Algorithm 1 Causal Explanation in Stochastic Processes

$$\begin{gathered} \begin{array}{cc}\hfill 1:& \mathbf{function}\text{EXPLAIN}(B,A)\hfill \\ \hfill 2:& (W_{\text{CF}},W_{\text{DEF}})\leftarrow \text{CHARACTERISTICSAMPLES}(A)\hfill \\ \hfill 3:& P_{\text{CF}}\leftarrow \frac{1}{n}\mid W_{\text{CF}}\cap B\mid \hfill \\ \hfill 4:& P_{\text{DEF}}\leftarrow \frac{1}{n}\mid W_{\text{DEF}}\cap B\mid \hfill \\ \hfill 5:& P_{\Delta }\leftarrow P_{\text{DEF}}-P_{\text{CF}}\hfill \\ \hfill 6:& p_P\leftarrow \text{LOGLIKELIHOODTEST}(n,n{P}_{\text{CF}},n{P}_{\text{DEF}})\hfill \\ \hfill 7:& t_{\text{CF}}\leftarrow \frac{1}{n}{\sum }_{w\in Wcf}{t}_B(w)\hfill \\ \hfill 8:& t_{\text{DEF}}\leftarrow \frac{1}{n}{\sum }_{w\in Wdef}{t}_B(w)\hfill \\ \hfill 9:& t_{\Delta }\leftarrow t_{\text{DEF}}-t_{\text{CF}}\hfill \\ \hfill 10:& p_t\leftarrow \text{WELCHSTTEST}(t_B(W_{\text{CF}}),t_B(W_{\text{DEF}}))\hfill \\ \hfill 11:& m_{\text{CF}}\leftarrow \frac{1}{n}{\sum }_{w\in Wcf}{m}_B(w)\hfill \\ \hfill 12:& m_{\text{DEF}}\leftarrow \frac{1}{n}{\sum }_{w\in Wdef}{m}_B(w)\hfill \\ \hfill 13:& m_{\Delta }\leftarrow m_{\text{DEF}}-m_{\text{CF}}\hfill \\ \hfill 14:& p_m\leftarrow \text{WELCHSTTEST}(m_B(W_{\text{CF}}),m_B(W_{\text{DEF}}))\hfill \\ \hfill 15:& \mathbf{return}(P_{\Delta },p_P),(t_{\Delta },p_t),(m_{\Delta },p_m)\hfill \\ \hfill 16:& \mathbf{function}\text{CHARACTERISTICSAMPLES}(A)\hfill \\ \hfill 17:& t\leftarrow t_A-\tau \hfill \\ \hfill 18:& W_{\text{CF}}\leftarrow \varnothing \hfill \\ \hfill 19:& W_{\text{DEF}}\leftarrow \varnothing \hfill \\ \hfill 20:& k\leftarrow 0\hfill \\ \hfill 21:& \mathbf{while}k<n\mathbf{do}\hfill \\ \hfill 22:& \\ \mathbf{repeat}\hfill \\ \hfill 23:& w_{\text{CF}}\leftarrow \text{sample from}{P}_t()\hfill \\ \hfill 24:& \\ \mathbf{until}{w}_{\text{CF}}\notin A\hfill \\ \hfill 25:& \\ \mathbf{repeat}\hfill \\ \hfill 26:& w_{\text{DEF}}\leftarrow \text{sample from}{P}_t()\hfill \\ \hfill 27:& \\ \mathbf{until}{w}_{\text{DEF}}\in A\hfill \\ \hfill 28:& \\ {W}_{\text{CF}}\leftarrow W_{\text{CF}}\cup w_{\text{CF}}\hfill \\ \hfill 29:& \\ {W}_{\text{DEF}}\leftarrow W_{\text{DEF}}\cup w_{\text{DEF}}\hfill \\ \hfill 30:& \\ k\leftarrow k+1\hfill \\ \hfill 31:& \mathbf{return}(W_{\text{CF}},W_{\text{DEF}})\hfill \end{array} \end{gathered}$$