Bounds on Causal Effects and Application to High Dimensional Data

Notably, if any observational data (e.g., $P(U)$) are unavailable in the above theorems, we can remove that term, and the rest of non-linear optimization problems still provide valid bounds for the causal effects. In general, midpoints of bounds on causal effects are effective estimates. However, the lower (upper) bounds are also informative, which can be interpreted as the minimal (maximal) causal effects. The proofs of Theorems 4 and 5 are provided in the appendix.

Drug manufacturers want to know the causal effect of recovery when a drug is taken. Thus, they conduct an observational study. Here, the recovery rates of $700$ patients were recorded. A total of $192$ patients chose to take the drug and $508$ patients did not. The results of the study are shown in Table 2. Blood type (type O or not) is not the only confounder of taking the drug and recovery. Another confounder is age (below the age of $70$ or not). The manufacturers have no data associated with age. They only know that $85.43\%$ of people in their region are below the age of $70$.

Because both age and blood type are confounders of taking the drug and recovery, and the observational data associated with age are unobservable, the causal effect is not identifiable.

Let $X=x$ denote the event that a patient took the drug, and $X=x^{\prime}$ denote the event that a patient did not take the drug. Let $Y=y$ denote the event that a patient recovered, and $Y=y^{\prime}$ denote the event that a patient did not recover. Let $W=w$ represent a patient with blood type O, and $W=w^{\prime}$ represent a patient without blood type O. Let $U=u$ represent a patient below the age of $70$, and $U=u^{\prime}$ represent a patient above the age of $70$. The causal diagram is shown in Figure 1(d).

An option for the manufacturers could be estimating the causal effect through the Tian-Pearl bounds in Equation 3 and the observational data from Table 2, where

Therefore, the bounds on the causal effect estimated using Equation 3 are $0.2257\leq P(y|do(x))\leq 0.9514$, where the causal information of the covariate $W$ and the prior information $P(U)$ are not used. These bounds are not sufficiently informative to conclude the actual causal effect. Although one may believe that we can use the midpoint of the bounds (i.e., $0.5886$), the gap (i.e., $0.9514-0.2257=0.7257$) between the bounds is not small; hence, this point estimate is unconvincing.

Now, considering the proposed bounds in Theorem 4 with the observational data from Table 2. $W\cup U$ satisfies the back-door criterion, and $P(X,Y,W)$ and $P(U)$ are available. We have $12$ optimal variables in each objective function, because $W$ and $U$ are binary. With the help of the “SLSQP” solver [Kraft(1988)] in the scipy package [SciPyCommunity(2020)], we obtain the bounds on the causal effect, which are $0.4728\leq P(y|do(x))\leq 0.9514$. The lower bound actually increased significantly, and reached close to $0.5$, which can help make decisions. The midpoint is $0.7121$. Our conclusion is then that the causal effect of recovery when taking the drug is $0.7121$. We show in the following section that this estimate of the causal effect is extremely close to the actual causal effect.

An informer with access to the fully observed observational data, as summarized in Table 2 (Note that although it can be verified that the data in Table 2 are compatible with those in Table 2, we will never know these numbers in reality), would easily calculate the causal effect of recovery when taking the drug using the adjustment formula in Equation 1 (shown in Equation 8). The error of the estimate of the causal effect using Theorem 4 is only $(0.7518-0.7121)/0.7518\approx 5.28\%$.

Here, we further illustrate that the midpoints of the proposed bounds on causal effects are sufficient for estimating the causal effects, and the midpoints of the proposed bounds in Theorem 4 are better than the midpoints of the Tian-Pearl bounds in Equation 3 based on a random simulation.

We employ the simplest causal diagram in Figure 1(a) with binary $W$, $U$, such that $W\cup U$ satisfies the back-door criterion. We randomly generated $1000$ sample distributions compatible with the causal diagram (the algorithm for generating the sample distributions is shown in the appendix). The average gap (upper bound $-$ lower bound) of the Tian-Pearl bounds among $1000$ samples is $0.487$, and the average gap of the proposed bounds among $1000$ samples is $0.383$. We then randomly picked $100$ out of $1000$ sample distributions to draw the graph of the actual causal effects, the midpoints of the Tian-Pearl bounds, and the midpoints of the proposed bounds. The results are shown in Figure 2(a).

From Figure 2(a), although both midpoints of the bounds on the causal effects are good estimates of the actual causal effects, the midpoints of the proposed bounds are much closer to the actual causal effects, particularly when the causal effects are close to $0$ and $1$. The average gap (upper bounds $-$ lower bounds), $0.383$, of the proposed bounds among $1000$ samples is much smaller than the average gap, $0.487$, of the Tian-Pearl bounds among $1000$ samples. This means that the midpoints of the proposed bounds are more convincing, because the bounds are narrower.

This equivalent tuple $(G^{\prime},O^{\prime})$ is easy to obtain. We can simply add two new nodes $W$ and $U$, and remove a node $Z$ in $G$ to obtain $G^{\prime}$. Let the arrows entering $Z$ in $G$ now enter both $W$ and $U$ in $G^{\prime}$, and let the arrows exiting $Z$ in $G$ now exit both $W$ and $U$ in $G^{\prime}$. Finally, add an arrow from $U$ to $W$. It is easy to show that $(G,O)$ and $(G^{\prime},O^{\prime})$ are equivalent if the states of $Z$ are the Cartesian product of the states of $W$ and the states of $U$. Formally, we have the following theorem (the proof of the theorem is provided in the appendix),

For example, consider the causal diagram in Figure 1(b) and the observational data (in the form of conditional probability tables (CPTs), where $X,Y$ are binary, and $Z$ has $4$ states.) in Table 4. The causal effect, $P(y|do(x))$, through the adjustment formula in Equation 1, is $0.47$. Based on the construction in Theorem 7 (see the appendix for details), we have the causal diagram in Figure 1(b) with the observational data in Table 4 is equivalent to the causal diagram in Figure 1(c) with the observational data in Table 4 (all nodes are binary), and we can verify that the causal effect, $P(y|do(x))$, in the causal diagram in Figure 1(c) with the observational data in Table 4 is also $0.47$.

Notably, the equivalent tuple is not unique and is transitive (i.e., if $(G,O)$ is equivalent to $(G^{\prime},O^{\prime})$, and $(G^{\prime},O^{\prime})$ is equivalent to $(G^{\prime\prime},O^{\prime\prime})$, then $(G,O)$ is equivalent to $(G^{\prime\prime},O^{\prime\prime})$).

Now, considering the problem in the beginning of Section 5. First, we transform the causal diagram $G$ with the compatible observational data $O$ into an equivalent tuple $(G^{\prime},O^{\prime})$ using Algorithm 1 based on the construction in Theorem 7 (note that the algorithm only construct the structure of the $G^{\prime}$ and assigning the meaning of the states for $W,U$, the corresponding observatioal data $O^{\prime}$ are then easy to obtain), then the new problem $(G^{\prime},O^{\prime})$ has the same causal effects of $X$ on $Y$ as in $(G,O)$. By picking the dimensionality of $W$ ($p$ in Algorithm 1), we can control the dimensionality of the new problem.

Note that, if $Z=(Z_{1},Z_{2},...,Z_{m})$ in $G$ is a set of variables, we can repeat Algorithm 1 for each variable in $Z$, and finally obtain $W=(W_{1},W_{2},...,W_{m})$ and $U=(U_{1},U_{2},...,U_{m})$, where the multiplication of the number of states in $W$ is equal to $p$.

We then treat the new problem $(G^{\prime},O^{\prime})$ as a partially observable back-door or front-door variables problem in Sections 3.1 and 3.2, where $P(X,Y,W)$ and $P(U)$ are given, and we can then obtain the bounds of the causal effects through Theorems 4 and 5. We claim that the midpoints of the bounds are good estimates of the original causal effects. In addition, the bounds themselves will help make decisions.

Consider the problem in Figure 1(b), where $X$ and $Y$ are binary and $Z$ has $256$ states. We randomly generated a distribution $P(X,Y,Z)$ that is compatible with the causal diagram. Because we know the exact distribution, we can easily obtain the causal effects through Equation 1. The causal effect $P(y|do(x))$ is $0.5527$ (the algorithm for generating the distribution is shown in the appendix).

Now, we transform the causal diagram with the observational data into an equivalent tuple $(G^{\prime},O^{\prime})$ ($G^{\prime}$ is shown in Figure 1(c)) using Algorithm 1 ($p=16$). We obtain the variable $W$ of $16$ states and the variable $U$ of $16$ states in $G^{\prime}$ ($(w_{j},u_{k})$ is equivalent to $z_{(j-1)*16+k}$). We are then forced to use only observational data $P(X,Y,W)$ and $P(U)$ (the construction of $P(X,Y,W)$ and $P(U)$ is shown in the appendix), and based on Theorem 4, with the “SLSQP” solver, we obtain the bounds on the causal effect $p(y|do(x))$, which are $0.4595\leq P(y|do(x))\leq 0.7012$. We see the midpoint, $0.5804$, is extremely close to the actual causal effect, $0.5527$.

Finally, lets consider how many samples are required for each method. According to [Roscoe(1975)], each state needs at least $30$ samples, and therefore, the exact solution by Equation 1 requires $2\times 2\times 256\times 30=30720$ samples. However, the proposed bounds based on Theorem 4 only requires $max(2\times 2\times 16,16)\times 30=1920$ samples. If the sample size is still unacceptable, we can use another equivalent tuple with $W$ having $8$ states and $U$ having $32$ states, we then only require $max(2\times 2\times 8,32)\times 30=960$ samples to obtain the bounds on the causal effects.

Similarly to the previous simulation, we further illustrate that the bounds on the causal effects of the proposed framework are sufficient for estimating the original causal effects.

Once again, by employing the simplest causal diagram in Figure 1(b), where $X$ and $Y$ are binary and $Z$ has $256$ states. We randomly generated $100$ sample distributions compatible with the causal diagram (the algorithm for generating the distributions are shown in the appendix). The average gap (upper bound $-$ lower bound) of the Tian-Pearl bounds among $100$ samples is $0.5102$, and the average gap of the proposed bounds through Theorems 7 and 4 among $100$ samples is $0.0676$. We then draw the graph of the actual causal effects, the midpoints of the Tian-Pearl bounds, and the midpoints of the proposed bounds through Theorems 7 and 4. The results are shown in Figure 2(b).

From Figure 2(b), both midpoints of the bounds on the causal effects are good estimates of the actual causal effects, whereas the midpoints of the proposed bounds are slightly closer to the actual causal effects, particularly when the causal effects are close to $0$ and $1$. Although the trend of the Tian-Pearl bounds is also close to the actual causal effects, the Tian-Pearl bounds are more likely to be parallel with the x-axis. Here, the Tian-Pearl bounds perform well because, in high-dimensionality cases, the randomly generated distributions are more likely to yield causal effects of approximately $0.5$. However, the average gap of the proposed bounds among $100$ samples, $0.0676$, is much smaller than the average gap of the Tian-Pearl bounds among $100$ samples, $0.5102$. This means that the midpoints of the proposed bounds are more convincing, because the bounds are narrower.