# 4.2.2 variable that simultaneously correlates with the cause

4.2.2 Integrating Confounders

A target pollutant is likely to have several

different causal pathways under different environmental conditions, which

indicate the causal pathways we learn may be biased and may not reflect the

real reactions or propagations of pollutants. To overcome this, it is necessary

to model the environmental factors (humidity, wind, etc.) as extraneous

variables in the causality model, which simultaneously influence the cause and

effect we will elaborate how to integrate the environmental factors into the

GBN-based graphical model, to minimize the biases in causality analysis and

guarantee the causal pathways are faithful for the government’s decision

making. We first introduce the definition of confounder and then elaborate the

integration.

Confounder. A confounder is defined as a third

variable that simultaneously correlates with the cause and effect, e.g. gender

K may affect the effect of recovery P given a medicine Q, Ignoring the

confounders will lead to biased causality analysis. To guarantee an unbiased

causal inference, the cause-and-effect is usually adjusted by averaging all the

sub-classification cases of K 11,integrating environmental

factors as confounders, denoted as Et = {E(1) t ;E(2) t …..}g, into the GBN-based causal pathways, one challenge is there can

be too many sub-classifications of environmental statuses. For example, if

there are 5 environmental factors and each factor has 4 statuses, there will

exist 45 = 1024 causal pathways for each sub-classification case. Directly

integrating Et as confounders to the cause and effect will result in unreliable

causality analysis due to very few sample data conditioned on each

sub-classification case. Therefore, we introduce a discrete hidden confounding

variable K, which determines the

probabilities of different causal pathways from Qt to Pt, . The environmental factors Et are further integrated into K, where K = 1; 2……K. In this ways, the large number of sub-classification cases of

confounders will be greatly reduced to a small number K, as K clusters of the environmental factors. Based on Markov

equivalence (DAGs which share the same joint probability distribution 10), we

can reverse the arrow Et K to K Et, as shown in the right part of K determines the distributions of P,Qt;Et, thus enabling us to learn the distribution of the graphical

model from a generative process. To help us learn the hidden variable K, the generative process further introduces a hyper-parameter that

determines the distribution of K. Thus the graphical model

can be understood as a mixture model under K clusters. We learn the parameters

of the graphical model by maximizing the new log likelihood: