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@@ -269,7 +269,7 @@ Figure 2: The data-generating process for the observational data $\left\{ X ^ {
 214 Gibbs sampling is a well-known MCMC algorithm that allows one to sample posterior distributions.   
 215 For convenience, we introduce the following notations. Let parameters $\pmb { \theta } = \{ \theta _ { u } | \forall U \in U , \forall u \}$   
 216 and ⇠ = n ⇠ (paV ,uV )V | $\bigstar \bigstar = \Big \{ \xi _ { V } ^ { ( p a _ { V } , u _ { V } ) } | \forall V \in V , \forall p a _ { V } , u _ { V } \Big \}$ . The set $\bar { U } = \left\{ U ^ { ( n ) } \right\} _ { n = 1 } ^ { N }$ are exogenous variables   
-217 affecting N observations V¯ =  V (n) N n=1; we use u¯ to represent their realizations. Our blocked   
+217 affecting N observations V¯ =  V (n) N n=1; we use u¯ to represent their realizations. Our blocked   
 218 Gibbs sampler works by iteratively drawing values from the conditional distributions of variables as   
 219 follows $\lVert \hat { 2 2 } \rVert$ . Detailed derivations of complete conditional distributions are shown in Appendix F.   
 220 Sampling $P \left( \bar { \pmb { u } } | \bar { \pmb { v } } , \pmb { \theta } , \pmb { \xi } \right)$ . Exogenous variables $U ^ { ( n ) }$ , $n = 1 , \ldots , N$ , are mutually independent   
@@ -351,7 +351,7 @@ $$
 260 observations $\underline { { \| 2 0 \| } } , \boxed { 4 7 } , \boxed { 3 7 } , \boxed { 8 } , \boxed { 4 6 } $ . As the number of observational data $N$ grows (to infinite), the $1 0 0 \%$   
 261 credible interval $[ l _ { 0 } , r _ { 0 } ]$ eventually converges to the optimal asymptotic bound $[ l , r ]$ in Eq. $( 6 )$ [11].
 
-Let  ✓(t) T be $T$ samples drawn from $P \left( \theta _ { \mathrm { c t f } } \mid \bar { \mathbf { v } } \right)$ . One could compute the $1 0 0 ( 1 - \alpha ) \%$ credible interval for $\bar { \theta _ { \mathrm { c t f } } }$ using the following consistent estimators $\pmb { \| 3 9 \| }$ :
+Let  ✓(t) T be $T$ samples drawn from $P \left( \theta _ { \mathrm { c t f } } \mid \bar { \mathbf { v } } \right)$ . One could compute the $1 0 0 ( 1 - \alpha ) \%$ credible interval for $\bar { \theta _ { \mathrm { c t f } } }$ using the following consistent estimators $\pmb { \| 3 9 \| }$ :
 
 $$
 \hat { l } _ { \alpha } ( T ) = \theta ^ { ( \lceil ( \alpha / 2 ) T \rceil ) } , \hat { r } _ { \alpha } ( T ) = \theta ^ { ( \lceil ( 1 - \alpha / 2 ) T \rceil ) } ,
@@ -417,7 +417,7 @@ generated SCM. Fig. 4a shows samples drawn from the posterior distribution of th
 311 unobserved confounding between $X$ and $Y$ has been acknowledged in [5]. For binary $X , Y , Z$ , [2]   
 312 derived closed-form, sharp bounds over $P ( y _ { x } )$ (labelled as opt). We collect $N = 1 0 ^ { 5 }$ observational   
 313 samples $\bar { \cal V } = \{ X ^ { ( n ) } , Y ^ { ( n ) } , Z ^ { ( n ) } \} _ { n = 1 } ^ { N }$ from a randomly generated SCM instance. Fig. $4  { \mathbf { b } }$ shows   
-314 samples drawn from the posterior distribution of  $P ( Y _ { x = 0 } = 1 ) \mid \bar { V } )$ . As a baseline, we also include   
+314 samples drawn from the posterior distribution of  $P ( Y _ { x = 0 } = 1 ) \mid \bar { V } )$ . As a baseline, we also include   
 315 the optimal bound opt, and posterior samples obtained from the Gibbs sampler of $\mathbb { \ m }$ , which utilizes   
 316 the canonical partitions of exogenous domains in $\pmb { \Vert 2 \Vert }$ $( b p )$ . The analysis reveals that our algorithm   
 317 derives the valid bound over the actual probability $P ( Y _ { x = 0 } = 1 ) = 0 . 3 9 5 4$ ; the $1 0 0 \%$ credible