Conformal Bayesian Computation
Year: 2021

Abstract:
We develop scalable methods for producing conformal Bayesian predictive intervals with finite sample calibration guarantees. Bayesian posterior predictive distributions, p(y | x), characterize subjective beliefs on outcomes of interest, y, conditional on predictors, x. Bayesian prediction is well-calibrated when the model is true, but the predictive intervals may exhibit poor empirical coverage when the model is misspecified, under the so calledM-open perspective. In contrast, conformal inference provides finite sample frequentist guarantees on predictive confidence intervals without the requirement of model fidelity. Using ‘add-one-in’ importance sampling, we show that conformal Bayesian predictive intervals are efficiently obtained from re-weighted posterior samples of model parameters. Our approach contrasts with existing conformal methods that require expensive refitting of models or data-splitting to achieve computational efficiency. We demonstrate the utility on a range of examples including extensions to partially exchangeable settings such as hierarchical models.

heading: 1 Introduction
text: We consider Bayesian prediction using training data Z1:n = {Yi, Xi}i=1:n for an outcome of interest Yi and covariates Xi ∈ Rd. Given a model likelihood fθ(y | x) and prior on parameters, π(θ) for θ ∈ Rp, the posterior predictive distribution for the response at a new Xn+1 = xn+1 takes on the form
p(y | xn+1, Z1:n) = ∫ fθ(y | xn+1)π(θ | Z1:n) dθ , (1)
where π(θ | Z1:n) is the Bayesian posterior. Asymptotically exact samples from the posterior can be obtained through Markov chain Monte Carlo (MCMC) and the above density can be computed through Monte Carlo (MC), or by direct sampling from an approximate model. Given a Bayesian predictive distribution, one can then construct the highest density 100× (1−α)% posterior predictive credible intervals, which are the shortest intervals to contain (1 − α) of the predictive probability. Alternatively, the central 100 × (1 − α)% credible interval can be computed using the α/2 and 1− α/2 quantiles. Posterior predictive distributions condition on the observed Z1:n and represent subjective and coherent beliefs. However, it is well known that model misspecification can lead Bayesian intervals to be poorly calibrated in the frequentist sense (Dawid, 1982; Fraser et al., 2011), that is the long run proportion of the observed data lying in the (1− α) Bayes predictive interval is not necessarily equal to (1− α). This has consequences for the robustness of such approaches and trust in using Bayesian models to aid decisions.
Alternatively, one can seek intervals around a point prediction from the model, ŷ = µ̂(x), that have the correct frequentist coverage of (1 − α) . This is precisely what is offered by the conformal prediction framework of Vovk et al. (2005), which allows the construction of prediction bands with finite sample validity without assumptions on the generative model beyond exchangeability of the
35th Conference on Neural Information Processing Systems (NeurIPS 2021).
data. Formally, for Zi = {Yi, Xi}i=1:n, Zi iid∼ P and miscoverage level α, conformal inference allows us to construct a confidence set Cα(Xn+1) from Z1:n and Xn+1 such that
P(Yn+1 ∈ Cα(Xn+1)) ≥ 1− α. (2) Note that P is over Z1:n+1, that is we also treat Xn+1 as random; see Lei and Wasserman (2014) for a discussion on the difficulties of attaining coverage conditional on Xn+1. In this paper we develop computationally efficient conformal inference methods for Bayesian models including extensions to hierarchical settings. A general theme of our work is that, somewhat counter-intuitively, Bayesian models are well suited for the conformal method.
Conformal inference for calibrating Bayesian models was previously suggested in Melluish et al. (2001), Vovk et al. (2005), Wasserman (2011) and Burnaev and Vovk (2014), where it is referred to as “de-Bayesing”, “frequentizing” and “conformalizing”, but only in the context of conjugate models. Here, we present a scalable MC method for conformal Bayes, implementing full conformal Bayesian prediction using an ‘add-one-in’ importance sampling algorithm. The automated method can construct conformal predictive intervals from any Bayesian model given only samples of model parameter values from the posterior θ ∼ π(θ | Z1:n), up to MC error. Such samples are readily available in most Bayesian analyses from probabilistic programming languages such as Stan (Carpenter et al., 2017) and PyMC3 (Salvatier et al., 2016). We also extend conformal inference to partially exchangeable settings which utilize the important class of Bayesian hierarchical models, and note the connection to Mondrian conformal prediction (Vovk et al., 2005, Chapter 4.5). Previously, the extension of conformal prediction to random effects was introduced in Dunn et al. (2020) in a non-Bayesian setting, with a focus on prediction in new groups, as well as within-group predictions without covariates. We will see that the Bayesian hierarchical model allows for a natural sharing of information between groups for within-group predictions with covariates. We discuss the motivation behind using the Bayesian posterior predictive density as the conformity measure for both the Bayesian and the frequentist, and demonstrate the benefits in a number of examples.

heading: 1.1 Background
text: The conformal inference framework was first introduced by Gammerman et al. (1998), followed by the thorough book of Vovk et al. (2005). Full conformal prediction is computationally expensive, requiring the whole model to be retrained at each test covariate xn+1 and for each value in a reference grid of potential outcomes, e.g. y ∈ R for regression. This makes the task computationally infeasible beyond a few special cases where we can shortcut the evaluation along the outcome reference grid, e.g. ridge regression (Vovk et al., 2005; Burnaev and Vovk, 2014) and Lasso (Lei, 2019). Shrinking the search grid is possible, but still requires many refittings of the model (Chen et al., 2016). The split conformal prediction method (Lei et al., 2018) is a useful alternative method which only requires a single model fit, but increases variability by dividing the data into a training and test set that includes randomness in the choice of the split, and has a tendency for wider intervals. Methods based on cross-validation such as cross-conformal prediction (Vovk, 2015) and the jacknife+ (Barber et al., 2021) lie in between the split and full conformal method in terms of computation and average length of the intervals. A detailed comparison of various conformal methods are provided in Barber et al. (2021, Section 4). A review of recent advances in conformal prediction is given in Zeni et al. (2020), and interesting extensions have been developed by works such as Tibshirani and Foygel (2019); Romano et al. (2019); Candès et al. (2021).

heading: 2 Conformal Bayes

heading: 2.1 Full Conformal Prediction
text: We begin by summarizing the full conformal prediction algorithm discussed in Vovk et al. (2005); Lei et al. (2018). Firstly, a conformity (goodness-of-fit) measure,
σi := σ(Z1:n+1;Zi),
takes as input a set of data points Z1:n+1, and computes how similar the data point Zi is for i = 1, . . . , n + 1. A typical conformity measure for regression would be the negative squared error arising from a point prediction −{yi − µ̂(xi)}2, where µ̂(x) is the point predictor fit to the augmented dataset Z1:n+1, assumed to be symmetric with respect to the permutation of the input
dataset. The key property of any conformity measure is that it is exchangeable in the first argument, i.e. the conformity measure for Zi is invariant to the permutation of Z1:n+1. Under the assumption that Z1:n+1 is exchangeable, we then have that σ1:n+1 is also exchangeable, and its rank is uniform among {1, . . . , n+ 1} (assuming continuous σ1:n+1). From this, we have that the rank of σn+1 is a valid p-value. If we now consider a plug-in value Yn+1 = y (where Xn+1 is known), we can define the rank of σn+1 among σ1:n+1 as
r(y) = 1
n+ 1 n+1∑ i=1 1 (σi ≤ σn+1) .
For miscoverage level α, the full conformal predictive set, Cα(Xn+1) = {y ∈ R : r(y) > α}, (3)
satisfies the desired frequentist coverage as in (2) under the exchangeability assumptions above. Intuitively, we are reporting the values of y which conform better than the fraction α of observed conformity scores in the augmented dataset. A formal proof can be found in Vovk et al. (2005, Chapter 8.7). For continuous σ1:n+1, we also have from Lei et al. (2018, Theorem 1) that the conformal predictive set does not significantly over-cover.
In practice, beyond a few exceptions, the function r(y) must be computed on a fine grid y ∈ Ygrid, for example of size 100, in which case the model must be retrained 100 times to the augmented dataset to compute σ1:n+1, with plug-in values for yn+1 on the grid. This is illustrated in Algorithm 1 below. We note here that the grid method only provides approximate coverage, as y may be selected even if it lies between two grid points that are not selected. This is formalized in Chen et al. (2018), but we do not discuss this further. In Appendix D.6, we provide an empirical comparison of the grid effects. This is also valid for binary classification where we now have a finite Ygrid = {0, 1}, and so the grid method for full conformal prediction is exact and feasible.
Observed data is Z1:n, Xn+1; Specify miscoverage level α for each y ∈ Ygrid do
Fit model to augmented dataset {Z1, . . . , Zn, {y,Xn+1}} Compute σ1:n and σn+1 Store the rank, r(y), of σn+1 among σ1:n+1
end Return the set Cα(Xn+1) = {y ∈ Ygrid : r(y) > α}.
Algorithm 1: Full Conformal Prediction

heading: 2.2 Conformal Bayes and Add-One-In Importance Sampling
text: In a Bayesian model, a natural suggestion for the conformity score, as noted in Vovk et al. (2005); Wasserman (2011), is the posterior predictive density (1), that is
σ(Z1:n+1;Zi) = p(Yi | Xi, Z1:n+1) for i ∈ {1 . . . , n+ 1}, where
p(Yi | Xi, Z1:n+1) = ∫ fθ(Yi | Xi)π(θ | Z1:n+1) dθ. (4)
We highlight that the datum Zi = {Yi, Xi} also lies in the conditioning Z1:n+1. This is a valid conformity score, as we have π(θ | Z1:n+1) ∝ π(θ) ∏n+1 i=1 fθ(Yi | Xi), and so σ is indeed invariant to the permutation of Z1:n+1. We refer to this method as conformal Bayes (CB) - this is a special case of full conformal prediction, and we will see shortly that the exchangeability structure of Bayesian models is key to constructing conformity scores in the partial exchangeability scenario.
Beyond conjugate models, we are usually able to obtain (asymptotically exact) posterior samples θ(1:T ) ∼ π(θ | Z1:n), e.g through MCMC, where T is a large integer. Such samples are typically available as standard output from Bayesian model fitting. The posterior predictive can then be computed up to Monte Carlo error through
p̂(Yn+1 | Xn+1, Z1:n) = 1
T T∑ t=1 fθ(t)(Yn+1 | Xn+1).
The key insight is that refitting the Bayesian model with {Z1, . . . , Zn, {y,Xn+1}} is well approximated through importance sampling (IS), as only {y,Xn+1} changes between refits. This leads immediately to an IS approach to full conformal Bayes, where we just need to compute ‘add-one-in’ (AOI) predictive densities. Here AOI refers to the inclusion of {Yn+1, Xn+1} into the training set, named in relation to ‘leave-one-out’ (LOO) cross-validation. Specifically, for Yn+1 = y and θ(1:T ) ∼ π(θ | Z1:n), we can compute
p̂(Yi | Xi, Z1:n+1) = T∑ t=1 w̃(t)fθ(t)(Yi | Xi) (5)
where w̃(t) are our self-normalized importance weights of the form
w(t) = fθ(t)(y | Xn+1), w̃(t) = w(t)∑T t′=1 w (t′) · (6)
We see that the unnormalized importance weights have the intuitive form of the predictive likelihood at the reference point {y,Xn+1} given the model parameters θ(t). The use of AOI importance sampling has similarities to the computation of Bayesian leave-one-out (LOO) predictive densities for cross-validation (Vehtari et al., 2017), which is also used in accounting for model misspecification. An interesting aspect of AOI in comparison with LOO is that AOI predictive densities are less vulnerable to importance weight instability for the following reasons:
• In LOO, the target π(θ | Z−i) generally has thicker tails than the proposal π(θ | Z1:n), leading to importance weight instability. In contrast, AOI uses the posterior π(θ | Z1:n) as a proposal for the thinner-tailed π(θ | Z1:n+1). For LOO the importance weights are proportional to 1/fθ(y | x), in contrast to the typically bounded fθ(y | x) for AOI.
• For AOI, we are predicting Zi given Z1:n+1 which is always in-sample unlike in LOO where the datum is out-of-sample, so we can expect greater stability with AOI.
• The IS weight stability is governed by Yn+1 = y, which is not random as we select it for the grid, unlike in LOO cross-validation. For sufficiently large α, we will not need to compute the AOI predictive density for extreme values of y.
The above discussion is a natural analogy to the discussion provided in Vehtari et al. (2017, Section 2.1.1) for computing LOO predictives. Note that the LOO predictive density is also a valid conformity score; we provide more details and verify the above comparisons between LOO and AOI empirically in Appendix B. We also provide some IS weight diagnostics in Section 4.1.1 and find that they are stable. In difficult settings such as very high-dimensions, one can make use of the recommendations of Vehtari et al. (2015) for assessing and Pareto-smoothing the importance weights if necessary.

heading: 2.3 Computational Complexity
text: Given the posterior samples, we must compute the likelihood for each θ(t) at Z1:n, as well at {y,Xn+1} for y ∈ Ygrid. The additional computation required for CB for each Xn+1 is thus T × (n + ngrid) likelihood evaluations, which is relatively cheap. This is then followed by the dot product of an (n + 1) × T matrix with a T vector for each y, which is O(nT ), so the overall complexity is O(ngridTn). The large matrices involved in computing the AOI predictives suggests we can take advantage of GPU computation, and machine learning packages such as JAX (Bradbury et al., 2018) are highly suitable for this application.
The values ngrid and T are constants we select, and we now briefly provide guidance on this. Selecting a grid is a general problem for full conformal prediction (beyond just CB), and this has been investigated in Chen et al. (2016, 2018). Here, we opt for a default choice of ngrid = 100 in Section 4.1 and we show in Appendix D.6 that this is sufficient for accurate coverage. For the selection of T , there are standard methods based on the effective sample size (ESS) which we compute in Section 4.1.1. One could then approximate the Monte Carlo standard error in the density estimates which is proportional to 1/ √ ESS, or estimate the IS variance directly. The ESS will depend both on MCMC mixing and IS stability, which may further depend on the dimensionality of the model. More details can be found in Owen (2013, Chapter 9.3)

heading: 2.4 Motivation
text: Much has been written on the contrasting foundations and interpretation of Bayes versus frequentist measures of uncertainty (Little, 2006; Shafer and Vovk, 2008; Bernardo and Smith, 2009; Wasserman, 2011), and we provide a summary in Appendix A. Here we motivate CB predictive intervals from both a Bayesian and frequentist perspective.
The pragmatic Bayesian, aware of the potential for model misspecification in either the prior or likelihood, may be interested in conformal inference as a countermeasure. CB predictive intervals with guaranteed frequentist coverage can be provided as a supplement to the usual Bayesian predictive intervals. The difference between the Bayesian and conformal interval may also serve as an informal diagnostic for model evaluation (e.g. Gelman et al. (2013)). Posterior samples through MCMC or direct sampling are typically available, and so CB through automated AOI carries little overhead.
The frequentist may also wish to use a Bayesian model as a tool for constructing predictive confidence intervals. Firstly, the likelihood can take into account skewness, heteroscedasticity unlike the usual residual conformity score. Secondly, features such as sparsity, support, and regularization can be included through priors, while CB ensures correct coverage. Thirdly, CB provides a computationally feasible method to compute full conformal prediction intervals for a larger class of models. Finally, a subtle issue that arises in full conformal prediction is that we lose validity if hyperparameter selection is not symmetric with respect to Zn+1, e.g. if we estimate the Lasso penalty λ using only Z1:n before computing the full conformal intervals with said λ(Z1:n). For CB, a prior on hyperparameters induces weighting of the hyperparameter values by implicit cross-validation for each refit (Gneiting and Raftery, 2007; Fong and Holmes, 2020). We highlight here that this issue does not affect the split conformal method, e.g. Lei and Wasserman (2014, Section 5).

heading: 3 Partial Exchangeability and Hierarchical Models
text: A setting of particular interest is for grouped data, which corresponds to a weakening of exchangeability often referred to as partial exchangeability (Bernardo and Smith, 2009, Chapter 4.6). Assume that we observe data from J groups, each of size nj , where again Zi,j = {Yi,j , Xi,j}. We write the full dataset as Z = {Zi,j : i = 1, . . . , nj , j = 1, . . . , J}. We may not expect the entire sequence Z to be exchangeable, instead only that data points are exchangeable within groups. Formally, this means that p(Z1:n1,1, . . . , Z1:nJ ,J) = p(Zγ1(1):γ1(n1),1, . . . , ZγJ (1):γJ (nJ ),J) (7) for any permutations γj of 1, . . . , nj , for j = 1, . . . , J . Alternatively, we can enforce the usual definition of exchangeability but only consider permutations of 1, . . . , n such that the groupings are preserved. A simple example of this partial exchangeability is if Zi,j
iid∼ Pj for i = 1, . . . , nj , j = 1, . . . , J , where Pj can now be distinct.
Partial exchangeability is useful in multilevel modelling, e.g. where Z1:nj ,j records exam results on students within school j, for schools j = 1, . . . , J . Students may be deemed exchangeable within schools, but not between schools. Further examples may be found in Gelman and Hill (2006).

heading: 3.1 Group Conformal Prediction
text: Given a new Xnj+1,j belonging to group j for j ∈ {1, . . . , J}, we seek to construct a (1 − αj) confidence interval for Ynj+1,j . We define a within-group conformity score as
σi,j := σZ−j (Z1:nj+1,j ;Zi,j)
for i = 1, . . . , nj + 1. We write Z−j as the dataset without group j, and Z−j in the subscript indicates the dependence of the conformity score on this, which we motivate in the next subsection. For each Z−j , we require the score to be invariant with respect to the permutation of Z1:nj+1,j . For Znj+1,j = {y,Xnj+1,j}, the conformal predictive set is then defined
rj(y) = 1
nj + 1 nj+1∑ i=1 1 ( σi,j ≤ σnj+1,j ) , Cαj ( Xnj+1,j ) = {y ∈ R : rj(y) > αj) (8)
In other words, we rank the conformity scores σ1:nj+1,j within the group j, and compute the conformal interval as usual with Algorithm 1. The interval is valid from the following.
Proposition 1. Assume that {Z,Znj+1,j} is partially exchangeable as in (7), and the conformity measure σi,j for group j is invariant to the permutation of Z1:nj+1,j . We then have
P ( Ynj+1,j ∈ Cαj ( Xnj+1,j )) ≥ 1− αj
where Cαj ( Xnj+1,j ) is defined in (8), and P is over {Z,Znj+1,j}.
Proof. Conditional on Z−j , the observations Z1:nj+1,j are still exchangeable, and thus so are σ1:nj+1,j from the invariance of the conformity measure. The usual conformal guarantee then holds:
P ( Ynj+1,j ∈ Cαj ( Xnj+1,j ) | Z−j ) ≥ 1− αj .
Taking the expectation with respect to Z−j gives us the result.
It is interesting to note that the above group conformal predictor coincides with the attributeconditional Mondrian conformal predictor of Vovk et al. (2005, Chapter 4.5), with the group allocations as the taxonomy. Validity under the relaxed Mondrian-exchangeability of Vovk et al. (2005, Chapter 8.4) is key for us here.

heading: 3.2 Conformal Hierarchical Bayes
text: Under this setting, a hierarchical Bayesian model can be defined of the form
[Yi,j | Xi,j , θj , τ ] iid∼ fθj ,τ (· | Xi,j) i = 1, . . . , nj , j = 1, . . . , J
[θj | φ] iid∼ π(· | φ) j = 1, . . . , J
φ ∼ π(φ), τ ∼ π(τ).
Here τ is a common parameter across groups (e.g. a common standard deviation for the residuals under homoscedastic errors). The desired partial exchangeability structure is clearly preserved in the Bayesian model (Bernardo, 1996). De Finetti representation theorems are also available for partially exchangeable sequences (when defined in a slightly different manner to the above), which motivate the specification of hierarchical Bayesian models (Bernardo and Smith, 2009, Chapter 4.6).
The posterior predictive is once again a natural choice for the conformity measure. Denoting Z̄y as the entire dataset augmented with Znj+1,j = {y,Xnj+1,j}, we have
σi,j = p(Yi,j | Xi,j , Z̄y) = ∫ fθj ,τ (Yi,j | Xi,j)π(θj , τ | Z̄y) dθj dτ (9)
for i = 1, . . . , nj + 1. The within-group permutation invariance follows as the likelihood is exchangeable within groups, and thus so is the posterior and resulting posterior predictive. Practically, this structure allows for independent coefficients θj for each group, but partial pooling through π(θ | φ) allows information to be shared between groups. A fully pooled model, whilst still valid, is usually too simple and predicts poorly, whereas a no-pooling conformity score ignores information sharing between groups. More details on hierarchical models can be found in Gelman et al. (2013, Chapter 5). We point out that we can select a separate coverage level αj for each group, which will be useful when group sizes nj vary - we provide a demonstration of this in Appendix D.4.2. Computation of σi,j is again straightforward, where MCMC now returns [θ (1:T ) 1:J , φ
(1:T ), τ (1:T )] ∼ π(θ1:J , φ, τ | Z). We can then estimate (9) using AOI importance sampling as in (5) and (6) using the marginal samples {θ(1:T )j , τ (1:T )} ∼ π(θj , τ | Z) and weights w(t) = fθ(t)j ,τ(t)(y | Xnj+1,j).
In the above, we consider predictive intervals within groups with covariates, extending the withingroup approach of Dunn et al. (2020). Predictive intervals for new groups are possible with the Bayesian model, but a conformal predictor would require additional stronger assumptions of exchangeability to ensure validity. The relevant assumptions and methods based on pooling and subsampling for new group predictions are discussed in Dunn et al. (2020), but would require rerunning MCMC in our case. We leave this for future work, noting that utilizing the Bayesian predictive density directly here seems nontrivial due to different group sizes.

heading: 4 Experiments
text: We run and time all examples on an Azure NC6 Virtual Machine, which has 6 Intel Xeon E5-2690 v3 vCPUs and a one-half Tesla K80 GPU card. We use PyMC3 (Salvatier et al., 2016) for MCMC and sklearn (Pedregosa et al., 2011) for the regular conformal predictor; both are run on the CPU. Computation of the CB and Bayes intervals is implemented in JAX (Bradbury et al., 2018), and run on the GPU. The code is available online1 and further examples are provided in Appendix D.

heading: 4.1 Sparse Regression
text: We first demonstrate our method under a sparse linear regression model on the diabetes dataset (Efron et al., 2004) considered by Lei (2019). The dataset is available in sklearn, and consists of n = 442 subjects, where the response variable is a continuous diabetes progression and the d = 10 covariates consist of patient readings such as blood serum measurements. We standardize all covariates and the response to have mean 0 and standard deviation 1.
The Bayesian model we consider is
fθ(y | x) = N (y | θTx+ θ0, τ2) π(θj) = Laplace(0, b), π(θ0) ∝ 1, π(b) = Gamma(1, 1) π(τ) = N+(0, c)
(10)
for j = 1, . . . , d, and where b is the scale parameter andN+ is the half-normal distribution. Note that a hyperprior on b has removed the need for cross-validation that is required for Lasso. We consider two values of c for the hyperprior on τ , which correspond to a well-specified (c = 1) and poorlyspecified (c = 0.02) prior; in the latter case our posterior on τ will be heavily weighted towards a small value. This model is well-specified for the diabetes dataset (Jansen, 2013, Chapter 4.5) under a reasonable prior (c = 1). We compute the central (1 − α) credible interval from the Bayesian posterior predictive CDF estimated using Monte Carlo and the same grid as for CB.
To check coverage, we repeatedly divide into a training and test dataset for 50 repeats, with 30% of the dataset in the test split. We evaluate the conformal prediction set on a grid of size ngrid = 100 between [ymin − 2, ymax + 2], where ymin, ymax is computed from each training dataset. The average coverage, length and run-times (excluding MCMC) with standard errors are given in Table 1 for α = 0.2. MCMC induced an average overhead of 21.9s for a = 1 and 26.8s for c = 0.02 for the Bayes and CB interval, where we simulate T = 8000 posterior samples. The CB intervals are only slightly slower than the Bayes intervals, and still a small fraction of the time required for MCMC, and is thus an efficient post-processing step. For c = 1, the Bayesian intervals have coverage close to (1− α) with the smallest expected length, with CB slightly wider and more conservative. However, when the prior is misspecified with c = 0.02, the Bayes intervals severely undercover, whilst the CB coverage and length remain unchanged from the c = 1 case.
As baselines, we compare to the split and full conformal method using the non-Bayesian Lasso as the predictor, with the usual residual as the nonconformity score. For the split method, we fit Lasso with cross-validation on the subset of size ntrain/2 to obtain the Lasso penalty λ. For the full conformal method, we use the grid method for fair timing, as other estimators beyond Lasso would not have the shortcut of Lei (2019). As setting a default λ = 1 gives poor average lengths, we estimate λ = 0.004 on cross-validation on one of the training sets, and use this value over the 50 repeats. However, we must emphasize again that this is somewhat misleading, as discussed in Section 2.4. A fairer approach would involve fitting Lasso with CV for each of the 100 grid values and 133 test values, but this is infeasible as each fit requires around 80ms, resulting in a total run-time of 17 minutes. On the other hand, the AOI scheme of CB is equivalent to refitting b for each grid/test value. In terms of performance, the split method has wider intervals than CB/full, but performs well given the low computational costs. The full conformal method performs as well as CB, but is comparable in time as MCMC + CB, whilst not refitting λ. We note that the value of c does not affect the split/full method.
1https://github.com/edfong/conformal_bayes

heading: 4.1.1 Importance Weights
text: For the diabetes dataset, we look at the ESS of the self-normalized importance weights (6), which can be computed as ESS = 1/ ∑T t=1{w(t)}2 for each xn+1 and y. The ESS as a function of y for a single xn+1 is shown in Figure 1 for the two cases c = 1, 0.02, with the CB conformal bands given for α = 0.2, 0.5. We have scaled the ESS plots by ESSMCMC/T , where T = 8000 is the number of posterior samples and ESSMCMC is the minimum ESS out of all posterior parameters return by PyMC3. We observe the ESS is well behaved and stable across the range of y values. In both cases, the ESS for α = 0.2 is sufficiently large for a reliable estimate of the conformity scores. However, for c = 0.02, the ESS decays more quickly with y as the Bayes predictive intervals are too narrow, which the CB corrects for. Other values of xn+1 produce similar behaviour.

heading: 4.2 Sparse Classification
text: In this section, we analyze the Wisconsin breast cancer (Wolberg and Mangasarian, 1990), again available in sklearn. The dataset is of size 569, where the binary response variable corresponds to a malignant or benign tumour. The 30 covariates consist of measurements of cell nuclei. Again, we standardize all covariates to have mean 0 and standard deviation 1.
We consider the logistic likelihood fθ(y = 1 | x) = [1 + exp {− (θTx+ θ0)}]−1, with the same priors for θ, θ0 as in (10). The Bayesian predictive set is the smallest set from {0}, {1}, {0, 1} that contains at least (1− α) of the posterior predictive probability. The conformal baselines are as above but with L1-penalized logistic regression, and for the full conformal method we have λ = 1. We again have 50 repeats with 70-30 train-test split, and set α = 0.2. The grid method is now exact, and the size of the CB intervals can take on the values {0, 1, 2}. The results are provided in Table 2, where MCMC required an average of 45.4s to produce T = 8000 samples. We see that even with reasonable priors, Bayes can over-cover substantially, which CB corrects in roughly the same amount of time as it takes to compute the usual Bayes interval. However, we point out that CB may produce empty prediction sets, whereas Bayes cannot, and we investigate this in Appendix D.3.2.

heading: 4.3 Hierarchical Model
text: We now demonstrate Bayesian conformal inference using a hierarchical Bayesian model for multilevel data. We stick to the varying intercept and varying slope model (Gelman et al., 2013), that is for j = 1, . . . , J :
fθj ,τ (yi,j) = N (yi,j | θTjXi,j + θ0,j , τ2) π(θj) = N (φ, s2), π(θ0,j) = N (φ0, s20)
(11)
with hyperpriors N (0, 1) on the location parameters φ, φ0 and Exp(1) on the standard deviations s, s0, τ . We now apply this to a simulated example, and an application to the radon dataset of Gelman and Hill (2006) is given in Appendix D.4.2.
We consider two simulation scenarios, with J = 5 groups and nj = 10 elements per group:
1. Well-specified: We generate group slopes θj iid∼ N (0, 1) for j = 1, . . . , J . For each j, we
generate Xi,j ∼ N (0, 1) and Yi,j ∼ N (θjXi,j , 1).
2. Misspecified: We generate group slopes and variances θj iid∼ N (0, 1), τj iid∼ Exp(1) for j = 1, . . . , J . For each j, we generate Xi,j ∼ N (0, 1) and Yi,j ∼ N ( θjXi,j , τ 2 j ) .
The first scenario has homoscedastic noise between groups as assumed in the model (11) whereas the second scenario is heteroscedastic between groups. To evaluate coverage, we only draw θ1:J , τ1:J once (and not per repeat), giving us the values
θ1:J = [1.33,−0.77,−0.32,−0.99,−1.07], τ1:J = [1.24, 2.30, 0.76, 0.28, 1.11].
For each of the 50 repeats, we draw nj = 10 training and test data points from each group using the above θ1:J (and τ1:J for scenario 2), and report test coverage and lengths within each group. We use a grid of size 100 between [−10, 10]. The group-wise average lengths and coverage are given in Table 3 again with α = 0.2. Again run-times are given post-MCMC, where MCMC required an average of 90.1s and 78.4s for scenarios 1 and 2 respectively to generate T = 8000 samples. The Bayes interval is again the central (1− α) credible interval. The CB and Bayes methods have comparable run-times, likely due to the small n. As a reference, fitting a linear mixed-effects model in statsmodels (Seabold and Perktold, 2010) to the dataset takes around 200ms, so the full conformal method, which requires refitting for each of the 100 grid value and 50 test values, would take a total of 17 minutes. For scenario 1, both Bayes and CB provide close to (1− α) coverage, with the Bayes lengths being smaller. This is unsurprising, as the Bayesian model is well-specified. In scenario 2, the Bayes intervals noticeably over/under-cover depending on the value of τ1:J in relation to the Bayes posterior mean τ̄ ≈ 1.3. CB is robust to this, adapting its interval lengths accordingly (in particular for Groups 2 and 4) and providing within-group validity.

heading: 5 Discussion
text: In this work, we have introduced the AOI importance sampling scheme for conformal Bayesian computation, which allow us to construct frequentist-valid predictive intervals from a baseline Bayesian model using the output of an MCMC sampler. This extends naturally to the partially exchangeable setting and hierarchical Bayesian models.
Under model misspecification, or theM-open scenario (Bernardo and Smith, 2009), CB can produce calibrated intervals from the Bayesian model. In the partially exchangeable case, CB can remain valid within groups. We find that even under reasonable priors, Bayesian predictives can over-cover, and CB can help reduce the length of intervals to get closer to nominal coverage. Diagnosing Bayesian miscalibration is in general non-trivial, but CB intervals may provide a diagnostic tool while also automatically correcting the miscalibration. When posterior samples of model parameters are available, AOI importance sampling is only a minor increase in computation, and interestingly is much faster than the split method (with the Bayesian model) which would require another run of MCMC. For the frequentist, CB intervals enjoy the tightness of the full conformal method, for a single expensive fit with MCMC followed by a cheap refitting process. We are also free to incorporate prior information, and use more complex likelihoods or priors, as well as automatically fitting hyperparameters.
There are however limitations to our approach, dictated by the realities of MCMC and IS. Firstly, the intervals are only approximate up to MC error, although we expect some robustness as the CB intervals are computed from the ranks and not the exact values. The stability of AOI importance sampling also depends on the posterior predictive being a good proposal, which may break down if the addition of the new datum {y,Xn+1} has very high effect on the posterior. Finally, if our Bayesian model is grossly misspecified and the predictive densities are too narrow, the IS weights may have low ESS and the AOI estimates may have high variance.
If only approximate posterior samples are available, e.g. through variational Bayes (VB), then an AOI scheme may still be feasible, where one includes an additional correction term in the IS weights for the VB approximation, e.g. in Magnusson et al. (2019). However, this remains to be investigated, as the IS weight stability is strongly reliant on the quality of the approximate posterior. Combining this with the Pareto-smoothed IS method of Vehtari et al. (2015) may lead to a feasible solution. In our experience, CB intervals tend to be a single connected interval, which may allow for computational shortcuts in adapting the search grid. It would also be interesting to pursue the theoretical connections between the Bayesian and CB intervals, in a similar light to Burnaev and Vovk (2014).

heading: Acknowledgements
text: We thank the anonymous reviewers for their feedback on our paper. Fong is funded by The Alan Turing Institute Doctoral Studentship, under the EPSRC grant EP/N510129/1. Holmes is supported by The Alan Turing Institute, the Health Data Research, U.K., the Li Ka Shing Foundation, the Medical Research Council, and the U.K. Engineering and Physical Sciences Research Council through the Bayes4Health programme grant EP/R018561/1.

Review summaries:
This paper proposes a scalable method for conformal Bayesian prediction, by estimating the modified posterior predictive density using "add-one-in" importance sampling.  On several tabular datasets with sparse and hierarchical models, estimation is shown to be efficient for non-extreme miscoverage levels, and the produced credible intervals have good coverage.  **Post-rebuttal update:** Thank you for the response.  I am keeping my score unchanged: on the plus side, the method is sensible and easy to implement; on the other hand, there are some (understandable) limitations when scaling to large datasets or applied on approximate posterior samples; and some reviewers raised questions about novelty, which I'm not most suited to judge, as I'm not an expert in this field.  In aggregate, the rating of "marginally above the acceptance threshold" still describes my opinions well.

The authors present an approach to using a Bayesian model to provide predictive intervals with  frequentist coverage within the framework of conformal prediction.  This work generalizes existing Bayesian methods for conformal inference that apply only to conjugate models.  The authors provide a straightforward generalization of their approach to partially exchangeable data.  Empirical validation on several datasets, and comparison to a baseline approach is provided.

This article proposes a novel approach to constructing prediction intervals with approximately correct frequentist coverage, even on misspecified models.  The basic idea is to employ the conformal prediction technique using the Bayesian posterior predictive density as the conformity (goodness-of-fit) measure.  The main innovation of the article is to use an importance sampling approximation to the posterior predictive density at candidate values of outcome $y$.  This leads to a computationally fast and generally applicable algorithm for using samples from the standard Bayesian posterior to construct well-calibrated prediction intervals, even under misspecification of the Bayesian model.  An extension to hierachical models is also provided, and empirical results are presented demonstrating the performance of the method.

This paper introduces Bayesian and importance sampling techniques into a previously known conformal prediction framework.  I had troubles identifying what exactly is the novel contribution of this paper. It could use a brief section stating "our own contributions are A, B , C". In my understanding the conformal prediction framework was known before and the novel items are using the Bayesian posterior predictive density as a conformity score and additionally estimating the predictive distribution given new data from a weighted importance sampling where the weights are just the likelihoods of the prameter draws based on the new data point.  There are modelling extensions from completely exchangeable models to partial exchangeability, i.e., where we have groups and complete exchangeability is still valid in the groups, but not outside the groups. From this flat hierarchical structure there is a further extension to hierarchical models that may model deeper hierarchies and via hyperpriors may share information between groups.  

 The authors develop a fast conformal Bayesian computation approach.  The original conformal prediction method can be computationally intensive:  It often requires refitting the model for every augmented value y. The authors avoid this refitting step by using importance sampling. The authors also show the extension to conditionally exchangeable data, which is especially useful for hierarchical data. The resulted conformal interval has finite sample correct coverage and is immune to model misspecification or distribution assumptions.