach derived from SNPE-C ( Greenberg et al. , 2019 ) . The paper is organized as follows . First , we formalize our estimation problem by introducing the notion of global and local parameters , and instantiate it on a motivating example amenable to analytic posterior estimates allowing for quantitative evaluation . Then , we propose a neural posterior estimation technique based on a pair of normalizing flows and a deepset architecture ( Zaheer et al. , 2017 ) for conditioning on the set X of observations sharing the global parameters ; we call our method ‘ hierarchical neural posterior estimation ’ , or simply HNPE . Results on an application with time series produced by a non-linear model from computational neuroscience ( Ableidinger et al. , 2017 ) demonstrate the gain in statistical power of our approach thanks to the use of auxiliary observations . We also use this model to analyse real brain signals , giving a full demonstration of the power of LFI to relate parameters from theoretical models to real experimental recordings . 2 Hierarchical models with global parameters . Motivating example . Consider a stochastic model with two parameters , ↵ and , that generates as output x = ↵ + `` , where `` ⇠ N ( 0 , 2 ) . We assume that both parameters have uniform prior distribution ↵ , ⇠ U [ 0 , 1 ] and that is known and small . Our goal is to obtain the posterior distribution of ( ↵ , ) for a given observation x0 = ↵0 0+ '' . This simple example describes common situations where indeterminacy emerges . For instance , x0 could be the radiation power measured by a sensor , ↵ the intensity of the emitting source , and the inverse squared distance of the sensor to the source . In this case , a given measurement may have been due to either close weak sources ( ↵ # and `` ) or far strong ones ( ↵ `` and # ) . Using Bayes ’ rule and considering small we can write ( see Appendix A for more details ) p ( ↵ , |x0 ) ⇡ e ( x0 ↵ ) 2/2 2 p 2⇡ 2 1 [ 0,1 ] ( ↵ ) 1 [ 0,1 ] ( ) log ( 1/x0 ) , ( 2 ) where 1 [ a , b ] ( x ) is an indicator function that equals one for x 2 [ a , b ] and zero elsewhere . Note that the first term in the product converges to ( x0 ↵ ) as ! 0 and that the joint posterior distribution has an infinite number of pairs ( ↵ , ) with the same probability , revealing the parameter indeterminacy of this example . Indeed , for x 2 [ 0 , 1 ] and 2 [ x , 1 ] , all pairs of parameters ( x , ) yield the same observations and the likelihood function p ( ·| x , ) is constant . Thus , the posterior distribution has level sets with a ridge or “ banana shape ” along these solutions . The top row of Figure 1 on Page 4 portrays the joint and the marginal posterior distributions when ( ↵0 , 0 ) = ( 0.5 , 0.5 ) and = 0 . Exploiting the additional information in X . Our motivating example illustrates a situation where two parameters are related in such a way that one may not be known without the other . In practice , however , it is possible that one of these parameters is shared with other observations . For instance , this is the case when a single source of radiation is measured with multiple sensors located at different unknown distances . The power of the source is fixed across multiple measurements and its posterior can be better inferred by aggregating the information from all sensors . Our goal in this section is to formalize such setting so as to leverage this additional information and obtain a posterior distribution that ‘ breaks ’ parameter indeterminacy . Note that the root cause of the statistical challenge here is not the presence of noise , but rather the intrinsic structure of the observation model . To tackle the inverse problem of determining the posterior distribution of parameters ( ↵0 , ) given an observation x0 of a stochastic model , we consider the following scenario . We assume that the model ’ s structure is such that ↵0 is a parameter specific to each observation ( local ) , while is shared among different observations ( global ) . Yet , both are unknown . We consider having access to a set X = { x1 , . . . , xN } of additional observations generated with the same as x0 . Taking the model ’ s hierarchical structure into account we use Bayes ’ rule to write p ( ↵0 , |x0 , X ) = p ( ↵0| , x0 , X ) p ( |x0 , X ) / p ( ↵0| , x0 ) p ( x0 , X| ) p ( ) / p ( ↵0| , x0 ) p ( ) QN i=0 p ( xi| ) / p ( ↵0 , |x0 ) p ( ) N QN i=1 p ( |xi ) ( 3 ) which shows how the initial posterior distribution p ( ↵0 , |x0 ) is modified by additional observations from X sharing the same as x0 . In Section 3 , we present a strategy for approximating such posterior distribution when the likelihood function of the stochastic model of interest is intractable and , therefore , the posterior distributions p ( ↵0| , x0 ) and p ( |x0 , X ) have to be approximated with conditional density estimators . Motivating example with multiple observations . We now detail the effect of X on the posterior distribution of our motivating example . The N + 1 observations in { x0 } [ X are such that xi = ↵i 0 + `` for i = 0 , . . . , N with ↵i ⇠ U [ 0 , 1 ] drawn from the same prior . The posterior distribution may be written as ( see Appendix A ) p ( ↵0 , |x0 , X ) ⇡ p ( ↵0 , |x0 ) 1 [ µ,1 ] ( ) N N log ( 1/x0 ) ( 1/µN 1 ) , ( 4 ) where µ = max ( { x0 } [ X ) . This expression shows how the initial full posterior distribution ( 2 ) changes with the extra information conveyed by X . It can be also shown that as N ! 0 ( no additional observations ) the posterior distribution converges back to p ( ↵0 , |x0 ) . Figure 1 portrays the joint and marginal posterior distributions with N = 10 and N = 100 .

Summary: The authors propose a new likelihood-free inference method for Bayesian hierarchical models, called h-Flow, that exploits additional information provided by a set of auxiliary observations with shared, global parameters. h-Flow works by approximating two relevant posterior distributions, i.e. that of the global parameter beta and the local parameters alpha given beta, with separate normalizing flows, which are then multiplied to yield the joint posterior distribution. The normalizing flows are trained by minimising the KL-divergence between the true and approximated posterior distribution and the training data is refined over several rounds by means of SNPE-C (also known as APT).