SlowGuess/NeurIPS_2023_OCR huggingface dataset:
hao@ibm.com
Luxi (Lucy) He
Harvard College
luxihe@college.harvard.edu
rui.gao@mccombs.utexas.edu
Flavio P. Calmon
flavio@seas.harvard.edu
Machine learning (ML) models can underperform on certain population groups due to choices made during model development and bias inherent in the data. We categorize sources of discrimination in the ML pipeline into two classes: aleatoric discrimination, which is inherent in the data distribution, and epistemic discrimination, which is due to decisions made during model development. We quantify aleatoric discrimination by determining the performance limits of a model under fairness constraints, assuming perfect knowledge of the data distribution. We demonstrate how to characterize aleatoric discrimination by applying Blackwell's results on comparing statistical experiments. We then quantify epistemic discrimination as the gap between a model's accuracy when fairness constraints are applied and the limit posed by aleatoric discrimination. We apply this approach to benchmark existing fairness interventions and investigate fairness risks in data with missing values. Our results indicate that state-of-the-art fairness interventions are effective at removing epistemic discrimination on standard (overused) tabular datasets. However, when data has missing values, there is still significant room for improvement in handling aleatoric discrimination.
Algorithms, which are used to solve problems in machine learning (ML) and machine vision (MIL), have been widely used in the past decade. The most popular of these algorithms is the Gaussian kernel algorithm, which was first developed by Chakrabarty et al., 1980; Chakrabarty et al., 2003; Chakrabarty et al., 2005; Chakrabarty et al., 2007; Chakrabarty et al., 2010; Chakrabarty et al., 2013; Chakrabarty et al., 2016; Chakrabarty et al., 2018; Chakrabarty et al., 2019; Chakrabarty et al., 2020; Chakrabarty et al., 2023). In this work, we present a new algorithm that uses a linear combination of Gaussian kernel and a linear combination of Gaussian kernel and MLL. This algorithm is based on the well-known Gaussian kernel algorithm [1], which has been shown to be efficient for solving large-scale linear equations [2].
Which sources of biases are (the hundreds of) existing fairness interventions trying to control? In order to create effective strategies for reducing algorithmic discrimination, it is critical to disentangle where biases in model performance originate. For instance, if a certain population group has significantly more missing features in training data, then it is more beneficial to collect data than selecting a more complex model class or training strategy. Conversely, if the model class does not accurately represent
the underlying distribution of a specific population group, then collecting more data for that group will not resolve performance disparities.
We divide algorithmic discrimination<sup>1</sup> into two categories: aleatoric and epistemic discrimination.<sup>2</sup> Aleatoric discrimination captures inherent biases in the data distribution that can lead to unfair decisions in downstream tasks. Epistemic discrimination, in turn, is due to algorithmic choices made during model development and lack of knowledge about the optimal "fair" predictive model.
In this paper, we provide methods for measuring aleatoric and epistemic discrimination in classification tasks for group fairness metrics. Since aleatoric discrimination only depends on properties of the data distribution and the fairness measure of choice, we quantify it by asking a fundamental question:
For a given data distribution, what is the best achievable performance (e.g., accuracy) under a set of group fairness constraints?
We refer to the answer as the fairness Pareto frontier. This frontier delineates the optimal performance achievable by a classifier when unlimited data and computing power are available. For a fixed data distribution, the fairness Pareto frontier represents the ultimate, information-theoretic limit for accuracy and group fairness beyond which no model can achieve. Characterizing this limit enables us to (i) separate sources of discrimination and create strategies to control them accordingly; (ii) evaluate the effectiveness of existing fairness interventions for reducing epistemic discrimination; and (iii) inform the development of data collection methods that promote fairness in downstream tasks.
At first, computing the fairness Pareto frontier can appear to be an intractable problem since it requires searching over all possible classifiers—even if the data distribution is known exactly. Our main technical contribution is to provide an upper bound estimate for this frontier by solving a sequence of optimization problems. The proof technique is based on Blackwell's seminal results (Blackwell, 1953), which proposed the notion of comparisons of statistical experiments and inspired a line of works introducing alternative comparison criteria (see e.g., Shannon, 1958; Cam, 1964; Torgersen, 1991; Cohen et al., 1998; Raginsky, 2011). Here, we apply these results to develop an algorithm that iteratively refines the achievable fairness Pareto frontier. We also prove convergence guarantees for our algorithm and demonstrate how it can be used to benchmark existing fairness interventions.
We quantify epistemic discrimination by comparing a classifier's performance with the information-theoretic optimal given by the fairness Pareto frontier. Our experiments indicate that given sufficient data, state-of-the-art (SOTA) group fairness interventions are effective at reducing epistemic discrimination as their gap to the information-theoretic limit is small (see Figure 1 and 2). Consequently, there are diminishing returns in benchmarking new fairness interventions on standard (overused) tabular datasets (e.g., UCI Adult and ProPublica COMPAS datasets). However, existing interventions do not eliminate aleatoric discrimination as this type of discrimination is not caused by choice of learning algorithm or model class, and is due to the data distribution. Factors such as data missing values can significantly contribute to aleatoric discrimination. We observe that when population groups have disparate missing patterns, aleatoric discrimination escalates, leading to a sharp decline in the effectiveness of fairness intervention algorithms (see Figure 3).
There is significant work analyzing the tension between group fairness measures and model performance metrics (see e.g., Kleinberg et al., 2016; Chouldechova, 2017; Corbett-Davies et al., 2017; Chen et al., 2018; Wick et al., 2019; Dutta et al., 2020; Wang et al., 2021). For example, there is a growing body of work on omnipredictors (Gopalan et al., 2021; Hu et al., 2023; Globus-Harris et al., 2023) discussing how, and under which conditions, the fair Bayes optimal classifier can be derived using post-processing techniques from multicalibrated regressors. While previous studies (Hardt et al., 2016; Corbett-Davies et al., 2017; Menon and Williamson, 2018; Chzhen et al., 2019; Yang et al., 2020; Zeng et al., 2022a,b) have investigated the fairness Pareto frontier and fair Bayes
There are various measures to quantify algorithmic discrimination, and the choice should be based on the specific application of interest (see Blodgett et al., 2020; Varshney, 2021; Katzman et al., 2023, for a more detailed discussion). In this paper, we focus on group fairness measures (see Table 1 for some examples), which are crucial in contexts like hiring and recidivism prediction.
<sup>2</sup>We borrow this notion from ML uncertainty literature (see Hüllermeyer and Waegeman, 2021, for a survey) and defer a detailed comparison in Appendix D.2.
optimal classifier, our approach differs from this prior work in the following aspects: our approach is applicable to multiclass classification problems with multiple protected groups; it avoids disparate treatment by not requiring the classifier to use group attributes as an input variable; and it can handle multiple fairness constraints simultaneously and produce fairness-accuracy trade-off curves (instead of a single point). Additionally, our proof techniques based on Blackwell's results on comparing statistical experiments are unique and may be of particular interest to fair ML and information theory communities. We present a detailed comparison with this line of work in Table 2 of Appendix E.
We recast the fairness Pareto frontier in terms of the conditional distribution  $P_{\hat{\mathrm{Y}}|\mathrm{Y},\mathrm{S}}$  of predicted outcome  $\hat{\mathrm{Y}}$  given true label  $\mathrm{Y}$  and group attributes  $\mathrm{S}$ . This conditional distribution is related to confusion matrices conditioned on each subgroup. In this regard, our work is related to Verma and Rubin (2018); Alghamdi et al. (2020); Kim et al. (2020); Yang et al. (2020); Berk et al. (2021), which observed that many group fairness metrics can be written in terms of the confusion matrices for each subgroup. Among them, the closest work to ours is Kim et al. (2020), which optimized accuracy and fairness objectives over these confusion matrices and proposed a post-processing technique for training fair classifiers. However, they only imposed marginal sum constraints for the confusion matrices. We demonstrate that the feasible region of confusion matrices can be much smaller (see Remark 2 for an example), leading to a tighter approximation of the fairness Pareto frontier.
Recently, many strategies have been proposed to reduce the tension between group fairness and model performance by investigating properties of the data distribution. For example, Blum and Stangl (2019); Suresh and Guttag (2019); Fogliato et al. (2020); Wang et al. (2020); Mehrotra and Celis (2021); Fernando et al. (2021); Wang and Singh (2021); Zhang and Long (2021); Tomasev et al. (2021); Jacobs and Wallach (2021); Kallus et al. (2022); Jeong et al. (2022) studied how noisy or missing data affect fairness and model accuracy. Dwork et al. (2018); Ustun et al. (2019); Wang et al. (2021) considered training a separate classifier for each subgroup when their data distributions are different. Another line of research introduces data pre-processing techniques that manipulate data distribution for reducing its bias (e.g., Calmon et al., 2017; Kamiran and Calders, 2012). Among all these works, the closest one to ours is Chen et al. (2018), which decomposed group fairness measures into bias, variance, and noise (see their Theorem 1) and proposed strategies for reducing each term. Compared with Chen et al. (2018), the main difference is that we characterize a fairness Pareto frontier that depends on fairness metrics and a performance measure, giving a complete picture of how the data distribution influences fairness and accuracy.
Next, we introduce notation, overview the key results in Blackwell (1953) on comparisons of experiments, and outline the fair classification setup considered in this paper.
Notation. For a positive integer  $n$ , let  $[n] \triangleq \{1, \dots, n\}$ . We denote all probability distributions on the set  $\mathcal{X}$  by  $\mathcal{P}(\mathcal{X})$ . Moreover, we define the probability simplex  $\Delta_m \triangleq \mathcal{P}([m])$ . When random variables A, X, Z form a Markov chain, we write  $A - X - Z$ . We write the mutual information between A, X as  $I(A; X) \triangleq \mathbb{E}_{P_{A,X}}\left[\log \frac{P_{A,X}(A,X)}{P_A(A)P_X(X)}\right]$ . Since  $I(A; X)$  is determined by the marginal distribution  $P_A$  and the conditional distribution  $P_{X|A}$ , we also write  $I(A; X)$  as  $I(P_A; P_{X|A})$ . When A, X are independent, we write  $A \parallel X$ .
If a random variable  $\mathbf{A} \in [n]$  has finite support, the conditional distribution  $P_{\mathrm{X|A}}: [n] \to \mathcal{P}(\mathcal{X})$  can be equivalently written as  $P \triangleq (P_1, \dots, P_n)$  where each  $P_i = P_{\mathrm{X|A} = i} \in \mathcal{P}(\mathcal{X})$ . Additionally, if  $\mathcal{X}$  is a finite set  $[m]$ , then  $P_{\mathrm{X|A}}$  can be fully characterized by a transition matrix. We use  $\mathcal{T}(m|n)$  to denote all transition matrices from  $[n]$  to  $[m]$ :  $\left\{\boldsymbol{P} \in \mathbb{R}^{n \times m} \mid 0 \leq P_{i,j} \leq 1, \sum_{j=1}^{m} P_{i,j} = 1, \forall i \in [n]\right\}$ .
Comparisons of Experiments
Given two statistical experiments (i.e., conditional distributions)  $P$  and  $Q$ , is there a way to decide which one is more informative? Here  $P$  and  $Q$  have the common input alphabet  $[n]$  and potentially different output spaces. Blackwell gave an answer in his seminal work (Blackwell, 1953) from a decision-theoretic perspective. We review these results next.
FAIRNESS METRIC	ABBR.	DEFINITION
EXPRESSION W.R.T. P
Statistical Parity	SP ≤ αsp	|Pr(Ŷ =帽子|S = s) - Pr(Ŷ =帽子|S = s')| ≤ αsp
|ΣC y=1 (μs,y / μs P(s,y),帽子 - μs',y / μs' P(s',y),帽子)| ≤ αsp
Equalized Odds	EO ≤ αEO	|Pr(Ŷ =帽子|S = s, Y = y) - Pr(Ŷ =帽子|S = s', Y = y)| ≤ αEO
|P(s,y),帽子 - P(s',y),帽子| ≤ αEO
Overall Accuracy Equality	OAE ≤ αOAE	|Pr(Ŷ = Y|S = s) - Pr(Ŷ = Y|S = s')| ≤ αOAE
|ΣC y=1 (μs,y / μs P(s,y),帽子 - μs',y / μs' P(s',y),帽子)| ≤ αOAE
Table 1: Standard group fairness metrics under multi-group and multi-class classification tasks. Here  $\alpha_{\mathrm{SP}}, \alpha_{\mathrm{EO}}, \alpha_{\mathrm{OAE}}, \in [0,1]$  are threshold parameters,  $\hat{y}, y \in [C], s, s' \in [A]$ , and  $\mu_{s,y}, \mu_s$  are defined in Proposition 1. Our analysis can be extended to many other group fairness metrics (see e.g., Table 1 in Kim et al., 2020).
Let  $\mathcal{A}$  be a closed, bounded, convex subset of  $\mathbb{R}^n$ . A decision function  $\pmb{f}(x) = (a_1(x), \dots, a_n(x))$  is any mapping from  $\mathcal{X}$  to  $\mathcal{A}$ . It is associated with a loss vector:
$$
\boldsymbol {v} (\boldsymbol {f}) = \left(\int a _ {1} (x) \mathrm {d} P _ {1} (x), \dots , \int a _ {n} (x) \mathrm {d} P _ {n} (x)\right). \tag {1}
$$
The collection of all  $\pmb{v}(\pmb{f})$  is denoted by  $\mathcal{B}(P, \mathcal{A})$ . Blackwell defined that  $P$  is more informative than  $Q$  if for every  $\mathcal{A}$ ,  $\mathcal{B}(P, \mathcal{A}) \supseteq \mathcal{B}(Q, \mathcal{A})$ . Intuitively, this result means any risk achievable with  $Q$  is also achievable with  $P$ . Moreover, Blackwell considered the standard measure  $P^*$  which is the probability distribution of  $p(\bar{\mathrm{X}})$  where  $p(x): \mathcal{X} \to \Delta_n$  is a function defined as
$$
\left(\frac {\mathrm {d} P _ {1}}{\mathrm {d} P _ {1} + \cdots + \mathrm {d} P _ {n}}, \dots , \frac {\mathrm {d} P _ {n}}{\mathrm {d} P _ {1} + \cdots + \mathrm {d} P _ {n}}\right). \tag {2}
$$
and  $\overline{X}$  follows the probability distribution  $\frac{P_1 + \cdots + P_n}{n}$ . One of the most important findings by Blackwell in his paper is to discover the following equivalent conditions.
Lemma 1 (Blackwell (1951, 1953)). The following three conditions are equivalent:
-  $P$  is more informative than  $Q$ ;
- for any continuous and convex function  $\phi : \Delta_n \to \mathbb{R}$ ,  $\int \phi(\pmb{p}) dP^*(\pmb{p}) \geq \int \phi(\pmb{p}) dQ^*(\pmb{p})$ ;
- there is a stochastic transformation  $\top$  s.t.  $\top P_{i} = Q_{i}$ . In other words, there exists a Markov chain  $\mathrm{A} - \mathrm{X} - \mathrm{Z}$  for any distributions on  $\mathrm{A}$  such that  $\mathbf{P} = P_{\mathrm{X|A}}$  and  $\mathbf{Q} = P_{\mathrm{Z|A}}$ .
If  $P = P_{\mathrm{X|A}}$  is more informative than  $Q = P_{\mathrm{Z|A}}$ , by the third condition of Lemma 1 and the data processing inequality,  $I(P_{\mathrm{A}};P_{\mathrm{X|A}})\geq I(P_{\mathrm{A}};P_{\mathrm{Z|A}})$  holds for any marginal distribution  $P_{\mathrm{A}}$ . However, the converse does not hold in general—even if the above inequality holds for any  $P_{\mathrm{A}}$ ,  $P$  is not necessarily more informative than  $Q$  (Rauh et al., 2017). In this regard, Blackwell's conditions are "stronger" than the mutual information based data processing inequality.
Group Fair Classification
Consider a multi-class classification task, where the goal is to train a probabilistic classifier  $h: \mathcal{X} \to \Delta_C$  that uses input features  $X$  to predict their true label  $Y \in [C]$ . Additionally, assume the classifier produces a predicted outcome  $\hat{Y} \in [C]$  and let  $S \in [A]$  represent group attributes (e.g., race and sex). Depending on the domain of interest,  $X$  can either include or exclude  $S$  as an input to the classifier. Our framework can be easily extended to the setting where multiple subgroups overlap (Kearns et al., 2018). Throughout this paper, we focus on three standard group fairness measures: statistical parity (SP) (Feldman et al., 2015), equalized odds (EO) (Hardt et al., 2016; Pleiss et al., 2017), and overall accuracy equality (OAE) (Berk et al., 2021) (see Table 1 for their definitions) but our analysis can be extended to many other group fairness metrics, including the ones in Table 1 of Kim et al. (2020), as well as alternative performance measures beyond accuracy.
3 Fairness Pareto Frontier
In this section, we introduce our main concept—fairness Pareto frontier (FairFront). We use it to measure aleatoric discrimination and quantify epistemic discrimination by comparing a classifier's performance to the FairFront. We recast FairFront in terms of the conditional distribution  $P_{\hat{Y} |S,Y}$  and apply Blackwell's conditions to characterize the feasible region of this conditional distribution. This effort converts a functional optimization problem into a convex program with a small number of variables. However, this convex program may involve infinitely many constraints. Hence, we introduce a greedy improvement algorithm that iteratively refines the approximation of FairFront and tightens the feasible region of  $P_{\hat{Y} |S,Y}$ . Finally, we establish a convergence guarantee for our algorithm.
Recall that we refer to aleatoric discrimination as the inherent biases of the data distribution that can lead to an unfair or inaccurate classifier. As its definition suggests, aleatoric discrimination only relies on properties of the data distribution and fairness metric of choice—it does not depend on the hypothesis class nor optimization method. Below we introduce FairFront that delineates a curve of optimal accuracy over all probabilistic classifiers under certain fairness constraints for a given data distribution  $P_{\mathrm{S,X,Y}}$ . We use FairFront to quantify aleatoric discrimination.
Definition 1. For  $\alpha_{\mathrm{SP}}, \alpha_{\mathrm{EO}}, \alpha_{\mathrm{OAE}} \geq 0$  and a given  $P_{\mathrm{S,X,Y}}$ , we define FairFront  $(\alpha_{\mathrm{SP}}, \alpha_{\mathrm{EO}}, \alpha_{\mathrm{OAE}})$  by
$$
\operatorname {F a i r F r o n t} \left(\alpha_ {\mathrm {S P}}, \alpha_ {\mathrm {E O}}, \alpha_ {\mathrm {O A E}}\right) \triangleq \max  _ {h} \mathbb {E} \left[ \mathbb {I} _ {\hat {\mathrm {Y}} = \mathrm {Y}} \right] \tag {3a}
$$
$$
s. t. S P \leq \alpha_ {S P}, E O \leq \alpha_ {E O}, O A E \leq \alpha_ {O A E} \tag {3b}
$$
where  $\mathbb{I}$  is the indicator function;  $\hat{\mathrm{Y}}$  is produced by applying the classifier  $h$  to  $\mathbf{X}$ ; the maximum is taken over all measurable  $h$ ; and the definitions of SP, EO, and OAE are in Table 1. As a special case, if  $\alpha_{\mathrm{SP}}, \alpha_{\mathrm{EO}}, \alpha_{\mathrm{OAE}} \geq 1$ , then FairFront  $(\alpha_{\mathrm{SP}}, \alpha_{\mathrm{EO}}, \alpha_{\mathrm{OAE}})$  is the accuracy of the Bayes optimal classifier.
Solving this functional optimization problem is difficult since it optimizes over all measurable classifiers. There is a line of works that proposed different fairness-intervention algorithms for training group-fair classifiers (see e.g., Menon and Williamson, 2018; Zhang et al., 2018; Zafar et al., 2019; Celis et al., 2019; Yang et al., 2020; Wei et al., 2021; Alghamdi et al., 2022). They restrict the model class and vary loss functions and optimizers to find classifiers that approach FairFront as close as possible. However, these algorithms only describe a lower bound for FairFront. They do not determine what is the best achievable accuracy for a given set of fairness constraints.
We circumvent the above-mentioned challenges by rewriting FairFront in terms of the conditional distribution  $P_{\hat{\mathrm{Y}}|\mathrm{S},\mathrm{Y}}$ . The caveat is that although each classifier yields a  $P_{\hat{\mathrm{Y}}|\mathrm{S},\mathrm{Y}}$ , not every conditional distribution corresponds to a valid classifier. Hence, we introduce the following definition which characterizes all feasible  $P_{\hat{\mathrm{Y}}|\mathrm{S},\mathrm{Y}}$ .
Definition 2. Given  $P_{\mathrm{X|S,Y}}$ , we define  $\mathcal{C}$  as the set of all conditional distributions  $P_{\hat{\mathrm{Y}}|\mathrm{S},\mathrm{Y}}$  where  $\hat{\mathrm{Y}}$  is produced by some probabilistic classifier  $h$ . In other words,
$$
\mathcal {C} \triangleq \left\{P _ {\hat {\mathrm {Y}} | \mathrm {S}, \mathrm {Y}} \mid (\mathrm {S}, \mathrm {Y}) - \mathrm {X} - \hat {\mathrm {Y}} \right\}. \tag {4}
$$
Throughout this paper, we write  $P_{\hat{\mathrm{Y}}|\mathrm{S},\mathrm{Y}}$  or its corresponding transition matrix  $\pmb{P} \in \mathcal{T}(C|AC)$  interchangeably. Specifically, the  $(C(s - 1) + y)$ -th row,  $\hat{y}$ -th column of  $\pmb{P}$  represents  $P_{\hat{\mathrm{Y}}|\mathrm{S},\mathrm{Y}}(\hat{y}|s,y)$  and we denote it by  $P_{(s,y),\hat{y}}$ .
Remark 1. We demonstrate the connection between the conditional distribution  $P_{\hat{\mathrm{Y}}|\mathrm{S},\mathrm{Y}}$  and confusion matrices in the setting of binary classification with binary groups. We define  $\hat{\mathcal{C}}$  as the counterpart of  $\mathcal{C}$  when we replace  $P_{\mathrm{X|S,Y}}$  with an empirical distribution  $\hat{P}_{\mathrm{X|S,Y}}$  computed from a dataset. The confusion matrix for group  $s \in \{0,1\}$  consists of four numbers: True Positive ( $\mathsf{TP}_s$ ), False Positive ( $\mathsf{FP}_s$ ), False Negative ( $\mathsf{FN}_s$ ), True Negative ( $\mathsf{TN}_s$ ). Assume that the number of positive-label data  $n_s^+ = \mathsf{TP}_s + \mathsf{FN}_s$  and negative-label data  $n_s^- = \mathsf{TN}_s + \mathsf{FP}_s$  are given—these numbers do not depend on the classifier. Then there is a one-to-one mapping from each element in  $\hat{\mathcal{C}}$  to a confusion matrix:
$$
\hat {P} _ {\mathrm {Y} | \mathrm {S}, \mathrm {Y}} (1 | s, 1) = \frac {1}{n _ {s} ^ {+}} \mathsf {T P} _ {s}, \quad \hat {P} _ {\mathrm {Y} | \mathrm {S}, \mathrm {Y}} (1 | s, 0) = \frac {1}{n _ {s} ^ {-}} \mathsf {F P} _ {s},
$$
$$
\hat {P} _ {\hat {Y} | \mathrm {S}, \mathrm {Y}} (0 | s, 1) = \frac {1}{n _ {s} ^ {+}} \mathsf {F N} _ {s}, \quad \hat {P} _ {\hat {Y} | \mathrm {S}, \mathrm {Y}} (0 | s, 0) = \frac {1}{n _ {s} ^ {-}} \mathsf {T N} _ {s}.
$$
Hence,  $\hat{\mathcal{C}}$  essentially characterizes all feasible confusion matrices and  $\mathcal{C}$  is the population counterpart of  $\hat{\mathcal{C}}$ . Note that  $\mathcal{C}$  is determined by the underlying data distribution while  $\hat{\mathcal{C}}$  (and confusion matrices) are tailored to a specific dataset.
Proposition 1. FairFront $(\alpha_{SP}, \alpha_{EO}, \alpha_{OAE})$  in (3) is equal to the solution of the following convex optimization:
$$
\max  _ {\boldsymbol {P} \in \mathbb {R} ^ {A C \times C}} \sum_ {s = 1} ^ {A} \sum_ {y = 1} ^ {C} \mu_ {s, y} P _ {(s, y), y} \tag {5a}
$$
$$
s. t. \mathsf {S P} \leq \alpha_ {\mathrm {S P}}, \mathrm {E O} \leq \alpha_ {\mathrm {E O}}, \mathrm {O A E} \leq \alpha_ {\mathrm {O A E}} \tag {5b}
$$
$$
\boldsymbol {P} \in \mathcal {C}. \tag {5c}
$$
Here the constants  $\mu_{s,y} \triangleq \operatorname{Pr}(\mathbf{S} = s, \mathbf{Y} = y)$  and  $\mu_s \triangleq \operatorname{Pr}(\mathbf{S} = s)$  for  $s \in [A]$ ,  $y \in [A]$  and  $P_{(s,y),\hat{y}}$  denotes the  $(C(s - 1) + y)$ -th row,  $\hat{y}$ -th column of the transition matrix  $\pmb{P}$ , which is  $P_{\hat{Y}|\mathbf{S},\mathbf{Y}}(\hat{y}|s,y)$ .