btje@dZddlmZddlmZddlZddlmZm Z gdZ e dej dZ e dej ddd d Z e dej ddd d Ze dd ZdS) ul' Algorithm for testing d-separation in DAGs. *d-separation* is a test for conditional independence in probability distributions that can be factorized using DAGs. It is a purely graphical test that uses the underlying graph and makes no reference to the actual distribution parameters. See [1]_ for a formal definition. The implementation is based on the conceptually simple linear time algorithm presented in [2]_. Refer to [3]_, [4]_ for a couple of alternative algorithms. The functional interface in NetworkX consists of three functions: - `find_minimal_d_separator` returns a minimal d-separator set ``z``. That is, removing any node or nodes from it makes it no longer a d-separator. - `is_d_separator` checks if a given set is a d-separator. - `is_minimal_d_separator` checks if a given set is a minimal d-separator. D-separators ------------ Here, we provide a brief overview of d-separation and related concepts that are relevant for understanding it: The ideas of d-separation and d-connection relate to paths being open or blocked. - A "path" is a sequence of nodes connected in order by edges. Unlike for most graph theory analysis, the direction of the edges is ignored. Thus the path can be thought of as a traditional path on the undirected version of the graph. - A "candidate d-separator" ``z`` is a set of nodes being considered as possibly blocking all paths between two prescribed sets ``x`` and ``y`` of nodes. We refer to each node in the candidate d-separator as "known". - A "collider" node on a path is a node that is a successor of its two neighbor nodes on the path. That is, ``c`` is a collider if the edge directions along the path look like ``... u -> c <- v ...``. - If a collider node or any of its descendants are "known", the collider is called an "open collider". Otherwise it is a "blocking collider". - Any path can be "blocked" in two ways. If the path contains a "known" node that is not a collider, the path is blocked. Also, if the path contains a collider that is not a "known" node, the path is blocked. - A path is "open" if it is not blocked. That is, it is open if every node is either an open collider or not a "known". Said another way, every "known" in the path is a collider and every collider is open (has a "known" as a inclusive descendant). The concept of "open path" is meant to demonstrate a probabilistic conditional dependence between two nodes given prescribed knowledge ("known" nodes). - Two sets ``x`` and ``y`` of nodes are "d-separated" by a set of nodes ``z`` if all paths between nodes in ``x`` and nodes in ``y`` are blocked. That is, if there are no open paths from any node in ``x`` to any node in ``y``. Such a set ``z`` is a "d-separator" of ``x`` and ``y``. - A "minimal d-separator" is a d-separator ``z`` for which no node or subset of nodes can be removed with it still being a d-separator. The d-separator blocks some paths between ``x`` and ``y`` but opens others. Nodes in the d-separator block paths if the nodes are not colliders. But if a collider or its descendant nodes are in the d-separation set, the colliders are open, allowing a path through that collider. Illustration of D-separation with examples ------------------------------------------ A pair of two nodes, ``u`` and ``v``, are d-connected if there is a path from ``u`` to ``v`` that is not blocked. That means, there is an open path from ``u`` to ``v``. For example, if the d-separating set is the empty set, then the following paths are open between ``u`` and ``v``: - u <- n -> v - u -> w -> ... -> n -> v If on the other hand, ``n`` is in the d-separating set, then ``n`` blocks those paths between ``u`` and ``v``. Colliders block a path if they and their descendants are not included in the d-separating set. An example of a path that is blocked when the d-separating set is empty is: - u -> w -> ... -> n <- v The node ``n`` is a collider in this path and is not in the d-separating set. So ``n`` blocks this path. However, if ``n`` or a descendant of ``n`` is included in the d-separating set, then the path through the collider at ``n`` (... -> n <- ...) is "open". D-separation is concerned with blocking all paths between nodes from ``x`` to ``y``. A d-separating set between ``x`` and ``y`` is one where all paths are blocked. D-separation and its applications in probability ------------------------------------------------ D-separation is commonly used in probabilistic causal-graph models. D-separation connects the idea of probabilistic "dependence" with separation in a graph. If one assumes the causal Markov condition [5]_, (every node is conditionally independent of its non-descendants, given its parents) then d-separation implies conditional independence in probability distributions. Symmetrically, d-connection implies dependence. The intuition is as follows. The edges on a causal graph indicate which nodes influence the outcome of other nodes directly. An edge from u to v implies that the outcome of event ``u`` influences the probabilities for the outcome of event ``v``. Certainly knowing ``u`` changes predictions for ``v``. But also knowing ``v`` changes predictions for ``u``. The outcomes are dependent. Furthermore, an edge from ``v`` to ``w`` would mean that ``w`` and ``v`` are dependent and thus that ``u`` could indirectly influence ``w``. Without any knowledge about the system (candidate d-separating set is empty) a causal graph ``u -> v -> w`` allows all three nodes to be dependent. But if we know the outcome of ``v``, the conditional probabilities of outcomes for ``u`` and ``w`` are independent of each other. That is, once we know the outcome for ``v``, the probabilities for ``w`` do not depend on the outcome for ``u``. This is the idea behind ``v`` blocking the path if it is "known" (in the candidate d-separating set). The same argument works whether the direction of the edges are both left-going and when both arrows head out from the middle. Having a "known" node on a path blocks the collider-free path because those relationships make the conditional probabilities independent. The direction of the causal edges does impact dependence precisely in the case of a collider e.g. ``u -> v <- w``. In that situation, both ``u`` and ``w`` influence ``v``. But they do not directly influence each other. So without any knowledge of any outcomes, ``u`` and ``w`` are independent. That is the idea behind colliders blocking the path. But, if ``v`` is known, the conditional probabilities of ``u`` and ``w`` can be dependent. This is the heart of Berkson's Paradox [6]_. For example, suppose ``u`` and ``w`` are boolean events (they either happen or do not) and ``v`` represents the outcome "at least one of ``u`` and ``w`` occur". Then knowing ``v`` is true makes the conditional probabilities of ``u`` and ``w`` dependent. Essentially, knowing that at least one of them is true raises the probability of each. But further knowledge that ``w`` is true (or false) change the conditional probability of ``u`` to either the original value or 1. So the conditional probability of ``u`` depends on the outcome of ``w`` even though there is no causal relationship between them. When a collider is known, dependence can occur across paths through that collider. This is the reason open colliders do not block paths. Furthermore, even if ``v`` is not "known", if one of its descendants is "known" we can use that information to know more about ``v`` which again makes ``u`` and ``w`` potentially dependent. Suppose the chance of ``n`` occurring is much higher when ``v`` occurs ("at least one of ``u`` and ``w`` occur"). Then if we know ``n`` occurred, it is more likely that ``v`` occurred and that makes the chance of ``u`` and ``w`` dependent. This is the idea behind why a collider does no block a path if any descendant of the collider is "known". When two sets of nodes ``x`` and ``y`` are d-separated by a set ``z``, it means that given the outcomes of the nodes in ``z``, the probabilities of outcomes of the nodes in ``x`` are independent of the outcomes of the nodes in ``y`` and vice versa. Examples -------- A Hidden Markov Model with 5 observed states and 5 hidden states where the hidden states have causal relationships resulting in a path results in the following causal network. We check that early states along the path are separated from late state in the path by the d-separator of the middle hidden state. Thus if we condition on the middle hidden state, the early state probabilities are independent of the late state outcomes. >>> G = nx.DiGraph() >>> G.add_edges_from( ... [ ... ("H1", "H2"), ... ("H2", "H3"), ... ("H3", "H4"), ... ("H4", "H5"), ... ("H1", "O1"), ... ("H2", "O2"), ... ("H3", "O3"), ... ("H4", "O4"), ... ("H5", "O5"), ... ] ... ) >>> x, y, z = ({"H1", "O1"}, {"H5", "O5"}, {"H3"}) >>> nx.is_d_separator(G, x, y, z) True >>> nx.is_minimal_d_separator(G, x, y, z) True >>> nx.is_minimal_d_separator(G, x, y, z | {"O3"}) False >>> z = nx.find_minimal_d_separator(G, x | y, {"O2", "O3", "O4"}) >>> z == {"H2", "H4"} True If no minimal_d_separator exists, `None` is returned >>> other_z = nx.find_minimal_d_separator(G, x | y, {"H2", "H3"}) >>> other_z is None True References ---------- .. [1] Pearl, J. (2009). Causality. Cambridge: Cambridge University Press. .. [2] Darwiche, A. (2009). Modeling and reasoning with Bayesian networks. Cambridge: Cambridge University Press. .. [3] Shachter, Ross D. "Bayes-ball: The rational pastime (for determining irrelevance and requisite information in belief networks and influence diagrams)." In Proceedings of the Fourteenth Conference on Uncertainty in Artificial Intelligence (UAI), (pp. 480–487). 1998. .. [4] Koller, D., & Friedman, N. (2009). Probabilistic graphical models: principles and techniques. The MIT Press. .. [5] https://en.wikipedia.org/wiki/Causal_Markov_condition .. [6] https://en.wikipedia.org/wiki/Berkson%27s_paradox )deque)chainN) UnionFindnot_implemented_for)is_d_separatoris_minimal_d_separatorfind_minimal_d_separator undirectedc |vr|hn|}|vr|hn|}|vr|hn|}||zp ||zp||z}|rtjd|||z|z}|jz r tjd|jz dn"#t$rtjdwxYwtjstjdt g}t}t |}t} tjfd|D|z|z} |s|rI|r| } | | | |vrdS| |vr;| j |  | z | j|  |z |r| } | | | |vrdS| | vr5| j |  | z | |vr5| j|  |z |F|IdS) apReturn whether node sets `x` and `y` are d-separated by `z`. Parameters ---------- G : nx.DiGraph A NetworkX DAG. x : node or set of nodes First node or set of nodes in `G`. y : node or set of nodes Second node or set of nodes in `G`. z : node or set of nodes Potential separator (set of conditioning nodes in `G`). Can be empty set. Returns ------- b : bool A boolean that is true if `x` is d-separated from `y` given `z` in `G`. Raises ------ NetworkXError The *d-separation* test is commonly used on disjoint sets of nodes in acyclic directed graphs. Accordingly, the algorithm raises a :exc:`NetworkXError` if the node sets are not disjoint or if the input graph is not a DAG. NodeNotFound If any of the input nodes are not found in the graph, a :exc:`NodeNotFound` exception is raised Notes ----- A d-separating set in a DAG is a set of nodes that blocks all paths between the two sets. Nodes in `z` block a path if they are part of the path and are not a collider, or a descendant of a collider. Also colliders that are not in `z` block a path. A collider structure along a path is ``... -> c <- ...`` where ``c`` is the collider node. https://en.wikipedia.org/wiki/Bayesian_network#d-separation -The sets are not disjoint, with intersection The node(s)  are not found in Gz6One of x, y, or z is not a node or a set of nodes in G graph should be directed acyclicc:g|]}tj|Snx ancestors.0nodeGs ^/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/networkx/algorithms/d_separation.py z"is_d_separator..0s%"G"G"GT2<4#8#8"G"G"GFT)r NetworkXErrornodes NodeNotFound TypeErroris_directed_acyclic_graphrsetunionpopleftaddextendpredkeyssucc) rxyz intersectionset_v forward_dequeforward_visitedbackward_dequebackward_visitedancestors_or_zrs ` rrrs^X66QCCq66QCCq66QCCq1u.A.Q  "N NN A  17? W/"U"U"U"UVV V W XXXoVWWWX  ' * *CABBB"IIMeeO1XXNuu SUU["G"G"G"GQ"G"G"GH1LqPN L>L  H!))++D   & & &qyyuqyy  ! !!&,"3"3"5"58H"H I I I  !2!2!4!4!F G G G  L ((**D    % % %qyyu~%%%%afTl&7&7&9&9L8 4s A5A99B)included restrictedctjstjd |vr|hn|}|vr|hn|}|t}n|vr|h}|t}n|vr|h}||z|z|z}|jz r tjd|jz dn"#t $rtjdwxYw||kstjd|d|||zp ||zp||z}|rtjd|||z|z}|jfd |D}||||zz z} t||| } | |zrdS| | |zz} t||| } | | |zzS) uhReturns a minimal d-separating set between `x` and `y` if possible A d-separating set in a DAG is a set of nodes that blocks all paths between the two sets of nodes, `x` and `y`. This function constructs a d-separating set that is "minimal", meaning no nodes can be removed without it losing the d-separating property for `x` and `y`. If no d-separating sets exist for `x` and `y`, this returns `None`. In a DAG there may be more than one minimal d-separator between two sets of nodes. Minimal d-separators are not always unique. This function returns one minimal d-separator, or `None` if no d-separator exists. Uses the algorithm presented in [1]_. The complexity of the algorithm is :math:`O(m)`, where :math:`m` stands for the number of edges in the subgraph of G consisting of only the ancestors of `x` and `y`. For full details, see [1]_. Parameters ---------- G : graph A networkx DAG. x : set | node A node or set of nodes in the graph. y : set | node A node or set of nodes in the graph. included : set | node | None A node or set of nodes which must be included in the found separating set, default is None, which means the empty set. restricted : set | node | None Restricted node or set of nodes to consider. Only these nodes can be in the found separating set, default is None meaning all nodes in ``G``. Returns ------- z : set | None The minimal d-separating set, if at least one d-separating set exists, otherwise None. Raises ------ NetworkXError Raises a :exc:`NetworkXError` if the input graph is not a DAG or if node sets `x`, `y`, and `included` are not disjoint. NodeNotFound If any of the input nodes are not found in the graph, a :exc:`NodeNotFound` exception is raised. References ---------- .. [1] van der Zander, Benito, and Maciej Liśkiewicz. "Finding minimal d-separators in linear time and applications." In Uncertainty in Artificial Intelligence, pp. 637-647. PMLR, 2020. rNr rzFOne of x, y, included or restricted is not a node or set of nodes in GIncluded nodes z must be in restricted nodes z3The sets x, y, included are not disjoint. Overlap: c:g|]}tj|Srrrs rrz,find_minimal_d_separator..s%,W,W,WtR\!T-B-B,W,W,Wr) rr rr!rrrr" _reachable) rr)r*r3r4set_yr,nodesetancestors_x_y_includedz_init x_closure z_updated y_closures ` rr r Qs1r  ' * *CABBB 66QCCq66QCCq  uuHH ]] zH  QJJ 1__$JA :- 17? W/"U"U"U"UVV V W    o T    z ! ! Qh Q QZ Q Q   q58AL8ALL  P, P P   !ehG*W],W,W,W,Ww,W,W,WX 1QU; >> G = nx.path_graph([0, 1, 2, 3], create_using=nx.DiGraph) >>> G.add_node(4) >>> nx.is_minimal_d_separator(G, 0, 2, {1}) True >>> # since {1} is the minimal d-separator, {1, 3, 4} is not minimal >>> nx.is_minimal_d_separator(G, 0, 2, {1, 3, 4}) False >>> # alternatively, if we only want to check that {1, 3, 4} is a d-separator >>> nx.is_d_separator(G, 0, 2, {1, 3, 4}) True Raises ------ NetworkXError Raises a :exc:`NetworkXError` if the input graph is not a DAG. NodeNotFound If any of the input nodes are not found in the graph, a :exc:`NodeNotFound` exception is raised. References ---------- .. [1] van der Zander, Benito, and Maciej Liśkiewicz. "Finding minimal d-separators in linear time and applications." In Uncertainty in Artificial Intelligence, pp. 637-647. PMLR, 2020. Notes ----- This function works on verifying that a set is minimal and d-separating between two nodes. Uses criterion (a), (b), (c) on page 4 of [1]_. a) closure(`x`) and `y` are disjoint. b) `z` contains all nodes from `included` and is contained in the `restricted` nodes and in the union of ancestors of `x`, `y`, and `included`. c) the nodes in `z` not in `included` are contained in both closure(x) and closure(y). The closure of a set is the set of nodes connected to the set by a directed path in G. The complexity is :math:`O(m)`, where :math:`m` stands for the number of edges in the subgraph of G consisting of only the ancestors of `x` and `y`. For full details, see [1]_. rNr rzIOne of x, y, z, included or restricted is not a node or set of nodes in Gr6z& must be in proposed separating set z zSeparating set z% must be contained in restricted set r c:g|]}tj|Srr)rnrs rrz*is_minimal_d_separator..:s%,Q,Q,QAR\!Q-?-?,Q,Q,QrFT) rr rr!rrrr,r"r8) rr)r*r+r3r4r9r,r:r;r=r?s ` rrrsh  ' * *CABBB 66QCCq66QCCq66QCCq  uuHH ]] zH  QJJ 1__$JA :- 17? W/"U"U"U"UVV V W    o W    q== Qh Q Qa Q Q    ?? Ra R Rj R R   >>!$$Nq(9(9NQ^^A=N=NL  JL J J   !ehG*W],Q,Q,Q,Q,Q,Q,QR1a!7;;I1}u ' ' 'u1a!7;;I \y94 5 5u 4s BB--C cfd}tg}|D]d}t|j|r|d|ft|j|r|d|fe|}t |r|\}} d|j| D} d|j| D} t| | } | D]G\} }| |f|vr<||| | |r.|| |f|| |fHt |d|DS)uModified Bayes-Ball algorithm for finding d-connected nodes. Find all nodes in `a` that are d-connected to those in `x` by those in `z`. This is an implementation of the function `REACHABLE` in [1]_ (which is itself a modification of the Bayes-Ball algorithm [2]_) when restricted to DAGs. Parameters ---------- G : nx.DiGraph A NetworkX DAG. x : node | set A node in the DAG, or a set of nodes. a : node | set A (set of) node(s) in the DAG containing the ancestors of `x`. z : node | set The node or set of nodes conditioned on when checking d-connectedness. Returns ------- w : set The closure of `x` in `a` with respect to d-connectedness given `z`. References ---------- .. [1] van der Zander, Benito, and Maciej Liśkiewicz. "Finding minimal d-separators in linear time and applications." In Uncertainty in Artificial Intelligence, pp. 637-647. PMLR, 2020. .. [2] Shachter, Ross D. "Bayes-ball: The rational pastime (for determining irrelevance and requisite information in belief networks and influence diagrams)." In Proceedings of the Fourteenth Conference on Uncertainty in Artificial Intelligence (UAI), (pp. 480–487). 1998. c&|v}|vp|o| }|o|S)uZWhether a ball entering node `v` along edge `e` passes to `n` along `f`. Boolean function defined on page 6 of [1]_. Parameters ---------- e : bool Directed edge by which the ball got to node `v`; `True` iff directed into `v`. v : node Node where the ball is. f : bool Directed edge connecting nodes `v` and `n`; `True` iff directed `n`. n : node Checking whether the ball passes to this node. Returns ------- b : bool Whether the ball passes or not. References ---------- .. [1] van der Zander, Benito, and Maciej Liśkiewicz. "Finding minimal d-separators in linear time and applications." In Uncertainty in Artificial Intelligence, pp. 637-647. PMLR, 2020. r)evfrBis_element_of_Acollider_if_in_Zar+s r_passz_reachable.._passss/6q&A:6!+A3#33rTFc3K|]}d|fV dS)FNrrrBs r z_reachable..s&//%//////rc3K|]}d|fV dS)TNrrMs rrNz_reachable..s&..q$......rch|]\}}|Srr)r_ws r z_reachable..s & & &&1aA & & &r) rboolr&appendr(copyanyr#r)rr)rJr+rKqueuer processedrErFpredssuccs f_n_pairsrGrBs `` rr8r8LsN444444@ "IIE(( t   ' LL$ & & & t   ( LL% ' ' ' I e**)}}1//QVAY///..AF1I...%''  ) )DAq1vY&&55Aq!+<+<& aV$$$  !Q((( e**) ' &I & & &&r)__doc__ collectionsr itertoolsrnetworkxrnetworkx.utilsrr__all__ _dispatchablerr rr8rrrrdspUUn99999999   \""gg#"gT\""264f.f.f.f.#"f.R\""37DKKKK#"K\\""X'X'#"X'X'X'r