Joint Gaussian Mixture Model for Versatile Deep Visual Model Explanation
Authors: Zhouyang Xie
Duanbing Chen
Venue: Submitted to ICLR 2023 (2023)
Keywords: Explainable AI
interpretable model
pixel attributing
convolutional neural networks

Abstract: Post-hoc explanations of deep neural networks improve human understanding on the learned representations, decision-making process and uncertainty of the model with faithfulness. Explaining deep convolutional neural networks(DCNN) is especially challenging, due to the high dimensionality of deep features and the complexity of model inference. Most post-hoc explaining methods serve a single form of explanation, restricting the diversity and consistency of the explanation. This paper proposes joint Gaussian mixture model(JGMM), a probabilistic model jointly models inter-layer deep features and produces faithful and consistent post-hoc explanations. JGMM explains deep features by Gaussian mixture model and inter-layer deep feature relations by posterior distribution on the latent component variables. JGMM enables a versatile explaining framework that unifies interpretable proxy model, global or local explanatory example generation or mining. Experiments are performed on various DCNN image classifiers in comparison with other explaining methods. It shows that JGMM can efficiently produce versatile, consistent, faithful and understandable explanations.

Under review as a conference paper at ICLR 2023
JOINT GAUSSIAN MIXTURE MODEL FOR VERSATILE DEEP VISUAL MODEL EXPLANATION
Anonymous authors Paper under double-blind review
ABSTRACT
Post-hoc explanations of deep neural networks improve human understanding on the learned representations, decision-makiin process and uncertainty of the model with faithfulness. Explaining deep convolutional neural networks (DCNN) is especially challenging, due to the high dimensionality of deep features and the complexity of model inference. Most post-hoe explaining methods serve a sin- gle form of explanation, restricting the diversity and consistency of the explana- tion. This paper proposes joint Gaussian mixture model (JGMM), probabilistic model jointly models inter-layer deep features and produces faithful and consis- tent post-hoc explanations. JGMM explains deep features by Gaussian mixture model and inter-layer deep feature relations by posterior distribution on the la- tent component variables. JGMM enables versatile explaining framework that unifies interpretable proxy model, global or local explanatory example genera- tion or mining. Experiments are performed on various DCNN image classifiers comparison with other explaining methods. It shows that JGMM can efficiently produce versatile, consistent, faithful and understandable explanations.
INTRODUCTION
Deep convolutional neural networks (DCNN) is a powerful type of machine learning model for visual recognition tasks. The key reasons give rise to the power of DCNN include the expressive vi- sual representations and the decision-making mechanism encoded in massive trainable convolution parameters. However, increasing model complexity incurs heavier burden for human to understand learned representations and decision making of the model. The high dimensionality and entangle- ment of deep features and the complexity of neural network inference are often considered the main hindrance explaining black-box DCNN.
recent proliferation of studies in post-hoc DCNN explainability show several effective and practi- cal DCNN explaining methods.
Proxy models are interpretable models (e.g. decision trees and linear models) that has approximate decision-making behavior the black-box model inference. Proxy models have inference process that can be intuitively understood by human, such as LIME (proposed byRibeiro et al. (2016)). linear classifier as local proxy model. To make sure the proxy model is an accurate surrogate, its faithfulness should be tested. The proxy model's predictions on unseen examples should be close to the black-box model's prediction, even if different from the ground truth.
The intermediate representations of DCNN are usually explained globally (not associated with spe- cific data point) by explanatory examples and association with semantic concepts. Prototypes, crit- icisms and influential examples are common types of global explanatory examples. Prototypes are representative examples of certain pattern deep features, illustrating a learned visual concept of the model. In contrast, criticisms (proposed by Kim al. (2016)) are examples not well-represented in the deep representations, i.c. outliers of the deep features, revealing the flaw of the learned repre- sentations. Influential examples are hard examples for the model training, having more influence on the final decision boundary than others. From a post-hoe view, influential examples lie close to the decision boundary. An example can be both influential and representative (or unrepresentative).
Local explanations are based on a specific query example, showing how the change of the query example features will affect the model prediction. Counterfactual examples offer an actionable

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course for the model decision. Counterfactual examples answer the 'what if" questions by minimal change of query example features and resulting different model decision against the query. Coun- terfactual examples reveal the sensitive features for the query. Semi-factual examples, in contrast, aim to answer the "even if" question. Semi-factual examples have significant change on a certain feature(s) from the query, but both have the same model prediction. Semi-factual examples reveal the insensitive features of the query. For example, Kenny & Keane| (2021) proposes method to generate counterfactual and semi-factual examples from one system.
Most DCNN explaining methods are single-purpose systems. Employing different explaining meth- ods is possible to give diverse explanations, but it's not guaranteed that different explanations are compatible and consistent with each other. For example, a counterfactual example generation sys- tem may suggest that the model is sensitive to a certain feature; but global explaining method, such as proxy model, may have conflicts with the former explanation. There is no hard rule to de- termine which explanation correct or more understandable. Thus generie explaining framework that enables various and consistent explaining forms has important value for the explainability of DCNN.
Higher Feature
Proxy Modiet P(FX)
Interpretability and Falthtiness Volidation
Black-box Modelipert)
Joint
Explanations
Example: Prototypes, Criticiams, Infleetiials
Lower Feature x
Local Explanations
Example: Counterfactuals and Somifactuals
Step 1: Probabilstic Madelling
Step 2: Explaining
Figure 1: Proposed probabilistic framework for versatile explaining method. The intermediate features (X and Y) are jointly modelled by a probabilistic model joint Gaussian mixture model (IGMM).
We propose a probabilistic framework for a versatile explaining method for DCNN. Figure demon strates the pipeline of the framework and the enabled explanation forms. The lower and the higher features are intermediate representations of DCNN from two different layers. In the computational graph of DCNN, the higher feature is dependent on the lower feature and the black-box model (part of the DCNN) between them. In an image classification setting, the higher feature can be the clas- sification probabilistic prediction of the black-box model, and the lower features can be the input of the DCNN, i.e. the raw image. The goal of the framework is to explain the learned representations of higher/lower features as well the black-bon model (either the whole DCNN or a part of DCNN) between them.
To serve this purpose, we propose joint Gaussian mixture model (JGMM) to jointly model the distribution of lower/higher features and produce a validatable proxy model and example-based global/local explanations. JGMM a probabilistic model based on GMM. JGMM learns two Gaus- sian mixture model (GMM) respectively for the lower features X and higher features Y. The latent categorical variables of both GMMs are connected by an estimated posterior probability matrix. JGMM is introduced in section B in details.
Compared with single purpose explaining methods, the proposed JGMM-based versatile explain- ing method has two advantages: (1) various forms of model explanations efficiently produced from one framework; (2) the consistency among different explanations is guaranteed, as they are computed from a common probabilistic model. The proposed explaining method is evaluated with various DCNN models and benchmarks in comparison with other explaining methods in section 4 The experiments show that JGMM can efficiently produce versatile, consistent, faithful and under- standable explanations.
2
RELATED WORKS
Proxy models widely leveraged method to produce model-agnostic explanations. Local inter- pretable model-agnostic explanations (LIME) by Ribeiro et al. (2016) is a typical method that learns linear classifier by sampling from data points from local region and the black-box classier. For
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deep networks, Zilke et al. (2016) propose to convert model parameters to hard rule-set but keep a computational approximation. To explain the visual concepts encoded in DCNN and how the decision is made, [Zhang al. (2018) and Zhang et al. 2019) respectively design explanatory graphs and decision trees to explain the model inference.
Pixel attributing, or saliency mapping, is another useful technique explain DCNN. Compared with traditional feature attributing of machine learning models on tabular data, pixel attributing is specialized the hierarchical structure of DCNN and visualizes the attention of the model. Methods like CAM by Zhou et al.] 2016). Integrated Gradients by SSnddraraaan et al. (2017), DeepLIFT ny Shrikumar et al. (2017). Grad-CAM ny [Selvaraju al. (2017) utilize different heuristies to back- propagate the feature importance from deep layers to shallow layers.
Example-based explaining is less explored field for deep visual model explanation. Studies like Goyal et al. (2019), Kenny& & Keane (2021) and |Black et al.] (2021) generate counterfactual ex- amples by simultaneous minimization of the feature distance after the query and maximization of model decision change. Prototypes, the representative examples of feature space cluster, are usu- ally employed to explain a certain understandable visual concept learned by the model. For example, Nauta et al. (2021) propose ProtoPNet to mine prototypical image patches and explain object parts. Semi-factual examples (e.g. Kenny Keane (202I)). criticisms (e.g. Kim et al. (2016)) and influ- ential examples (c.g. Koh & Liing (207)) can also help explaining the distribution of deep model
features.
3 METHOD
3.1 JOINT GAUSSIAN MIXTURE MODEL
E
14
q
+
Kr
K
K
Cat
X
2
Wn
Yn
Cat
N
N
Figure 2: Graphical model of joint Gaussian mixture model (JGMM) on lower feature X and higher feature Y.
We propose joint Gaussian mixture model, a probabilistic model base on GMM to estimate the posterior distribution connecting the latent variables behind the deep features. Figure 2 illustrates the model in plate notation. The model includes three parts: the GMM on observation Y. the conditioned GMM on observation X, and the probabilistic model between discrete latent variables and w.
The model on observation Y resembles the classical GMM. It includes the observed -yddimnnsionn real-value higher features Y ~ NN , ...). Data point y, is the n-th of N sampled higher features. The latent categorical variable E {1...., Ky) decides which Gaussian component Y generated from. dlyee--paramete Ky is the number of mixture components. Prior probability æ is a KD vector which represents the weight each K, mixture component.
The model on observation X is also Gaussian mixture model. The observed Di-dimension lower features X is generated by the z-th Gaussian component: X N(J22, M2z). Latent categorical variable E {1..., has hyper-parameter Kz, which is the number of mixture components. However, the key difference of our model from a traditional GMM is that the prior probability dependent on posterior probability p( p(2/w) parameterized by Q-
The connection between latent categorical variables w and z is modeled by the posterior probability
matrix Q RK,,K,, The entry Q is the conditional distribution of 2. i.c p(2 i/w j). In figure 2 9д the k-th column vector of Q, ie. the distribution of & conditioned on w k. whose sum is

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By the chain rule of Bayesian inference, the likelihood function of JGMM is(G is Gaussian distri- bution density function):
K
Ky
y(x,y)
=
pxx22 = ()p(y/u==)))) j)p(= = i/w j)p(w m
i=
Ky
pa E.cp E.) QiykT
(I)
The maximum likelihood estimation (MLE) problem of JGMM parameters be solved by expectation-maxxiizatiio (EM) algorithm. The optimization algorithm can be straightforwardly derived from the EM algorithm framework, whose details are described the appendix section A The complexity of the optimization is similar to separately learning two GMM for X and Y.
Equation D shows that the trained JGMM is still GMM with Ky components, preserving the desired properties of GMM. Discussions below use the following notations for simplicity. For a given observation x and y. E ((Kxx)) and By E R(Kxx)) are probability density on each component, and v E (K, xKy) is the conditional distribution of w:
[ = pl Σ.), loyla = plulig. 6,), = p(z i) = Qi
Qu,y i 1,... K =1, I...., Ky
G = p(w = j|z = -) E QUATTR
(2)
where Q jz Σ, δ. θ and स are parameters estimated by MLE. Qi,+ is the i-th row vector of Q
Trained JGMM can work as a bi-directional Bayesian classifier to predict latent variable and w. Conditioned on a known x or y, the distribution of the other feature's latent component variable w or is:
(of 2i (T QR)
(
p(w -]xx
p(z :ly)
(i
Posterior probability in equation 3 is naturally a Bayesian classifter to predict the component cate- gory of the given example. This enables a traditional training-test evaluation manner the proxy model to evaluate the faithfulness of modelling.
JGMM also makes handy example generator and miner to improve human understanding on the model by examples. When the conditional distribution w or z is computed by equation JGMM on y degrades to normal GMM, which is easy for sampling and interpretation.
3.2 MINING AND GENERATING EXPLANATORY EXAMPLES
This section introduces how to mine or generate explanatory examples (prototypes, criticisms, in- fluential examples, counterfactuals and semi-factuals) by trained JGMM. The example mining problem is to search for the best explanatory examples in the dataset, which requires a scoring func- tion fxx to evaluate the quality. Generation problem is the sampling of data that maximize the scoring function. We respectively define the scoring function for each type of explanatory examples and introduce the way to mine or generate examples.
Prototypes. From probabilistic aspect, representative example given = is at the center of the cluster, whose neighboring region has high cumulative probability on p(a|w = j), which is GMM. So define the prototype scoring function of the j-th higher feature component as equation
f fproto (x) p(z/w j) = GM, (R,
(4)
Sampling prototypes for the j-th higher feature component is trivially sampling from multivariate Gaussian distribution p(x|u wu
Criticisms. Criticisms can be considered as outliers that is hardly represented by any prototypes. So we search for examples that least likely sampled from any w, which is the opposite of the
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prototype score, equation S
Serit (x) = max f (x)
(5)
Sampling criticisms is supported in our method, because the realistic data distribution is often very sparse compared to the feature space. Most sampled outliers are not understandable examples. We only focus on criticisms found in realistic data.
Influential examples. Influential examples are close to the decision boundary and can lead to model decision with low confidence. From JGMM, we can measure the influence of an example by the entropy of the posterior probability of w conditioned on x. Higher the entropy, higher the uncertainty and influence. The scoring function is defined as equation 6
Ky
Sinnu
Σ j=1
p(w = llx) = j|x)
Influential examples can be generated by sampling from each components of X and select those with largest finu((X) (x). Sampled influential examples, though not actually in the training data of the black-box, can be useful for exploring uncertain examples for the black-box model.
Counterfactuals. A counterfactual example is close to the query example Xqueryy in the feature space but have very different posterior distribution on w. Therefore, the scoring function for counter- factuals is defined as the probability that x and Xquery are sampled from the same but different w:
N₂ Ky
Comnn (x)
-- p(w =1===)ppp ilx)p((uqurr # jl = i)p( ilsquery)
To maximize the counterfactual score, there actually exists an optimal distribution of p(z|x). Let this distribution be a, the maximization equation 7 is:
K2 a,
K2
maxx
p(w = j|z = i)p( Uquery #
i| jl = i)p(Equery i (Q )), s.t.
a
(8)
Note that all terms except a are constant, which makes it linear programming problem with sum constraint. Though it's still hard to derive optimal the solution still provides an efficient sampling approach. The approach to generate counterfactuals is to sample from the GMM under this optimal distribution a and select the best counterfactuals.
Semi-factuals. A semi-factual example is likely sampled from a different = from the query example Xquery but have probably the same posterior distribution which is the reverse of counterfactuals. Therefore, the scoring function for counterfactuals is defined as equation
Kr L p(w = j|z = i)p(2 = ilz)p [1==)(@Wuuerry = =Il * t)p( * il ((
Ssemi (I) = Σ
(9)
Similar as the counterfactuals, the maximization of semi-factual score leads to an optimal distri- bution of p(2|x). We sample from the GMM of this optimal distribution and select the best semi- factuals.
3.3 EXTENSION FOR DCNN
The deep feature maps of DCNN usually has additional spatial dimensions apart from the feature dimension, such as the time dimension signal data and the vertical and the horizontal dimensions of 2D images. The feature maps can be formulated with spatial encoding s E S. i.c. ( For example, for feature map of width W and height H. the spatial encoding is S = [(i,j)]1 < s W. و < H}. In later discussions, S and T respectively denote the spatial dimensions of and Y.
The addition of spatial dimensions makes the latent variable relation changes from one-to-one to many-to-many, i.e. modelling each pair of ((s, S, E T). Theoretically, JGMM can be
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extended to such scenario by estimating each of Q(s:9, and make GMM parameters Σ, θ shared by every s E S. However, both E-step and M-step in the EM algorithm JGMM requires enumeration at all possible latent variable assignment (see equation D where the combinatorial
space (K! K elements) so large that enumeration is infeasible.
However, most modern DCNN architectures have translation-invarian mapping of features, which means translation of lower features over spatial dimensions will cause the same translation (relative) of higher features. Also, though the relative receptive field (the region the lower features that affects a specific position (016)) shows that the central feature. Therefore we can extend the proposed probabilistic model to a one-to-many relation. Let this relative effective receptive field be R. R region cast the lower feature and connected to point higher feature. We train [R] posterior distribution matrix Q(r) between S and T. The size of is a hyper-parameter. Also, the parameters and E are shared by all positions. Figure 3 shows the graphical model of JGMM extended for deep features with spatial dimensions. The hyper-parameter [R] decides how many lower feature positions will affect the higher feature. Like the setting of convolution hyper-parameters, the size of the effective receptive field should satisfy |S] + [T' 2 RR].
higher feature) of T on can be very large, research by Luo et al. (the effective receptive field) has main influence the higher
Σj
Φj
K
Ky
Cat
Xn
Zn
Wn
Yn
A
Cat
i (1, N)
(r)
qk
E [1,2, Ky)
E R
Figure 3: Extension of JGMM relative spatial dimension R.
important property of the extended JGMM that (r) is conditionally independent from other rf E R on w:
p((x() r E R))w)
II
p(x(1))())
(10)
rER
So the inference the extended JGMM can be decomposed to the inference on each the [R] JGMMs:
RR) p((x(1)/r E R)(w)p(w) p(w) p(w) Iren p((x())\r R}) p((x(t) R})
(11)
Therefore, the inference and sampling methods of JGMM are directly applicable to the extended JGMM.
4 EXPERIMENTS
4.1 IMPLEMENTATION DETAILS
JGMM implementation. For a vanilla GMM, the covariance matrix contributes the most parame- ters to be estimated. To reduce the complexity of JGMM training, whiten the lower and higher

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features by PCA (principal component analysis). All PCA components are kept to make the trans- formation invertible and all dimensions are uncorrelated with each other empirically. Thus, training JGMM can be less complex by estimating two diagonal covariance matrices, ie. only the diagonal entries of Σ and o are estimated and others are constantly zero
Data splitting. To fairly test the faithfulness of JGMM, only use the training data of the black- box model to train JGMM and use the test data to evaluate JGMM's accuracy in predicting the black-box model's decision. The effectiveness test of counter/semi-factuall example generation is also performed with the test data. But the explaining step (the generation of all kinds of explanatory examples) makes use of all data.
Features. For a DCNN image classifier, the classification output (without softmax) is always se- lected as a higher feature explain. Because it directly links the understandable image classes with feature dimensions. For models with large input image resolution, the feature map is very large and, because JGMM sample from each position, makes the training of JGMM difficult. To balance the data size and training feasibility, the feature maps are average-pooled with 2 x 2 or 3 x 3 kernels to shrink the feature map. Following JGMM training and explaining on the pooled features.
The training of JGMM requires two hyper-parament nents of the lower and the higher features. The component number of the lower feature chosen by validation performance on the validation dataset (split from the training data). When the higher feature is the classification output of the model, the number of components is set to the number of classes; otherwise, the higher feature has already been modelled by JGMM where it's the lower feature, and the number of component is inherited from the last JGMM. The effective receptive field size is as large the computation resource allows capture as many positional information as possible. But when [R] is too large, the training of JGMM will be unacceptably slowed down. We set the effective receptive field size from 2 x x 5.
Slyper-parametee
the number of compo-
4.2 EXPERIMENTAL SETTINGS
Networks and data. The experiments are performed on VGG-16/19 Simonyan & Zisserman (0155), ResNet- H/ 1 trained on MNIST (LeCun et al. (1998), CIFAR- (iKiizekkk (2009)) and CUB-200( et al. (201T)) The dataset split is the same as the official split. We choose three feature maps from each network, named as 'high', 'middle" and 'low' in the tables and graphics below. In experiments, we train JGMM with 32 components on middle features and 128 components on low features for MNIST; 64 components on middle features and 256 components on low features for CIFAR- 10; 128 components on middle features and 1024 components on low features for CUB-200.
al. (2016) and MobileNe=VV//2(Sandeee
al. (2018)) respectively
Faithfulness evaluation. We quantitatively evaluate the faithfulness of JGMM and other inter- pretable proxy models by classification accuracy on the test dataset (note that the ground truth label for this evaluation is the prediction of the black-box model). As equation 3 shows, JGMM can pre- dict higher feature latent variable distribution given observation of the lower feature. The GMM of the higher feature is dependent on JGMM, which is biased prediction target, so we only choose the 'high' feature map (which is associated with black-box model classification output) for faithfulness evaluation. By associating each component of the 'high' feature neuron with dataset class, the JGMM between the 'middle' and the 'high' layers can directly predict class label with uncertainty. The counterpart interpretable proxy models include decision tree (DT), logistic regression (LR), k-nearest neighbor (kNN) and Gaussian process classifier (GP).
Counterfactuals quality. The quality of counterfacutal examples is evaluated by the ratio of pre- diction change from the query input (higher is better) and the distance between the query input and the generated data (lower is better). To mitigate the conflict between the two metrics, we set differ- ent constraints on the 22 distance between query feature and the generated counterfacutals. For any tested methods, a generated counterfacutal is rejected it breaks the constraint. The examples in the test dataset are queried, and we randomly sample 1000 data points by the local explanatory example sampling method in section JGMM is compared with MAD Wachter et al. (2017). DiCEMothilal & Klaise (2021) and K-d TreeGoyal et al. (2019). The evalua- tion metric is the counterfactual effectiveness, i.c. the ratio of generated counterfactual examples that have different black-box prediction from the queried example, under specific distance constraint.
et al. (2020), Proto-Guided
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4.3 GLOBAL EXPLANATIONS
Pize (Wm)
Pawijeil
7
4 4
Ò
50
4
4 4 4 4 4
04
4
4
4
4
4 4 4
Critiismms
20
20
7
7vs9
OvsB
5
Svsu
Bvss
1
1 1 1
25
7
4ve6
2-14
Prototypes
Influeniaas
(a) Estimated posterior Q and Ap
(b) Global explanatory examples
Figure 4: Global explaining for VGG-19 trained on MNIST. (a) visualizes the posterior distribution of the latent variable, which interprets how JGMM makes predictions. (b) presents global explana- tory examples: prototypes (with its component index), criticisms and influentials (with its two most likely labels).
Table 1: Proxy model accuracy (%) evaluation 'middle' and 'high' layer features with various benchmarks and networks.
Data
Network
Proxy Model
DT
LR
GP
kNN
JGMM
MNIST
VGG-16
85.65
93.84
94.52
89.63
97.65
MNIST
VGG-19
89.43
89.91
93.59
92.43
98.13
CIFAR-10
ResNet-18
88.62
89.12
91.13
83.75
96.42
CIFAR-10
ResNet-50
82.31
90.45
93.73
90.89
98.01
CUB-200
MobileNet-V1
40.47
56.42
81.34
40.91
90.44
CUB-200
MobileNet-V2
30.93
69.34
85.92
64.95
94.10
Pdw-18)s>-0.43
26
Influeniaal example
8-
Protatypes
Criticions
PKw-5jp-00.47
Figure 5:
Global explaining for ResNet-18 18 trained on CUB-200.
Table presents proxy model accuracy JGMM and other interpretable proxy models various datasets and networks. It shows that JGMM is highly faithful to the black-box, which increases the credibility of the explanations. We also find that, the black-box classifier is tasked with more classes, the accuracy of proxy models is lower. This is mainly because the black-box model tends to encode more complex deep features, but fitting efficiency interpretable models is often not comparable with the deep network.
Figure presents the estimated posterior Q and the derived V JGMM and global explanatory examples mined from the test dataset. The visualization of Q suggests that each latent component z of the lower feature mainly mapped to a single higher feature component (though noisy). The visualization of & illustrates that each higher feature component (linked to digit class label) has multiple patterns of examples. As observed from V, we can find that components 2 E {2, 18, 19) learned patterns for digit "4", and E {4, 14) are learned patterns for digit "1". The prototypes
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of those components visualized. It can be found that each latent component z is encoded with a certain style of writing. The mined criticisms in figure can hardly be represented by any latent component, which is intuitively true by the examples and technically evaluated by equation The influentials are presented with two most likely p(w j|x) in equation 6 For example, the first influential example can probably be classified '7' and "9" with similar posterior probability of
Figure 5 presents global explanatory examples mined from ResNet-18 trained on CUB-200 with bird species classification task. The 'middle' feature map has encoded different visual concepts into the deep representations, which is revealed by the prototypes, such as 7 "long pointed beak', 16 'yellow body part', 2 26 'branches texture' and 2 82 'red body part'. An influential example z has close posterior probability on = and w = 48. The bird in the influential example has red feather at the top of the head and pointed beak, but the shapes of the red feather and the beak are not typical in either w 5 or = 48, which makes it near the decision boundary. Criticisms of CUB-200 are often rarely observed sub-species or whose image features are not sufficient for discrimination.
4.4 LOCAL EXPLANATIONS
Table 2: Counterfactual effectiveness (%) evaluation results with varying ly constraint (o). The experiment is performed MobileNet-v2 trained on CUB-200.
Counterfactual Generator
l2 constraint
0.1
0.5
1.0
0 2.0
3.0
MAD Wachter et al.
2017
21.68
24.73
52.41
55.91
84.22
DiCE Mothilal
2020
43.67
47.88
51.35
71.67
98.41
Proto-Gunded Looveren & Klaise 2021)
31.95
34.53
49.96
69.01
93.64
K-d Tree Goyal ct al. (2019
52.78
69.00
78.82
81.23
98.94
JGMM (ours)
78.41
85.67
92.36
97.19
99.17
Query
Counterfactuals
Semi-factuale
Query
Counterfactuals
Semi-factuals
6 66666
66 4.446 a
6 7
6 6 6
7 7 7
7
1
Figure 6: Mined counterfactual and semi-factuals for queries from CIFAR-10 10 and MNIST.
In table 2 we test the counterfactual generation effectiveness of JGMM and other methods on MobileNet-v2 trained CUB-200, with la constraint of feature distance varying from 0.1 to 3.0. JGMM's sampling method shows higher counterfactual generation effectiveness than the counter- parts, especially low distance-constraint settings. An advantage of JGMM on counterfactual effectiveness is that we can estimate the distance between the generated example and the queried example and reject distant examples by equation 7 which significantly reduce the number of distant examples.
Figure 6 presents query images for CIFAR- 10 and MNIST well some mined counterfacutals and semt-factuals. For example, the first CIFAR- 10 image query is side view of horse, but many deer images have very similar visual features the query. So most mined counterfacutals are deers viewed from one side, which has different label prediction as the query. The third CIFAR-10 query flying bird viewed from the top, which can be easily confused (similar deep features) with plane, such the counterfacutals. The semi-factuals provide examples of the same prediction but have very different deep features from the query. The second query of MNIST also demonstrates how the perturbation of deep features will change the prediction. The query image (the black-bor box model and JGMM predict '6') can be confused with '4' counterfacutals if the last draw is slightly longer. In contrast, other '6' semi-factuals have very different latent encoding.

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5 CONCLUSIONS
In this paper, joint Gaussian mixture model is proposed to explain the deep features and model in- ference of black-box DCNN. As probabilistic model that links two deep features, JGMM enable consistent and diverse explanation forms, including proxy models and example-based explaining. Experiments show that JGMM can faithfully interpret model inference and efficiently mine gen- erate explanatory examples. The computational efficiency of JGMM worth improvement, especially with large dataset and higher feature dimensionality, as training with EM algorithm in such scenarios is still very time and memory costly.
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Emily Black, Zifan Wang, and Matt Fredrikson. Consistent counterfactuals for deep models. In International Conference on Learning Representations. 2021.
Yash Goyal, Ziyan Wu, Jan Ernst, Dhruv Batra, Devi Parikh, and Stefan Lee. Counterfactual visual explanations. In International Conference on Machine Learning, pp. 2376-2386 PMLR, 2019.
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Been Kim, Rajiv Khanna, and Oluwasanmi O Koyejo. Examples are not enough, learn to criticize! criticism for interpretability. In D. Lee. M. Sugiyama, U. Luxburg. I. Guyon, and R. Garnett (eds.), Advances in Neural Information Processing Systems, volume 29, Curran Associates, Inc., 2016.
Pang Wei Koh and Percy Liang. Understanding black-box predictions influence functions. In Proceedings of the 34th International Conference Machine Learning- Learning-Volume 70, pp. 1885- 1894, 2017.
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Arnaud Van Looveren and Janis Klaise. Interpretable counterfactual explanations guided by pro- totypes. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, pp. 650-665. Springer, 2021.
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Ramprasaath R Selvaraju, Michael Cogswell, Abhishek Das, Ramakrishna Vedantam, Devi Parikh, and Dhruv Batra. Grad-cam: Visual explanations from deep networks via gradient- based local- ization. In Proceedings of the IEEE international conference computer vision, pp. 618-626, 2017.
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A EM ALGORITHM FOR JGMM
The maximum likelihood estimation (MLE) problem of JGMM is the optimization on the log-
likelihood of N observations (x1, Y1,..,, NN):
N
K=
K2 G(a: (,),,,,,,
max 8
log
(12)
n=1
(13)
Let the parameters at the t-th iteration step be:
(()) = {u
(t)
(0) sis K S Ky)
(14)
The expectation step (E-step) involves the estimation of the latent variables based on 0(1) Let Fn. and h / respectively be the expectation of the latent variable p(=n. i) and p(w, j) at the 1-th
(t)
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Under review as a conference paper at ICLR 2023
iteration. They computed by:
C G 200)
9n.1
K
Q G
(1) 1
19 G(y )
RJ ΣKn j C(ynlof or (0)
(1)
1si5 K2 $ - Kg
(15)
The maximization step (M-step) maximize the likelihood function by updating the estimation 000 to g(t+1), The update of Gaussian parameters is:
(0) In TT
(++1)
n=1
(16)
Σ N (t)
In.i
xx(+1)
Σ n= In.
- p4
(++ (Z,, M4 yr
(17)
>m (3)
In,s
Σm
(((-1)
nnJin
(18)
NV
Em=
Bin (yrr
of
duttlyr
(++
(19)
Σn m/
The optimization of latent distribution parameters Q and x are:
(++1)
In. Rnj
(c+1) ((+1)
(20)
N
(11) j. π
In.i
(1 (
(t+1) n.j
(21)
(1+1)
As above, the E-step and the M-step the EM algorithm for JGMM are defined. We first initialize Q and by uniform distribution parameters. Then, we cluster the data by K-means (K₃ clusters for X and Ky clusters for Y) and initialize Gaussian parameters by cluster sample mean and covariance. Last, the algorithm execute E-step and M-step until convergence.
12


Reviews:
[{"review_id": "kThIv9W2EVo", "reviewer": "Reviewer_DQ8v", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper proposed a probabilistic joint Gaussian mixture model (JGMM) that could simultaneously produce various forms of explanations for DCNN, including proxy models, prototypes, criticisms, influential examples, counter-factual examples and semi-factual examples. In this way, the compatibility and consistency among different explanations could be guaranteed. Experiments were conducted to demonstrate the effectiveness of the proposed JGMM framework.", "summary_of_the_review": "This paper is interesting and has significant contributions. It pointed out a potential problem in Explainable AI, i.e., different types of explanations were produced from different explanation methods/systems, which may result in incompatibility and inconsistency between these explanations. To address the problem, the authors proposed a probabilistic JGMM framework that can simultaneously produce various types of explanations. Experiments verified the effectiveness of the proposed JGMM framework. Besides, this paper is clearly written, and easy to understand. Nevertheless, I still have some concerns about this paper. Please refer to the weakness in my review. It would be great if my main concerns can be well addressed. \nIn summary, I would suggest to accept this paper. ", "strength_and_weaknesses": "[Strength]\n1. This paper pointed out a potential problem in Explainable AI, that is, different types of explanations were produced from different explanation methods/systems, which may result in incompatibility and inconsistency between these explanations. To address the problem, the authors proposed a probabilistic JGMM framework that can simultaneously produce various types of explanations. \n2. Experiments verified the effectiveness of the proposed JGMM framework.\n3. This paper is clearly written, and easy to understand.\n\n[Weakness]\n1. This paper was motivated by a potential problem in XAI, i.e., different types of explanations may be incompatible and inconsistent with each other. However, no theoretical or empirical evidences were provided to support such a claim. It would be great if the authors can give an example to show how different types of explanations will conflict with each other in a specific task. \n2. More experiments are encouraged to demonstrate the effectiveness of the proposed JGMM framework.\na)\tFor the faithfulness evaluation in Table 1, the authors only compared JGMM with four traditional proxy methods. However, there are many other interpretable models with better performance. A convincing demonstration should include a comparison with SOTA interpretable proxy models.\nb)\tFor many types of explanations like prototypes and criticisms, The authors only showed explanation results produced by the proposed JGMM. There lacks a comparison between the proposed JGMM and corresponding baseline methods. It would be nice to see such a comparison, to show whether the proposed JGMM produced better explanation results or not.\nc)\tSome additional experiments were needed to further verify the consistency among different types of explanations produced by the proposed JGMM. \n3. The proxy model accuracy in Table 1 used “middle features” for prediction, why not use “low features”? If the proxy model accuracy using “low features” was poor (i.e., low features may not be faithful), were other types of explanations based on “low features” faithful?\n\nMinors \n\n4. In experiments, the authors adopted “high feature”, “middle feature”, and “low feature”. What is the exact definition of “middle feature”? \n\n5. In Table 2, the evaluation metric “Counter-factual effectiveness” is not clearly defined in the paper.\n\n6. In Equation (2), the summation in the denominator should be from k=1 to k=K_y, rather than K_x.\n\n7. In Section 2, it is not appropriate to refer to LIME as a proxy model.\n\n8. It would be more clear to give a table to list hyper-parameters used in the proposed JGMM. \n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "Ybx4-0UgjgA", "reviewer": "Reviewer_XcXy", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "A new a probabilistic model, named joint Gaussian mixture model(JGMM), is proposed for post-hoc explanations of deep neural networks. By jointly modeling the latent features of lower and higher layers with a joint GMM, JGMM, trained with a post-hoc EM algorithm, is capable of delivering both global explanations (like Prototypes, Criticisms, and Influentials) and local explanations (like counterfactual and semifactual examples). Experiments on common networks and datasets are conducted to demonstrate the effectiveness of the proposed method.", "summary_of_the_review": "Explanations of deep neural networks are essential. The presented JGMM might serve as an explaining framework that produces versatile, consistent, faithful, and understandable explanations.\n\nNotations should be revised carefully. For example, those used in the figures are inconsistent with those in the text. In Eq (1), the function G() is not defined.\n\nI am quite concerned about using the GMM to model lower features (like the input images). Is GMM expressive enough? Or how does Section 3.3 help with that?", "strength_and_weaknesses": "Strength.\n(1) The paper is well-written with clear logic.\n(2) The presented method is sound.\n(3) The empirical experiments are convincing.\n\nWeaknesses.\n(1) The limit of the proposed JGMM is not fully demonstrated, e.g., under what circumstances will JGMM not work well?\n(2) Section 3.3 is believed crucial for improving the practicability of the JGMM. However, this section is not easy to follow.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "NcxZorrr1t", "reviewer": "Reviewer_pdiK", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "In this paper, the authors proposed a joint Gaussian mixture model (JGMM)-based post-hoc explanation method, which applies inter-layer deep features in a probabilistic model. The JGMM can explain deep features and inter-layer deep feature relationship on the latent component variables in its GMMs. The JGMM is applicable to any black-box DNN model for providing explanations for higher/lower level features and the black-box model between them. \n\nThe proposed method is interesting and technically sound. It can obtain different forms of model explanations with same trends. However, the writing of this paper is poor which should be improved by the authors. In addition, the complexity of the JGMM should be discussed.\n", "summary_of_the_review": "As described above, the authors should carefully check and revise their paper. The whole paper should be improved.", "strength_and_weaknesses": "Strength:\n1. The authors proposed a novel viewpoint of DNN model explanation by mixture models, which introduces the correlation between the features of two layers into the probabilistic model instead of independently modeling in some other methods. The joint modeling can explain the learned representations of higher/lower-level features as well as the black-box model between them.\n2. The JGMM is applicable to any black-box DNN model as a post-hoc interpreter.\n3. It can produce explanations in various image classification tasks, which has been shown in the experiments. The visualizations can clearly show their superiority. Various forms of model explanations can be efficiently produced from the framework. The consistency among different explanations is also demonstrated.\n\nWeaknesses:\n1. The writing of this paper is not good enough. Some sentences are hard to understand, e.g., “a probabilistic model jointly models inter-layer deep features and produces faithful and consistent post-hoc explanations” in abstract, “The deep feature maps of DCNN usually has spatial dimensions” in section 3.3.\n2. The position of variable Q is not clarified clearly. How to understand it under a probabilistic view? Besides, it is hard to understand how it works. Eq (22) shows its updating function, are there any references that can support it?\n3. The paper has many typos, e.g., \\hat{q} in eq (4) is undefined, a variable i in eq (9) is lost, many variables are unclearly defined. The authors should carefully check their paper.\n4. How to select the component number of the two GMM? How to select which two layers are used for modeling the JGMM? Do different values affect the results? Are there any ablation studies for clarifying the points?\n5. The definitions of global and local explanations should be clarified.\n6. The comparisons of complexity and parameter number between JGMM and others should be discussed in the paper.\n\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "6fwPYosqGyd", "reviewer": "Reviewer_gUvR", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper proposed a simple joint Gaussian mixture model to generate post-hoc model explanations for DCNN. Essentially the authors use two GMMs to model the higher and lower features Y and X, with a projection matrix Q connecting the component probabilities from these two GMMs. Experiments demonstrated the proposed method’s capability of generating local and global model explanations. ", "summary_of_the_review": "This paper proposed a simple joint Gaussian mixture model to generate post-hoc model explanations for DCNN. Essentially the authors use two GMMs to model the higher and lower features Y and X, with a projection matrix Q connecting the component probabilities from these two GMMs. \n\nOverall I feel the problem the paper explores, i.e., to develop a probabilistic model that is capable of making versatile explanations, is interesting. One of my major concerns is the limited technical merit of the proposed method. Besides, the chosen baselines are severely out of date, especially for the global part (Table 1). Another major concerns is that there are various inaccuracies and missing details that make one question the correctness and rigor of the paper. I feel these are difficult, if not impossible, to address in a single round of conference revision at ICLR. \n", "strength_and_weaknesses": "Strength \n\n+ Generating model explanations is an important and interesting problem for the ML community.  \n+ The proposed model can generate a wide range of explanation types including prototypes and counterfactuals. \n+ The definitions of prototypes, criticisms, etc. make sense to me, though (1) it is unclear how the definition of counterfactuals in Eq (7) is consistent with the standard definition of counterfactuals in Pearl’s do-calculus and (2) since Eq (9) is incomplete, it is unclear why the definition is the reverse of counterfactuals. \n\nWeaknesses\n\n- The proposed model is not principled; that is, it does not seem to be a rigorous probabilistic graphical model (PGM) formulation any more. \n- A lot of details on Equation (2) and (3) are missing/unclear in the main paper. \n- Equation (4) is confusing and potentially problematic. \n- Given that one of the focus in evaluation is to generate counterfactual explanations, some important baselines are missing, e.g., [3]. \n- The baselines (decision trees, logistic regression, etc.) used for evaluating proxy model accuracy are weak. The authors did not even include the 2016 method LIME as a baseline.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "1: The main claims of the paper are incorrect or not at all supported by theory or empirical results.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "1: The contributions are neither significant nor novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject