This paper considers the problem of inferring a discrete joint distribution from a sample subject to selection. Abstractly, we want to identify a distribution p(x; w) from its conditional p(x j w). We introduce new assumptions on the marginal model for p(x), under which generic identi cation is possible. These assumptions are quite general and can easily be tested; they do not require precise background knowledge of p(x) or p(w), such as proportions estimated from previous studies. We particularly consider conditional independence constraints, which often arise from graphical and causal models, although other constraints can also be used. We show that generic identi ability of causal e ects is possible in a much wider class of causal models than had previously been known.
Selection bias occurs when samples are obtained from a population in a manner which depends upon the attributes of the samples themselves. This can happen by accident|such as survey participants self-selecting in a way which is correlated with their responses|or by design, for example in a case-control study. In this paper we show that under plausible and testable structural hypotheses about the relationships between the variables being measured, we can recover from selection bias, even without having external information about the distribution of variables in the population. Example 1.1. Consider a case-control study in which participants are selected according to a disease status W with dw = 2 levels. We are interested in the e ect of a discrete measured treatment X on W which, assuming no confounding is the same as estimating the
X
Y
W conditional distribution p(w j x). Since our data are selected according to disease status, what we observe, ignoring sampling error, is the conditional distribution p(x j w). It is well known that we can use p(x j w) to obtain the causal odds-ratio for X on W , which is one measure of the causal e ect, but that the full conditional distribution p(w j x) is generally not identi able. However, suppose that we also measure a covariate Y , which is known to be marginally independent of X (see Figure 1). The marginal independence means that p(x; y) = p(x) p(y), so
X p(w)p(x; y j w) w
! =
X p(w)p(x j w) w
X p(w)p(y j w) w ! (1) for each value of x; y; letting dx; dy be the number of levels of X and Y respectively, this gives at most (dx 1)(dy 1) non-redundant equations. Since W is binary, each equation is quadratic in the single unknown p(w = 0) and can have at most two solutions. The true marginal distribution of W must be a solution to each equation, so the distribution is identi able up to at most two solutions1.
To be concrete, suppose we were given the following conditional distributions p(x; y j w).
1Provided that Y 6?? W j X; see the discussion in the next section.
W = 0 0 1 Since X and Y are positively correlated given W = 0, and negatively correlated given W = 1, it is clear that by taking some appropriate convex combination, we should nd a table for independence. In fact, choosing p(w = 0) = 14 gives the marginal table: Having recovered p(w = 0) we obtain the entire joint distribution p(x; y; w) = p(w) p(x; y j w) and therefore any causal e ects identi able from it, including p(w j x).
Note that no information external to the study was required (e.g. background knowledge of the distribution of p(x)) other than the marginal independence X ?? Y . In addition, we will see that this condition is in principle testable by equality and inequality constraints. In particular (1) may have no solution, so the model can be refuted.
Intuitively the above method works because the independence X ?? Y is `destroyed' if the marginal distribution of W is speci ed incorrectly (except in certain degenerate cases). Hence a correct speci cation of the marginal p(w) results in a distribution in the marginal model for (X; Y ), and an incorrect speci cation usually does not. 1.1
Selection bias has received much attention in the
statistical literature for many decades dating back at least
to [
Several of the above authors formalise the problem of selection by including a binary random indicator S of whether or not an individual is selected in the sample. The available data then comes from the conditional distribution p(x; wjs = 1), where X; W may be vectors of random variables. There is little we can do without further assumptions, but suppose that X ?? S j W , i.e. the selection only depends upon the variables in W ; in this case, provided p(w j s = 1) > 0, we can recover p(x j w) = p(x j w; s = 1). If the distribution of W in the population is known, say from a previous study, then we can obtain p(x; w) and in e ect `recover' from the selection bias. In this paper we will assume that no such information is available, and ask what assumptions allow p(x; w) to be recovered from p(x j w). We assume throughout that p(x; w) > 0.
The rest of the paper is structured as follows. After formally de ning generic identi ability in Section 2, we address in Section 3 the role of constraints on the marginal model p(x) for recovering p(x; w). In Section 4 we then consider special cases where the marginal constraints arise from various independence assumptions. While we use directed acyclic graphs (DAGs) throughout to depict marginal or conditional independences, Secion 5 formally addresses situations where the model is explicitly de ned by a DAG, such as is typical for causal models. It turns out that it is an important prerequisite for identi cation that the model be non-decomposable. Further implications and examples for causal inference are discussed in Section 6, and practical considerations given in Section 7. 2
We distinguish `strict' (sometimes called `global') and `generic' identi ability. The latter allows identi ability to fail on a lower dimensional subset O of the model M. It can be argued that, under certain assumptions, observations are `unlikely' to lie exactly in such a subset.
Consider random variables (X; W ) taking values in a nite discrete product space X W. Let =
X W fp > 0 : X p(x; w) = 1g x;w be the strictly positive probability simplex of distributions over X W. An algebraic model M is a collection of probability distributions satisfying a nite set of polynomial constraints in the probabilities p(x; w). For example, the model M under which X ?? W is the set of p 2 such that p(x; w)
! X p(x; w0) w0
X p(x0; w) x0 ! = 0; 8w; x: Consider the act of conditioning on W , de ned by the rational map : p(x; w) 7! p(x; w)=p(w). Let N = (M) be the image of the model under this operation, which is also an irreducible algebraic variety in the invariants p(x j w) [6, Proposition 4.5.6].
De ne the bre of
at p 2 M as the set
F (p) = fq 2 M : (q) = (p)g: An injective map is one for which all bres have cardinality one.
De nition 2.1. Let k 2 N. We say that M is generically k-identi able if the bres F (p) have cardinality at most k for all p 2 M n O, where O is some proper (i.e. lower dimensional) algebraic subset of M. In the case k = 1 then is just generically identi able. If a model is not generically k-identi ed for any k 2 N then it is unidenti able.
In other words, generic k-identi ability requires the map to be at most k-to-one on a set that contains `almost all' the distributions. Generic identi ability corresponds to the map being injective on M n O. Throughout this article we use `generically' to mean `at all but a strict sub-variety of the parameter values within the model (M)'.
If O = ; then this is sometimes referred to as strict (k-)identi ability. Strict identi ability of p(x; z) is a very stringent condition and essentially never occurs in our framework. If W ?? X, for example, it is clear that p(x j w) = p(x) does nothing to help us identify p(w), nor therefore p(x; w). We consider the weaker notion of generic identi ability so that identi ability may fail on a small (i.e. measure zero) subset O. We say a model on X; W is testable if it places nontrivial equality constraints on p(x j w), i.e. N is a strict sub-variety of the set of all conditional distributions p(x j w). We will work throughout the paper as though we can observe p(x j w) itself; in practice we would instead have a consistent estimator with asymptotically correct standard errors. 3
In this section we consider the possibility of using constraints on the marginal model of X to recover p(x; w) from p(x j w). The rst result gives us a necessary condition for identi ability.
Lemma 3.1. Suppose that MX be a marginal model for X, and let A be the (dx dw)-matrix with entries ax;w = p(x j w). Then p(x; w) is k-identi able from p(x j w) only if A has rank dw.
If W ?? X then, as previously noted, the rank condition will not be satis ed. It may fail in a more subtle way if, for example, there is a lower dimensional variable that mediates the relationship between X and W ; see Example 6.3.
In general it is di cult to characterize exactly when
a model will be generically identi able, but an
important special case comes when W 's dependence on X
is completely unrestricted: that is, our model makes
no restriction on p(w j x) no matter what the value of
p(x). This variation independence is known as a
`parameter cut', or simply a cut [
Theorem 3.2. Let M be a model for (X; W ) with a cut between X and W jX, such that p(w j x) is unrestricted and the marginal model for p(x) is a variety of dimension dx 1 l.
Then given p(xjw), the variety de ned by the bre fq(x; w) 2 M : q(x j w) = p(x j w)g generically has dimension max(0; dw 1 l). In particular, p(x; w) is generically k-identi able if and only if dw l + 1.
The quantity l is the number of independent constraints on the marginal model for X. We note that Theorem 3.2 implies that some constraints on p(x) are needed; when l = 0 the inequality in Theorem 3.2 is only satis ed if W is constant, meaning that there is in fact no selection.
Proof. Since p(x j w) is known, we just need (w) such that Pw (w)p(x j w) is contained in the marginal model for X, i.e. nd a point which is in the marginal model for X as well as the column span of the matrix P with (x; w)th entry p(x j w).
Let q(x) be a point in both the marginal model for X and the column span of P . Let the kernel of the Jacobian of constraints which de ne the marginal model for X be K, which is generically of rank dx 1 l by assumption.
The tangent space at q of the variety of distributions that satisfy both the marginal model and lie within P must lie within the intersection of the column spans of P and K. By Proposition A.1 any set of min(l + 1; dw) columns of P are generically linearly independent of K. The dimension of the tangent space of the intersection is then the number of remaining columns, which is max(0; dw l 1). Generically we may choose this point so that the dimensions are equal (in the language of algebraic geometry, the two manifolds are generically transverse).
It is important to keep in mind that generic identi ability allows for identi ability to fail on interesting sub-models. It may, therefore, be a serious problem if W is (conditionally) independent of some part of X. Example 3.3. Consider the graphical model in Figure 2, which implies
p(x; y; z; w) = p(y j x)p(z j x)p(x j w)p(w): In this case Y ?? Z j X is a marginal constraint, however W does not depend arbitrarily on X; Y; Z so we cannot apply Theorem 3.2. In fact W ?? Y; Z j X so p(x; y; z) = p(y j x)p(z j x) X p(w)p(x j w); w which satis es the required marginal constraint for any p(w). We cannot, therefore, use this constraint to identify p(w).
This phenomenon is generalized in the following lemma.
Lemma 3.4. Suppose distributions in M are of the form p(x; y; w) = p(x; w)p(y j x; w), where p(y j x; w) is variation independent of p(x; w). Then p(x; y; w) is identi able from p(x; y j w) if and only if p(x; w) is identi able from p(x j w).
Proof. We have
p(x; y j w) = p(y j x; w) p(x j w); so the two factors are recoverable from p(x; y j w). Since p(y j x; w) is variation independent of p(x; w), it is also variation independent of p(w) given p(x j w), which are functions of p(x; w). It follows that no restriction is placed on p(w) by p(y j x; w).
Our method relies strongly on variation dependence
between p(w) and p(x j w); the kind of variation
independence in the Lemma and in Example 3.3 is not
helpful for identifying p(x; w). Note that the special
case in which X is a trivial random variable shows
that we must not have a parameter cut between the
marginal model for W and the distribution of XjW ;
contrast this with the conditions of Theorem 3.2. This
is related to the observations of [
Marginal and conditional independence constraints often arise in statistical models, commonly from Markov assumptions such as those implied by causal models. In this section we shall focus on when p(x; w) is identi able in this context. 4.1
Consider again Example 1.1 in which X ?? Y but we can only directly observe p(x; y j w). Consider each p(x; y j w) for w 2 W as a single distribution for X; Y ; we need to nd a convex combination of these distributions which satis es the independence constraint. If all the variables are binary we have two 2 2-tables in the 3-dimensional probability simplex, and want to nd a point on the line segment between these tables which lies on the surface of marginal independence. This is illustrated in Figure 3. Assuming that neither 1. the two conditional tables have opposite signed correlations (or log-odds-ratios), in which case there is exactly one convex combination that satis es marginal independence; 2. the two tables have the same sign, and there is no solution which satis es the independence; 3. the two tables have the same sign, but there are one or two convex combinations which satisfy the independence.
Situation 2 allows the assumption of X ?? Y to be refuted, and corresponds to an inequality constraint on p(x; y j w). This is the only constraint on this model in the binary case.
The possibility of two solutions is due to the quadratic
nature of the independence constraint. A line segment
which intersects the surface of independence twice is
shown in Figure 3. If there are two distinct solutions
then there is no way to identify the true distribution
uniquely without further assumptions. However, this
situation can only occur if the joint distribution
exhibits a form of Simpson's paradox: X and Y are
positively (or negatively) correlated within each level of W ,
but are independent after collapsing across the levels
of W . It has been argued that this happens relatively
rarely in practice [
The method extends readily to non-binary X and Y ; we obtain (dx 1)(dy 1) separate quadratics in the unknown p(w = 0), and they will generically have distinct solutions; it follows that p(w = 0) is the common solution to these quadratics, and that the model is generically identi able. If W is non-binary then we obtain quadratic equations in multiple variables, but analogous results are available.
Lemma 4.1. Consider the model M on discrete variables X; Y; W such that X ?? Y . Then p(w; x; y) is generically k-identi able from p(x; y j w) if and only if dw 1 (dx 1)(dy 1). Identi ability fails if either X ?? Y; W or Y ?? X; W .
If dw = 2 then p(w; x; y) is 2-identi able from p(x; w j y) if X 6?? Y j W , and unidenti able otherwise. The proof is in the appendix.
Note that, under the causal graphical model in Figure 1, the causal e ect of X on W is given by p(w j do(x)) = p(w j x). This quantity is never strictly identi able from p(x j w), but it is generically identi able from p(x; y j w) under the marginal independence model.
Note that, although we lose identi ability of p(w) if (for example) X ?? W j Y , we can still test this as a causal null hypothesis because it is observable directly from p(x; y j w).
In the all-binary case, the marginal independence model is not `testable' in the sense that it places no equality constraints on p(x; y j w): thus for some distributions p(x; y; w) which do not satisfy the marginal independence, we can follow the procedure given above and fallaciously `recover' some other distribution q(w)p(x; y j w) which does satisfy the model. However in the case where dx = 3, there are two independent constraints, so we can use one to recover p(w) and the other to check the model. 4.2
Consider the model de ned by the graph in Figure 4, which implies that X ?? Y j Z. Under selection bias on W we can only observe the conditional distribution p(x; y; z j w). Following the same approach as in Example 1.1 we obtain p(z) p(x; y; z) p(x; z) p(y; z) = 0; 8x; y; z; if W is binary we obtain p(x; y; z) = p(x; y; zjw = 0) + (1 )p(x; y; zjw = 1), so each factor is linear in the unknown = p(w = 0). This gives us quadratic equations in , which are generically distinct for each level of Z.
Lemma 4.2. Let M be the model on X; Y; Z; W dened by X ?? Y j Z. Then p(x; y; z; w) is generically k-identi able from p(x; y; z j w) if and only if dw 1 (dx 1)(dy 1)dz.
If dw = 2 the model is unidenti able if and only if X ?? Y j Z; W .
The result is really just an application of Lemma 4.1 to separate models for each level of Z. If Z is trivial it reduces to the marginal independence case. Note that the condition X ?? Y j W; Z for non-identi ability is testable directly from the observed conditional distribution. The unidenti ability arises because any choice of p(w) will give the required conditional independence.
Particular cases of unidenti ability arise when either
X ?? Y; W j Z or Y ?? X; W j Z (see Figure 5(a),
Example 3.3); in fact if W is binary then unidenti ability
is equivalent to at least one of these constraints holding
[
As the next example illustrates, we can nd that extra independences give `reduced' identi ability by making some constraints redundant, without leading to full unidenti ability. In particular, although the allbinary case of a conditional independence under selection generically gives two non-redundant quadratic equations, in degenerate cases these equations may have the same roots.
Example 4.3. Consider the model in Figure 5(b), in which Z ?? W j X; Y as well as X ?? Y j Z. In this case p(x; y; z j w) = p(z j x; y) p(x; y j w): In order to have X ?? Y j Z we need p(x; y; z) = p(x; z)p(y j z). So since p(x; y; z) = p(z j x; y)p(x; y) = p(z j x; y) Pw p(x; y j w)p(w), we need to pick p(w) so that the odds-ratio between X and Y exactly cancels out the x; y factor of p(z j x; y). This gives a single quadratic equation; and solution to this gives p(w = 0) such that p(w)p(x; y; z j w) is inside the model, so we have generic 2-identi ability. Note that although there is only one equation to solve in one unknown, unlike the marginal independence model this one is testable, since Z ?? W j X; Y can be checked directly from the observed conditional p(x; y; z j w).
The example shows that no three-way interaction is present between X; Y; Z in p(x; y; z j w), and that this remains true for any weighted combination Pw (w)p(x; y; z j w). If the three-way interaction is zero then the two constraints X ?? Y j Z = 0 and X ?? Y j Z = 1 become equivalent, thus we have a partial degeneracy: there is only one non-redundant constraint to ful l. This issue is explored more generally in Section 5.1. 5
In the previous sections, we used directed acyclic
graphs (DAGs), also known as Bayesian networks,
to represent marginal and conditional independences.
Here we give explicit results for situations where the
model is de ned by a DAG, such as is typical for causal
models. A DAG model is de ned by a recursive
factorization according to the structure of a graph, or
equivalently a collection of conditional independence
constraints. Let G be a directed acyclic graph with
disjoint sets of vertices V [_W representing vectors of
random variables XV ; XW . We make use of the fairly
standard terminology of parents (paG ), children,
ancestors (anG ), etc. See, for example, [
We will assume that our distribution obeys the Markov property with respect to a DAG G, so that p(xV ; xW ) =
p(xv j xpa(v)):
Y
Lemma 5.1. Let p(xV j xW ) be a conditional distribution from a DAG. Then p(xV ; xW ) is identi able from p(xV j xW ) if and only if p(xan(W )nW ; xW ) is identi able from p(xan(W )nW j xW ).
Proof. If V 6= anG (W ) n W then there exists some childless v 2 V in G. Hence there is a parameter cut between W [ V n fvg and fvgjW [ V n fvg, so p(xV ; xW ) = p(xv j xV nv; xW ) p(xV nv; xW ) where p(v j xV nv; xW ) = p(xv j xpa(v)) is variation independent of p(xV nv; xW ). The result then follows from Lemma 3.4 and repeating over all v 62 anG (W ). In light of this result, we henceforth assume that all variables are ancestors of the selection variables W . 5.1
A hierarchical model over XV is the set of distributions p(xV ) which factorize as p(xV ) = Y
C (xC )
C2C for some collection of inclusion maximal sets C; undirected graphical models and decomposable models are special cases. Note that if p(xV ; xW ) is hierarchical with maximal sets C, then p(xV j xW ) = p(xV ; xW )=p(xW ) also factorizes into functions over the maximal sets C [ fW g. Any DAG model is contained in the hierarchical model with maximal sets fvg [ paG (v), which has consequences for our conditional distributions.
Lemma 5.2. Let p(xV ; xW ) obey the global Markov property for a DAG G on V [ W . Then p(xV j xW ) is of the form p(xV j xW ) =
W (xW ) v(xv; xpa(v)):
(2)
Y That is, a hierarchical function with maximal sets W and fvg [ paG (v); v 2 V [ W .
This follows directly from writing out the factorization for DAGs.
Note that (2) yields testable constraints on p(xV j xW ), but these constraints do not give any way to identify p(xW ). Clearly the true p(xV ; xW ) must satisfy the same hierarchical model, but so will any distribution of the form q(xW )p(xV j xW ).
Proposition 5.3. Let M be the model implied by a DAG G, and M the hierarchical model with maximal sets
C = fvg [ paG (v) : v 2 V [ W : Then p(xV ; xW ) is generically k-identi able from p(xV j xW ) only if dw 1 d(M) d(M). In particular p(xV ; xW ) is generically k-identi able from p(xV j xW ) only if G is not decomposable. The proof is given in the appendix. Proposition 5.3 is illustrated by Example 3.3 where the model is indeed decomposable, while in Example 4.3 it is not decomposable. We conjecture that the converse of Proposition 5.3 also holds, so that if dw 1 d(M) d(M) then generic k-identi ability follows. We can obtain the following weaker result from Theorem 3.2, however.
Proposition 5.4. Let G be a DAG with vertices V [ fwg such that paG (w) = V , and such that G imposes l independent constraints on XV . Then p(xV ; xw) is identi able from p(xV j xw) if and only if dw l + 1. 6
In the context of causal inference, our results will be
useful when it is known a priori that the causal e ect
of interest is identi ed from p(x; w), p(x), or p(wjx).
Before quantifying a causal e ect, one may rst want
to test the null hypothesis of no causal e ect. As
noticed before, when the null translates into a conditional
independence (or absence of a directed edge in a DAG),
this may actually hamper identi cation, but we
conjecture that the null can often be see from p(xjw) itself,
using e.g. result from [
Example 6.1. It may be plausible that the causal e ect of X on W , p(wjdo(x)), is identi ed given a su cient set of covariates C, e.g. by the back-door formula. In the context of a case control study, however, one would then often just estimate the (conditional) causal odds-ratio between X and W given C as the most straightforward causal quantity. Our results would be useful if data on additional covariates Z is available, where it is known that, for example, X ?? Z j C because subject matter tells us that X is only determined by C. The conditional independence then imposes the sort of constraints which may enable identi cation of p(x; c; z; w). Hence we can obtain other causal e ect measures, such as risk di erences or risk ratios. In particular we can obtain the intervention distribution p(yjdo(x)) by integrating out C, so that unconditional causal parameters can be reported which are more easily compared with results from randomized controlled trials. However, problems may occur: if the association between Z and W is weak, identi cation becomes unstable, as with a weak instrument in instrumental variable methods. Additionally, if W represents a very rare disease, then the marginal distribution of X is approximately the same as that for the controls, p(x) p(xjw = 0), so recovering p(x) exactly will typically not lead to interesting new insights. However, our method could still be used for sensitivity analysis in such situations.
Example 6.2. Consider the causal model represented by the graph in Figure 6 under selection on X4. This 1 2
4 model arises in the context of dynamic treatment regimes, where X1; X3 are treatments, X2; X4 outcomes, and X0 covariates. A typical quantity of interest is p(x4 j do(x1; x3)), which can be expressed as (for example) either of p(x4 j do(x1; x3)) =
X p(x4 j x1; x3; x0)p(x0) x0 = X p(x4 j x1; x2; x3)p(x2 j x1):
x2 However neither of these quantities is a function of p(x0123 j x4), so we need to recover the distribution p(x4) before proceeding. The graph exhibits the conditional independence constraints X0 ?? X1 and X0; X1 ?? X3 j X2 and these independences are not visible after selection on X4, so they provide a method for recovering p(x4).
The conditional distribution has the form of a hierarchical model p(x0123 j x4) = (x0; x1; x2) (x2; x3) (x0; x1; x3; x4); and so q(x4)p(x0123 j x4) factorizes in the same way for any margin q(x4). Notice that the constraints which are lost are precisely that X0 ?? X1 and X0 ?? X3 j X2. These constitute (d0 1)(d1 1) and (d0 1)(d3 1)d2 constraints respectively, but the three-way interaction X0; X2; X3 is not present in the hierarchical model, so in fact there are only (d0 1)(d3 1) additional constraints from the conditional independence (this is essentially the same as what happens in Example 4.3). Thus we can generically identify the model only if (d0 1)(d1 + d3 2) d4 1: In the all-binary case this corresponds to two constraints to resolve one parameter.
Example 6.3. The ability to identify distributions in DAG models may su er from `choke points' caused by lower dimensional variables. Consider the model in Figure 7 with selection on W . Then p(x; y; z; u j w) = p(x; y; z j u)p(u j w), so that provided du 1 (dx 1)(dy 1)dz we can generically identify p(u) from the rst factor. However, knowing p(u) and p(u j w) will not allow us to generically identify p(w) unless dw du. Such a lack of identi cation is because p(w j x; y; z) is not unrestricted if dw > du, leading to a rank de ciency of the matrix with entries p(x; y; z j w) and violating the conditions of Lemma 3.1. Note, however, that this rank problem is detectable even if U is unobserved. 7
We envisage that the practical application of our results will work as follows. In a given data situation, we would need to know that sampling was subject to selection and which variable(s) this a ected. Then, a set of marginal or conditional independences the true distribution should obey needs to be postulated. Identi ability can then be checked using our results and if it holds the full model can be t with appropriate numerical procedures. Note that if these postulated constraints are just enough to enable identi ability they cannot empirically be tested and must be based on subject matter knowledge which will typically be informed by causal assumptions.
In principle it is a relatively straightforward matter to t these models; to perform maximum likelihood estimation we need to maximize the conditional loglikelihood lXjW (p)
X nxw log p(x j w) x;w = X nxw log p(x; w)
x;w = lXW (p) lW (p):
X nxw log p(w)
x;w
The TM algorithm of [
All the variables in our causal models were assumed
to be observed, but in principle an extension could be
made to conditional independence or other constraints
which arise in the presence of latent variables. We
conjecture that distributions arising from the graph in
Figure 6 are recoverable even if X0 is unobserved, for
example (this is the Verma model [
p(x; yjw) + p(xjw)p(yjw)
p(x; yjw)
We have complete unidenti ability if and only if axy =
bxy = cxy = 0 for all x; y. From the forms of cxy and
bxy it is clear that this occurs only if X ?? Y j W .
Conversely, if X ?? Y j W and X ?? Y then for binary
W this implies that either X ?? W; Y or Y ?? W; X [
The map : p(xV ; xw) 7! p(xV j xw) maps M into the d(M) dw +1 dimensional space of hierarchical models after conditioning on Xw; therefore the image of M M also lies in a d(M) dw + 1 dimensional space. It follows that generic bres of when applied only to M must have dimension at least d(M) d(M) + dw 1, and in particular have positive dimension if d(M) d(M) < dw 1; it follows that d(M) d(M) dw 1 is necessary for generic identi ability.
In particular, if G is decomposable then M = M, so there is no identi ability for any dw 2.
A.1
Proof of Lemma 3.1. Suppose we nd p(w) such that p(x) = Pw p(w) p(x j w) is contained in MX . This is simply a sum over the columns of A, so if it is not of full column rank then we can clearly add any vector in the kernel of A to p(w) and obtain the same distribution. Hence p(w) is not identi able.
Proposition A.1. Let M be a model for (X; W ) in which there is a parameter cut between X and W jX, and such that p(w j x) is unrestricted. The matrix C with entries cxw = p(x j w) generically has rank r = min(dx; dw). Further, if we x dw 1 columns of C and p(x), the remaining column of C is generically not contained within the span of any r 1 dimensional vector space.
Proof. Since p(w j x) is unrestricted, the matrix A with entries axw = p(w j x) is generically of full rank min(dx; dw). Now, since cxw = axw p(x)=p(w) is just a rescaling of the rows and columns of A, the rank of C is the same.
For the last part, the column p(x j w) of C is obtained from A by multiplying it by the non-singular diagonal matrix with entries p(x)=p(w). Any restriction of p(x j w) to a subspace would imply a similar restriction on p(w j x), so by contradiction no such restriction exists.