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J Am Stat Assoc. Writer manuscript; available in PMC 2012 Dec 1.

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Instability, Sensitivity, and Degeneracy of Discrete Exponential Families

Abstract

In applications to dependent data, first and foremost relational information, a number of discrete exponential family models has turned out to exist near-degenerate and problematic in terms of Markov concatenation Monte Carlo simulation and statistical inference. We introduce the notion of instability with an centre to narrate, discover, and penalize detached exponential family unit models that are most-degenerate and problematic in terms of Markov chain Monte Carlo simulation and statistical inference. Nosotros show that unstable discrete exponential family models are characterized by excessive sensitivity and near-degeneracy. In special cases, the subset of the natural parameter infinite corresponding to not-degenerate distributions and mean-value parameters far from the boundary of the mean-value parameter infinite turns out to be a lower-dimensional subspace of the natural parameter space. These characteristics of unstable discrete exponential family models tend to obstruct Markov concatenation Monte Carlo simulation and statistical inference. In applications to relational data, we show that discrete exponential family unit models with Markov dependence tend to be unstable and that the parameter infinite of some curved exponential families contains unstable subsets.

Keywords: social networks, statistical exponential families, curved exponential families, undirected graphical models, Markov chain Monte Carlo

1 Introduction

We consider discrete exponential families (Barndorff-Nielsen 1978) with accent on applications to relational data (Wasserman and Faust 1994). Examples of relational data are social networks, terrorist networks, the world wide web, intra- and inter-organizational networks, trade networks, and cooperation and conflict betwixt nations. A common grade of relational information is discrete-valued relationships Y_ij between pairs of nodes i, j = 1, …, north. Let Y be the collection of relationships Y_ij given n nodes and be the sample space of Y . Whatever distribution with back up can be expressed in exponential family form (Besag 1974, Frank and Strauss 1986). Discrete exponential families of distributions with support were introduced by Frank and Strauss (1986), Wasserman and Pattison (1996), Snijders et al. (2006), Hunter and Handcock (2006), and others.

In terms of statistical computing, the most important obstacle is the fact that relational data tend to be dependent and discrete exponential families for dependent information come with intractable likelihood functions. Therefore, conventional maximum likelihood and Bayesian algorithms (e.k., Geyer and Thompson 1992, Snijders 2002, Handcock 2002a, Hunter and Handcock 2006, Møller et al. 2006, Koskinen et al. 2010) exploit draws from distributions with support to maximize the likelihood function and explore the posterior distribution, respectively. Every bit Markov chain Monte Carlo (MCMC) is the foremost ways to generate draws from distributions with support , MCMC is fundamental to both simulation and statistical inference.

In exercise, MCMC simulation from discrete exponential family unit distributions with support has brought to light some serious issues: first, Markov bondage may mix extremely slowly and inappreciably movement for millions of iterations (Snijders 2002, Handcock 2003a); and second, the extremely slow mixing of Markov bondage may exist rooted in the stationary distribution: the stationary distribution may be near-degenerate in the sense of placing near all probability mass on a small subset of the sample infinite (Strauss 1986, Jonasson 1999, Snijders 2002, Handcock 2003a, Hunter et al. 2008, Rinaldo et al. 2009). The almost troublesome observation, though, is that the subset of the natural parameter space corresponding to non-degenerate distributions may be a negligible subset of the natural parameter infinite. These troublesome observations raise at least two questions. Kickoff, why is the effective natural parameter space of some discrete exponential families (due east.g., Frank and Strauss 1986) negligible, while the effective natural parameter space of others (e.g., the Bernoulli model, under which the Y_ij are i.i.d. Bernoulli random variables) is non-negligible? 2nd, which sufficient statistics can induce such problematic behavior?

Handcock (2002a, 2003a,b) adjusted and extended results of Barndorff-Nielsen (1978, pp. 185–186) and pointed out that, as the natural parameters tend to the boundary of the natural parameter infinite, the probability mass is pushed to the boundary of the convex hull of the space of sufficient statistics (cf. Rinaldo et al. 2009, Geyer 2009, Koskinen et al. 2010). However, these results are applicable to both the Bernoulli model and Frank and Strauss (1986) and neither explain the hitting contrast between them nor clarify which sufficient statistics tin induce problematic behavior.

We introduce the notion of instability along the lines of statistical physics (Ruelle 1969) with an eye to characterize, detect, and penalize problematic discrete exponential families. Strauss (1986) was the start to notice that the problematic beliefs of the discrete exponential families of Frank and Strauss (1986) is related to lack of stability of betoken processes in statistical physics (Ruelle 1969, p. 33). We adapt the notion of stability of point processes in the sense of Ruelle (1969, p. 33) to discrete exponential families and introduce the notions of unstable discrete exponential family unit distributions and unstable sufficient statistics. We testify that unstable exponential family distributions are characterized by excessive sensitivity and near-degeneracy. In special cases, the subset of the natural parameter infinite corresponding to non-degenerate distributions and hateful-value parameters far from the boundary of the mean-value parameter space turns out to be a lower-dimensional subspace of the natural parameter infinite. In applications to relational information, it turns out that the parameter space of exponential families with Markov dependence (Frank and Strauss 1986) tends to be unstable and that the parameter space of some curved exponential families (Snijders et al. 2006, Hunter and Handcock 2006) contains unstable subsets.

We introduce the notion of instability and its implications in Department 2, hash out its impact on MCMC simulation and statistical inference in Sections iii and 4, respectively, and present applications to relational data and simulation results in Sections 5 and six, respectively.

2 Instability, sensitivity, and degeneracy

Permit Y _{Due north} be a discrete random variable with sample infinite = , where is a discrete ready of G elements and Northward is the number of degrees of freedom. In applications to relational data (Wasserman and Faust 1994), Y _Northward may correspond to Northward ≤ n ^two relationships among north nodes; in applications to spatial data (Besag 1974), North random variables located at N sites of a lattice; and in binomial sampling, Northward i.i.d. Bernoulli random variables.

We consider discrete exponential families of distributions {P _θ , θ ∈ Θ} with probability mass functions of the grade

(1)

where η _North : Θ ↦ ℝ ⁵⁰ is a vector of natural parameters and chiliad _Northward : ↦ ℝ ^L is a vector of sufficient statistics,

(2)

is the cumulant generating office, and Θ = { θ ∈ ℝ ^K : ψ_North ( θ ) < ∞} is the parameter infinite. The vector of natural parameters η _N ( θ ) may be a linear or non-linear function of parameter vector θ . If η North ( θ ) = A N T θ is a linear office of θ , where A _N is a K × Fifty matrix, the not-uniqueness of the canonical form of exponential families can be exploited to absorb A _N into g _{Due north} ( y _N ), so that η _N ( θ ) = θ can exist assumed without loss of generality. If η _N ( θ ) is a non-linear function of θ and Thou < Fifty, the exponential family unit is curved (Efron 1978).

Let q θ ( y N ) = η Due north T ( θ ) g North ( y North ), and I_N ( θ ) = min _{y Due north ∈} [q _θ ( y _N )] and Southward_Northward ( θ ) = max _{y N ∈} [q _θ ( y _N )] be the minimum and maximum of q _θ ( y _North ), respectively. Since p _θ ( y _N ) is invariant to translations of q _θ ( y _N ) by −I_North ( θ ), let I_N ( θ ) = 0 without loss of generality.

Definition: stable, unstable distributions

A detached exponential family distribution P _θ , θ ∈ Θ, is stable if there exist constants C > 0 and N_C > 0 such that

Due south _North ( θ ) ≤C N for all N >N _C ,

(3)

and unstable if, for any C > 0, however large, at that place exists N_C > 0 such that

Southward _N ( θ ) >C N for all North >Due north _C .

(4)

In full general, instability may be induced by η _N ( θ ) or 1000 _North ( y _N ). In the important special case where η _N ( θ ) is a linear function of θ , in which case η _N ( θ ) = θ can be assumed without loss of generality, chiliad _North ( y _N ) is the exclusive source of instability. Let η_{N , g} ( θ ) and g_{North , k} ( y _N ) exist the k-th coordinate of η _N ( θ ) and thousand _N ( y _North ), respectively, Fifty_{N , k} = min _{y N ∈} [grand_{N , m} ( y _North )] and U_{N , k} = max _{y N ∈} [g_{Northward , k} ( y _{Due north} )] exist the minimum and maximum of g_{N , k} ( y _N ), respectively, and L_{N , k} = 0 without loss of generality, attributable to the invariance of p _θ ( y _N ) to translations of q _θ ( y _N ) by −η_{N , k} ( θ ) L_{N , k} (k = i, …, L).

Definition: stable, unstable sufficient statistics

A sufficient statistic k_{N , thousand} ( y _N ) is stable if at that place exists constants C > 0 and N_C > 0 such that

U _{Due north,m}  ≤C N for all N >Due north _C ,

(5)

and unstable if, for any C > 0, however big, there exists Northward_C > 0 such that

U _N,k  >C Due north for all N >Northward _C .

(six)

While the notion of unstable detached exponential families holds intuitive appeal, the parameter space Θ of most detached exponential families of involvement includes subsets indexing stable distributions. With a wide range of applications in mind, it is therefore preferable to study the characteristics of unstable sufficient statistics and unstable distributions and to detect in applications unstable sufficient statistics and subsets of Θ indexing unstable distributions. It is worthwhile to note that Handcock (2002a, b, 2003a) discussed an alternative, but unrelated notion of stability, calling discrete exponential families stable if small changes in natural parameters result in pocket-sized changes of the probability mass function.

To demonstrate instability and its implications, we innovate ii classic examples in Section ii.one. In Sections two.2 and two.3, we bear witness that unstable exponential family distributions are characterized past excessive sensitivity and near-degeneracy.

ii.1 Examples

A elementary but common form of relational data is undirected graphs y _N , where the relationships y_ij ∈ {0, 1} satisfy the linear constraints y_ij = y_ji (all i < j) and y_ii = 0 (all i), which reduces the number of degrees of freedom Northward from n ^two to n(n−i)/ii. Two archetype models of undirected graphs are the Bernoulli model with natural parameter θ and stable sufficient statistic Σ _{i < j} y_ij and the 2-star model with natural parameter θ and unstable sufficient statistic Σ _{i, j < yard} y_ijy_ik . The Bernoulli model arises from the assumption that the random variables Y_ij are i.i.d. Bernoulli (all i < j), while the 2-star model can be motivated past Markov dependence (Frank and Strauss 1986). The Bernoulli model implies S_North (θ) = |θ|N and is therefore stable for all θ, while the 2-star model implies S_N (θ) = |θ| (n − 2)N and is therefore unstable for all θ ≠ 0.

ii.2 Instability and sensitivity

Unstable detached exponential family distributions are characterized by excessive sensitivity.

Consider the smallest possible changes of y _{Due north} , that is, changes of 1 element of y _North , and let

(vii)

exist the log odds of p _θ ( y _N ) relative to p _θ ( x _North ), where x _N ~ y _N ways that x _Northward and y _N are nearest neighbors in the sense that x _North and y _N match in all simply one element. The following theorem shows that, if an exponential family distribution is unstable, then the probability mass office is characterized past excessive sensitivity in the sense that the nearest neighbor log odds are unbounded and therefore even the smallest possible changes can upshot in extremely large log odds.

Theorem 1

If a detached exponential family distribution P _θ , θ ∈ Θ, is unstable, then there exist no constants C > 0 and N_C > 0 such that

(8)

Theorem 1 implies that some, only not necessarily all, nearest neighbor log odds are unbounded. It indicates that the probability mass office is excessively sensitive to small-scale changes in subsets of and that some elements of boss others in terms of probability mass. A walk through resembles a walk through a rugged, mountainous landscape: small-scale steps in tin result in dramatic increases or decreases in probability mass. An case is given by the 2-star model of Section 2.i: for all θ ≠ 0, the nearest neighbor log odds satisfy |Λ _North ( 10 _N , y _North ; θ)| ≤ two |θ| (n − ii) (all ten _Northward ~ y _Northward ) and are therefore O(n). The excessive sensitivity of the 2-star model is well-known (Handcock 2003a), but Theorem 1 indicates that all unstable exponential family distributions endure from excessive sensitivity.

Section three shows that the unbounded nearest neighbor log odds of unstable exponential family distributions accept a directly touch on MCMC simulation.

Theorem 2

If a discrete exponential family distribution P _θ , θ ∈ Θ, is unstable, then it is degenerate in the sense that, for any 0 < ε < 1, notwithstanding small-scale,

P _θ ( Y _N  ∈ ℳ_{ε,Due north} ) → 1 as Due north →  ∞ .

(nine)

A related result was reported by Strauss (1986) and Handcock (2003a). In general, the fact that nearly all probability mass tends to be concentrated on the modes of the probability mass role is troublesome: commencement, considering the effective support, the subset of the back up with non-negligible probability mass, is reduced; and second, considering in most applications the modes do not resemble observed data.

In the of import special case of exponential families with unstable sufficient statistics, it is possible to gain more than insight into near-degeneracy. Consider one-parameter exponential families {P _θ , θ ∈ Θ} with natural parameter η_N (θ) = θ and sufficient statistic one thousand_N ( y _N ). Let Fifty_N = 0 (without loss of generality) and U_Northward exist the minimum and maximum of g_{Due north} ( y _N ), respectively, and, for any 0 < ε < 1, allow = { y _N ∈ : g_N ( y _N ) < ε U_N } and = { y _{Due north} ∈ : g_Northward ( y _{Due north} ) > (1 − ε) U_N } be the subset of the sample space close to the minimum and maximum of g_{Due north} ( y _N ), respectively. The following result shows that one-parameter exponential families with unstable sufficient statistics k_{Due north} ( y _N ) tend to be degenerate with respect to g_N ( y _Northward ).

Theorem 3

A one-parameter exponential family {P _θ , θ ∈ Θ} with natural parameter θ and unstable sufficient statistic yard_North ( y _N ) is degenerate with respect to g_N ( y _N ) in the sense that, for any 0 < ε < 1, however small, and for whatsoever θ < 0,

P _θ( Y _{Due north}  ∈ ℒ_ε,N ) → 1 every bit N →  ∞

(x)

and, for whatsoever θ > 0,

P _θ( Y _North  ∈ 𝒰_ε,N ) → 1 equally North →  ∞ .

(xi)

Thus, the probability mass is pushed to the minimum of g_N ( y _North ) for all θ < 0 and the maximum of m_N ( y _North ) for all θ > 0, and the subset of the natural parameter space Θ corresponding to non-degenerate distributions is a lower-dimensional subspace of Θ: the betoken θ = 0. An example of a one-parameter exponential family unit with unstable sufficient statistic is given by the two-star model of Department 2.i.

Consider Chiliad-parameter exponential families {P _θ , θ ∈ Θ} with natural parameters η_{Northward ,1}( θ ) = θ ₁, …, η_{N , Yard} ( θ ) = θ_Yard and K − 1 stable sufficient statistics k_{N ,one}( y _N ), …, g_{N , K −1}( y _N ) as well as ane unstable sufficient statistic yard_{N , 1000} ( y _N ). In accordance with the preceding paragraph, let and be the subset of the sample space close to the minimum and maximum of the unstable sufficient statistic grand_{N , 1000} ( y _N ), respectively. The following result shows that Chiliad-parameter exponential families with G − 1 stable and i unstable sufficient statistic tend to be degenerate with respect to the unstable sufficient statistic.

Theorem 4

A K-parameter exponential family {P_θ , θ ∈ Θ} with natural parameters θ _i, …, θ₁₀₀₀ and Thousand − i stable sufficient statistics g_{Due north ,1}( y _N ), …, g_{Northward , M −1}( y _N ) also every bit one unstable sufficient statistic g_{North , K} ( y _N ) is degenerate with respect to yard_{Due north , One thousand} ( y _N ) in the sense that, for any 0 < ε < i, however modest, and for any θ_K < 0,

P _θ ( Y _N  ∈ ℒ_ε,N,K ) → ane every bit N →  ∞

(12)

and, for whatever θ_Thousand > 0,

P _θ ( Y _N  ∈ 𝒰_ε,N,K ) → i as N →  ∞ .

(13)

In general, it is not straightforward to see where the probability mass of K-parameter exponential families with multiple unstable sufficient statistics ends upwardly. In special cases, though, insight can be gained. Consider a G-parameter exponential family unit {P_θ , θ ∈ Θ} with natural parameters θ _one, …, θ_K and sufficient statistics chiliad_{N ,1}( y _N ), …, g_{N , Thousand} ( y _North ), where g_{N ,1}( y _N ), …, yard_{Northward , K −1}( y _Northward ) may be unstable while g_{N , Grand} ( y _N ) is unstable and dominates g_{N ,ane}( y _Northward ), …, g_{N , Chiliad −1}( y _N ) in the sense that, for any D > 0, notwithstanding large, there exists N_D > 0 such that

U N , K U Due north , k > D for all N > Due north D , k = 1 , … , Chiliad − 1.

(14)

A K-parameter exponential family with multiple unstable sufficient statistics, including an unstable, dominating sufficient statistic g_{N , K} ( y _N ), tends to be degenerate with respect to g_{Northward , 1000} ( y _N ).

Theorem 5

A Grand-parameter exponential family unit {P_θ , θ ∈ Θ} with natural parameters θ ₁, …, θ_K and sufficient statistics 1000_{N ,1}( y _N ), …, thousand_{N , Thousand} ( y _N ), where g_{Due north ,1}( y _North ), …, g_{N , K −1}( y _Northward ) may exist unstable while 1000_{N , K} ( y _N ) is unstable and dominates g_{N ,1}( y _N ), …, g_{N , G −1}( y _Northward ), is degenerate with respect to g_{N , K} ( y _Northward ) in the sense that, for whatever 0 < ε < 1, all the same pocket-sized, and for whatever θ_Grand < 0,

P _θ ( Y _{Due north}  ∈ ℒ_ε,N,1000 ) → 1 as Northward →  ∞

(15)

and, for any θ_K > 0,

P _θ ( Y _{Due north}  ∈ 𝒰_{ε,Due north,K} ) → 1 as N →  ∞ .

(sixteen)

Information technology is worthwhile to betoken out that whether virtually probability mass tends to exist concentrated on 1 chemical element of the sample space and the entropy of the distribution tends to 0 depends on the sufficient statistics. An exponential family that is degenerate with respect to sufficient statistics is as degenerate every bit it can be.

Equally nosotros will see in Section four, the degeneracy of exponential families with unstable sufficient statistics tends to push the mean-value parameters to the purlieus of the mean-value parameter space, which tends to obstruct statistical inference.

iii Impact of instability on MCMC simulation

If a Markov chain with unstable stationary distribution is constructed past MCMC methods, the excessive sensitivity and near-degeneracy of the stationary distribution tend to have a direct impact on MCMC simulation.

The excessive sensitivity of unstable stationary distributions, excessive in the sense that the nearest neighbor log odds are unbounded, affects the probabilities of transition between nearest neighbors: e.g., in applications to undirected graphs (cf. Section 2.one), Gibbs samplers sample elements y_ij from full conditional distributions of the form

Y _{i j} ∣ y _{−i j}  ∼ Bernoulli(π _{i j} ( y _{−i j} ; θ )),

(17)

where y _−ij denotes the collection of elements y _N excluding y_ij , and the log odds of π_ij ( y _−ij ; θ ) is given by

log π i j ( y − i j ; θ ) 1 − π i j ( y − i j ; θ ) = Λ N ( { y − i j , y i j = 0 } , { y − i j , y i j = 1 } ; θ ) .

(eighteen)

A Metropolis-Hastings algorithm moves from x _Northward to y _North , generated from a probability mass office f with support { y _N : y _N ~ x _N }, with probability

α ( ten Due north , y N ; θ ) = min { 1 , exp [ Λ N ( x N , y N ; θ ) ] f ( x N ∣ y N ) f ( y N ∣ ten North ) } .

(19)

Since the nearest neighbor log odds satisfy Λ _N ( x _N , y _N ; θ ) = −Λ _N ( y _N , x _N ; θ ) (all 10 _N ~ y _N ) and are unbounded past Theorem ane, Markov bondage with unstable stationary distributions can motility extremely fast from some subsets of the sample space to other subsets and extremely slowly back. In improver, if the mode of the probability mass function is non unique, multiple Markov chains may exist required, considering Theorems 1 and two betoken that one Markov chain may be trapped at 1 of the modes. Worse, Theorems 3 and iv suggest that MCMC simulation from exponential families with unstable sufficient statistics may be a waste of time and resources in the starting time place.

The most of import conclusion, though, is that mixing problems of MCMC algorithms tend to exist rooted in the unstable stationary distribution rather than the design of the MCMC algorithms, as is evident from the unbounded nearest neighbor log odds and the near-degeneracy of unstable stationary distributions. A related upshot and conclusion was reported by Handcock (2003a).

4 Impact of instability on statistical inference

The degeneracy of exponential families with unstable sufficient statistics tends to push the mean-value parameters to the purlieus of the mean-value parameter space and therefore tends to obstruct maximum likelihood estimation.

Permit μ_North: Θ ↦ int( ) be the map from parameter infinite Θ to the hateful-value parameter space int( ) (Barndorff-Nielsen 1978, p. 121) given past

μ _{Due north} ( θ ) =Due east _θ [thousand _N ( Y _{Due north} )] ∈ int(𝒞 _{Due north} ),

(20)

where int( ) denotes the interior of the convex hull of {g_{Due north} ( y _N ): y _N ∈ }.

We start with ane-parameter exponential families {P_{θ ,} θ ∈ Θ} with natural parameter θ and unstable sufficient statistic m_{Due north} ( y _{Due north} ). Let 50_N = 0 (without loss of generality) and U_N be the minimum and maximum of chiliad_N ( y _{Due north} ), respectively, and

μ N ( θ ) U N = E θ [ g Northward ( Y Northward ) ] U Northward ∈ ( 0 , ane )

(21)

be the mean-value parameter, where re-scaling by i/U_{Due north} ensures that the range of μ_N (θ)/U_N is (0, 1). The post-obit result shows that ane-parameter exponential families with unstable sufficient statistics yard_North ( y _N ) push the mean-value parameter μ_North (θ) to its infinum for all θ < 0 and its supremum for all θ > 0.

Corollary ane

The mean-value parameter μ_N (θ) of a one-parameter exponential family {P_θ , θ ∈ Θ} with natural parameter θ and unstable sufficient statistic g_{Due north} ( y _{Due north} ) tends to the boundary of the hateful-value parameter infinite in the sense that, for any θ < 0, notwithstanding small,

and, for any θ > 0, however minor,

By Corollary 1, the subset of the natural parameter space Θ corresponding to mean-value parameters far from the purlieus of the hateful-value parameter infinite tends to be a lower-dimensional subpace of Θ: the point θ = 0. In addition, the mean-value parameter μ_N (θ) can be expected to be extremely sensitive to changes of the natural parameter θ around 0.

The relationship betwixt the natural parameter θ and the mean-value parameter μ_N (θ) is problematic in terms of maximum likelihood interpretation. If g_{Due north} ( y _{Due north} ) ∈ int( ) denotes an observation in the interior of , the maximum likelihood estimate of θ exists and is unique (Barndorff-Nielsen 1978, p. 150) and is given by the root of the estimating part

δ _{Due north} (θ) =thousand _N ( y _Northward )−E _θ[one thousand _{Due north} ( Y _Northward )] =g _N ( y _N )−μ _{Due north} (θ).

(24)

The estimating part δ_N (θ) depends on θ through μ_N (θ), and since μ_North (θ) tends to be extremely sensitive to changes of θ around 0, and so does δ_N (θ). If the observation grand_North ( y _N ) is not close to the boundary of , the maximum likelihood estimate of θ tends to be shut to 0, since merely values of θ close to 0 map to values of μ_N (θ) which are non close to the boundary of . As a result, maximum likelihood algorithms tend to search for the maximum likelihood estimate of θ in a small neighborhood of 0, simply are hampered by the farthermost sensitivity of the estimating role δ_Northward (θ) around θ = 0 and tend to make small steps in the natural parameter space Θ around θ = 0 and large steps in the mean-value parameter space int( ) and struggle to converge. A related result and conclusion was reported past Handcock (2003a).

The behavior of K-parameter exponential families {P _θ , θ ∈ Θ} with natural parameters θ _ane, …, θ_K and Yard − 1 stable sufficient statistics g_{Northward ,1}( y _{Due north} ),…, g_{Northward , K −1}( y _{Due north} ) as well as i unstable sufficient statistic g_{N , K} ( y _N ) resembles the beliefs of ane-parameter exponential families with unstable sufficient statistic grand_{North , K} ( y _N ). Allow 50_{N , Grand} = 0 (without loss of generality) and U_{N , Thou} be the minimum and maximum of the unstable sufficient statistic g_{Northward , K} ( y _N ), respectively, and

μ N , K ( θ ) U North , G = Eastward θ [ grand North , Yard ( Y North ) ] U N , G ∈ ( 0 , 1 )

(25)

be the coordinate of the vector of hateful-value parameters μ_N ( θ ) corresponding to k_{Due north , K} ( y _N ).

Corollary 2

The vector of mean-value parameters μ_N ( θ ) of a G-parameter exponential family unit {P _θ , θ ∈ Θ} with natural parameters θ ₁, …, θ_Thousand and K − 1 stable sufficient statistics grand_{N ,1}( y _N ), …, g_{N , G −i}( y _North ) likewise equally one unstable sufficient statistic g_{North , K} ( y _Northward ) tends to the boundary of the hateful-value parameter infinite in the sense that, for any θ_Yard < 0, however pocket-size,

and, for any θ_Thousand > 0, nevertheless minor,

To conclude, while some maximum likelihood algorithms may outperform others, Corollaries one and 2 bespeak that all maximum likelihood algorithms tin can be expected to suffer from degeneracy with respect to sufficient statistics (cf. Handcock 2003a, Rinaldo et al. 2009).

five Applications to relational data

The intention of the nowadays section is to detect unstable subsets of the parameter infinite of detached exponential families, because unstable discrete exponential family distributions are characterized by excessive sensitivity and about-degeneracy (cf. Section 2), which tends to obstruct MCMC simulation (cf. Department three) as well as statistical inference (cf. Section 4).

We focus on applications to relational information, but note that in applications to lattice systems (Besag 1974) and binomial sampling, exponential family models (with suitable neighborhood assumptions) tend to be stable (Ruelle 1969). We consider undirected graphs and the most widely used exponential family models of undirected graphs, so-called exponential family unit random graph models (ERGMs) with Markov dependence and curved exponential family random graph models (curved ERGMs). Information technology is worthwhile to note that the number of degrees of liberty N is O(n ²) and is therefore large even when the number of nodes north is small, suggesting that the large-Northward results of Sections ii–4 shed light on the behavior of ERGMs fifty-fifty when north is not large.

A uncomplicated and appealing class of ERGMs with Markov dependence (Frank and Strauss 1986) is given past

q θ ( y N ) = ∑ m = 1 n − 1 η N , grand ( θ ) s N , k ( y N ) + η N , n ( θ ) ∑ i < j < k y i j y j m y i chiliad ,

(28)

where due south_{N , k} ( y _N ) = Σ _{i, j 1<…<j chiliad} y _{ij 1} · · ·y_ijk is the number of k-stars (1000 = ane, …, n − ane) and Σ _{i < j < k} y_ijy_jky_ik is the number of triangles. Since the number of natural parameters of (28) is north, it is common to impose linear or not-linear constraints on the natural parameters of (28) with an eye to reduce the number of parameters to be estimated. The following ERGMs are special cases of (28) obtained by imposing suitable linear constraints on the natural parameters of (28).

Result ane

ERGMs with 2-star terms of the form

q θ ( y Due north ) = θ 1 ∑ i < j y i j + θ 2 ∑ i , j < k y i j y i g

(29)

are unstable for all θ ₂ ≠ 0.

Result ii

ERGMs with triangle terms of the form

q θ ( y North ) = θ 1 ∑ i < j y i j + θ 2 ∑ i < j < thousand y i j y j one thousand y i k

(30)

are unstable for all θ _two ≠ 0.

Results 1 and 2 are in line with existing results: both ERGMs are known to be near-degenerate and problematic in terms of MCMC simulation and statistical inference (Strauss 1986, Jonasson 1999, Snijders 2002, Handcock 2003a, Rinaldo et al. 2009). The most hitting conclusion is that in both cases the subset of the natural parameter space ℝ² corresponding to not-degenerate distributions is a lower-dimensional subspace of ℝⁱⁱ: the line (θ ₁, 0). In terms of MCMC, the nearest neighborhood log odds |Λ _Northward ( x _N , y _N ; θ )| are O(north), which suggests that MCMC algorithms tend to suffer from extremely tedious mixing, as is well-known (Snijders 2002, Handcock 2002a, 2003a).

To reduce the problematic behavior of ERGMs of the course (29) and (thirty), it has sometimes been suggested to counterbalance positive instability-inducing terms by negative instability-inducing terms.

Result 3

ERGMs with two-star and triangle terms of the form

q θ ( y N ) = θ 1 ∑ i < j y i j + θ ii ∑ i , j < m y i j y i k + θ 3 ∑ i < j < chiliad y i j y j k y i m

(31)

are unstable for all θ ₂ and θ _three excluding θ _ii = θ ₃ = 0 and θ ₂ = − θ _three/three.

Result 3 demonstrates that counterbalancing instability-inducing terms does non, in general, piece of work: the subset of ℝⁱⁱⁱ corresponding to non-degenerate distributions is severely constrained by the linear constraints θ _two = θ ₃ = 0 and θ _ii = − θ ₃/3.

We turn to the curved ERGMs of Snijders et al. (2006) and Hunter and Handcock (2006), which were motivated by the problematic behavior of ERGMs with Markov dependence. Three of the all-time-known curved ERGM terms are geometrically weighted degree (GWD), geometrically weighted dyadwise shared partner (GWDSP), and geometrically weighted edgewise shared partner (GWESP) terms (cf. Hunter et al. 2008).

Result iv

Curved ERGMs with GWD terms of the form

q θ ( y N ) = θ one ∑ i < j y i j + θ ii exp [ θ 3 ] ∑ k = i northward − 1 [ one − ( 1 − exp [ − θ 3 ] ) k ] D Northward , g ( y North ) ,

(32)

where D_{Northward , k} ( y _{Due north} ) is the number of nodes i with degree Σ _{j ≠ i} y_ij = m, are unstable for all θ ₂ ≠ 0 and θ ₃ < −log two.

Result 5

Curved ERGMs with GWDSP terms of the form

q θ ( y N ) = θ i ∑ i < j y i j + θ 2 exp [ θ three ] ∑ thou = 1 due north − ii [ 1 − ( 1 − exp [ − θ iii ] ) k ] DSP Northward , k ( y Northward ) ,

(33)

where DSP_{North , k} ( y _North ) is the number of pairs of nodes {i, j} with Σ _{h ≠ i , j} y_ihy_jh = k dyadwise shared partners, are unstable for all θ _ii ≠ 0 and θ ₃ < −log ii.

Upshot six

Curved ERGMs with GWESP terms of the form

q θ ( y N ) = θ one ∑ i < j y i j + θ 2 exp [ θ 3 ] ∑ one thousand = 1 n − 2 [ ane − ( ane − exp [ − θ 3 ] ) thou ] ESP N , g ( y N ) ,

(34)

where ESP_{N , thousand} ( y _North ) is the number of pairs of nodes {i, j} with y_ij Σ _{h ≠ i , j} y_ihy_jh = k edgewise shared partners, are unstable for all θ ₂ ≠ 0 and θ ₃ < −log 2.

Thus, the parameter space of curved ERGMs with GWD, GWDSP, and GWESP terms contains unstable subsets. In terms of MCMC, in unstable subsets of the parameter space the curved ERGMs tend to exist worse than the ERGMs with Markov dependence: if θ _two ≠ 0 and θ ₃ < −log 2, the nearest neighborhood log odds |Λ _Northward ( x _N , y _{Due north} ; θ )| are O(exp[northward]). On the other hand, the curved ERGMs with GWD, GWDSP, and GWESP terms are stable provided θ ₂ ≠ 0 and θ ₃ ≥ −log two, which is encouraging and indicates that the effective parameter space is not-negligible, in contrast to ERGMs with Markov dependence. The unstable subsets of the parameter infinite of curved ERGMs should be penalized by specifying suitable penalties in a maximum likelihood framework and suitable priors in a Bayesian framework.

half dozen Simulation results

To demonstrate that unstable discrete exponential family distributions are characterized by excessive sensitivity and nigh-degeneracy (cf. Department two) and tend to obstruct MCMC simulation (cf. Department 3) and statistical inference (cf. Section 4), we resort to MCMC simulation of undirected graphs with northward = 32 nodes and N = 496 degrees of liberty from the ERGMs of Results 1–6 (cf. Section v). Since the computational cost of MCMC simulation is prohibitive, we exploit the fact that Results one–6 hold regardless of the value of θ ₁, the natural parameter corresponding to the sufficient statistic Σ _{i < j} y_ij , and gear up the value of θ _i at −one and the value of θ ₂ of the ERGMs of Results three–6 at 1. For every ERGM and every not-fixed parameter, nosotros consider 200 values in the interval [−5, 5]. At every such value, we generate an MCMC sample of size 2,000,000, discarding 1,000,000 draws as burn-in and recording every ane,000th mail service-burn down-in draw. The MCMC samples were generated past a Metropolis-Hastings algorithm of the form (19) (Hunter et al. 2008).

We start with two classic examples: the Bernoulli model with stable sufficient statistic 1000_{Due north} ( y _Northward ) = Σ _{i < j} y_ij and the 2-star model with unstable sufficient statistic grand_N ( y _Northward ) = Σ _{i, j < m} y_ijy_ik (cf. Section 2.1). Figure 1 plots the MCMC sample estimates of the hateful-value parameters μ_North (θ) = Due east_θ [thou_N ( Y _N )] of these models confronting the corresponding natural parameters θ. The MCMC sample guess of the hateful-value parameter μ_N (θ) of the Bernoulli model is shut to the verbal value μ_N (θ) = N/(1+exp[−θ]) (inside two standard deviations of the sample average based on random samples of size 1,000), demonstrating that MCMC simulation from the Bernoulli model is hardly problematic. The MCMC sample estimate of the mean-value parameter μ_North (θ) of the 2-star model is, in line with Corollary one, close to its infinum for all θ < 0 and close to its supremum for all θ > 0, and extremely sensitive to small changes of θ effectually 0.

MCMC sample estimate of mean-value parameter μ_Northward (θ) plotted against natural parameter θ of Bernoulli model and two-star model, where C_{Due north} ensures that the range of μ_N (θ)/C_N is (0, 1); shaded regions betoken unstable regions

The ERGMs with Markov dependence (Results i–3) are expected to be degenerate with respect to the unstable sufficient statistics, the number of 2-stars (Upshot 1), the number of triangles (Result 2), and the number of triangles (Outcome iii with 2-star parameter equal to 1), and the corresponding mean-value parameters are expected to be close to the purlieus of the mean-value parameter space. Figure ii plots the proportion of ii-stars (Result ane) and triangles (Results 2 and iii) against the corresponding natural parameter and confirms these considerations.

MCMC sample proportion of two-stars (Result one) and triangles (Results 2 and 3) plotted against corresponding natural parameter; shaded regions signal unstable regions

Concerning the curved ERGMs with GWD, GWDSP, and GWESP terms (Results four–six), since the number of sufficient statistics is linear in n, we focus on the sufficient statistic Σ _{i < j} y_ij , i of the virtually fundamental functions of undirected graphs y _N . We accept the coefficient of variation CV _N , defined as the standard divergence of Σ _{i < j} y_ij divided by the mean of Σ _{i < j} y_ij , as an indicator of mixing and nearly-degeneracy: low coefficients of variation indicate slow mixing and about-degeneracy. We dissever the coefficients of variation CV _{Due north} by the coefficient of variation CV _N (Bernoulli) under the corresponding ERGM with θ _ane = −ane and θ ₂ = 0, which corresponds to the Bernoulli model of Section 2.1 with θ = −1. Figure 3 plots the MCMC sample coefficients of variation CV _Northward /CV _N (Bernoulli) against the disquisitional parameter θ _three of the ERGMs of Results four–6. The simulation results indicate that in the unstable subset of the parameter infinite, corresponding to θ ₃ < −log 2, the coefficients of variation are close to 0, as expected, and around θ ₃ = −log ii, the coefficients of variation rise to a value comparable to the coefficient of variation CV _N (Bernoulli) under the respective Bernoulli model.

MCMC sample coefficient of variation CV _{Due north} of curved ERGM with GWD term (Event 4), GWDSP term (Effect v), and GWESP term (Consequence 6), re-scaled past 1/CV _North (Bernoulli); shaded regions betoken unstable regions

7 Discussion

Building on the piece of work of Strauss (1986) and Handcock (2002a, 2003a,b), we take introduced the notion of instability and shown that unstable discrete exponential family distributions are characterized past excessive sensitivity and near-degeneracy. In the important special example of exponential families with unstable sufficient statistics, the subset of the natural parameter space respective to non-degenerate distributions and mean-value parameters far from the boundary of the hateful-value parameter infinite turns out to exist a lower-dimensional subspace of the natural parameter space. These characteristics of instability tend to obstruct MCMC simulation and statistical inference. In applications to relational data, we detect that exponential families with Markov dependence tend to be unstable and that the parameter space of some curved exponential families contains unstable subsets. We conclude that unstable subsets of the parameter infinite of curved exponential families should exist penalized by specifying suitable penalties in a maximum likelihood framework and suitable priors in a Bayesian framework.

Information technology is worthwhile to point out that, while instability implies undesirable behavior such as well-nigh-degeneracy, stability is not—and cannot be—an insurance against well-nigh-degeneracy. Indeed, every discrete exponential family, with or without unstable sufficient statistics, includes near-degenerate distributions provided the natural parameters are sufficiently big (cf. Barndorff-Nielsen 1978, pp. 185–186, Handcock 2002a, 2003a,b). In add-on, while unstable sufficient statistics tin be stabilized, there are good reasons to be sceptical of elementary stabilization strategies. Consider one-parameter exponential families with natural parameter θ and unstable sufficient statistic grand_N ( y _North ). The unstable sufficient statistic yard_N ( y _N ) tin can be transformed into the stable sufficient statistic thousand_N ( y _Northward )/U_N past dividing g_N ( y _N ) by its maximum U_North . Since the canonical form of exponential families is not unique (Brown 1986, pp. 7–eight), mapping yard_N ( y _North ) to g_N ( y _N )/U_Northward is equivalent to mapping θ to θ/U_{Due north} and tin can therefore be regarded as a reparameterization of the exponential family with unstable sufficient statistic chiliad_N ( y _{Due north} ). Let η_N (θ) = θ/U_N . Past the parameterization invariance of maximum likelihood estimators, the maximum likelihood estimators $\widehat{\theta }$ and η ^ N = def η N ( θ ) ^ of θ respectively η_N (θ) satisfy $\widehat{\theta }$ = $\widehat{\eta }$ _{Due north} U_N . The probability of information under the maximum likelihood estimator is the same nether both parameterizations. The simple stabilization strategy therefore fails to accost the trouble of lack of fit: even nether the maximum likelihood estimator, the probability of data may be extremely depression relative to other elements of the sample space and the fit of the model thus unacceptable (cf. Hunter et al. 2008). The argument extends to K-parameter exponential families and linear transformations of sufficient statistics (Brown 1986, pp. seven–viii).

Last, while the weather under which maximum likelihood estimators of discrete exponential families for dependent data be and are unique are well-understood (cf. Barndorff-Nielsen 1978, p. 151, Handcock 2002a, 2003a,b, Rinaldo et al. 2009), information technology is an open question which conditions ensure consistency and asymptotic normality of maximum likelihood estimators (cf. Hunter and Handcock 2006, Rinaldo et al. 2009). An anonymous referee suggested semi-group construction (cf. Lauritzen 1988, pp. 140–146). Semi-group structure implies stability and holds hope.

Acknowledgments

Support is acknowledged from the Netherlands Organisation for Scientific Inquiry (NWO grant 446-06-029), the National Institute of Health (NIH grant 1R01HD052887-01A2), and the Function of Naval Research (ONR grant N00014-08-1-1015). The writer is grateful to David Hunter and two bearding referees for stimulating questions and suggestions.

A Appendix: proofs

Proof of Theorem ane

We testify Theorem 1 by contradiction. Given an unstable detached exponential family distribution, suppose that there exist C > 0 and North_C > 0 such that

(35)

Consider a given Due north ≥ 1. Let a _{Due north} ∈ { y _Northward ∈ : q_θ ( y _N ) = I_N ( θ )} and b _N ∈ { y _N ∈ : q _θ ( y _North ) = South_{Due north} ( θ )}, and let K_N ≤ N be the number of non-matching elements of a _N and b _N . By changing the non-matching elements of a _N and b _{Due north} one by i, it is possible to go from a _N to b _N within K_N ≤ Due north steps. Let y _{N ,0}, y _{North ,1}, …, y _{N,K N −1}, y _{N,K N} exist a path from a _N to b _N such that y _{Northward ,0} = a _Northward and y _{Due north,K Due north} = b _N and y _{Northward , thou −1} ~ y _{N , yard} (one thousand = 1, …, Chiliad_N ). By Jensen'southward inequality and (35), at that place exist C > 0 and North_C > 0 such that, for any N > North_C ,

| ∑ k = 1 K N Λ N ( y N , m − one , y N , k ; θ ) | ≤ ∑ thou = i One thousand Due north ∣ Λ North ( y N , k − 1 , y N , k ; θ ) ∣ ≤ C North .

(36)

The left-hand side of (36) is, by definition of a _North and b _N , given by

| ∑ k = ane K N Λ N ( y N , 1000 − i , y N , grand ; θ ) | = ∣ q θ ( b N ) − q θ ( a N ) ∣ = S N ( θ ) .

(37)

Thus, (35) implies that at that place exist C > 0 and N_C > 0 such that

S _N ( θ ) ≤C N for all N >North _C ,

(38)

which contradicts the supposition of instability.

Proof of Theorem two

For whatever 0 < δ < ε < one, however pocket-size, and whatsoever Due north ≥ 1,

P _θ ( Y _N  ∈ ℳ_ε,N ) ≥P _θ ( Y _Northward  ∈ ℳ_δ,Northward ) ≥ exp[(1 − δ)South _Northward ( θ )−ψ _N ( θ )]

(39)

using the fact that contains at least one element, and

1 −P _θ ( Y _N  ∈ ℳ_ε,N ) < exp[NorthlogM + (1 − ε)S _N ( θ )−ψ _N ( θ )]

(twoscore)

using the fact that \ contains at almost exp[N logM] −1 < exp[N logM] elements. Thus, the log odds of P_θ ( Y _N ∈ ) is given past

ω ε , N = log P θ ( Y Due north ∈ M ε , N ) ane − P θ ( Y N ∈ Thou ε , N ) > ( ε − δ ) S Due north ( θ ) − N log M .

(41)

By instability, for any C > 0, however large, there exists Due north_C > 0 such that

ω_ε,N  > (ε − δ)South _N ( θ )−NorthlogM > [(ε − δ)C − logM] N for all N >N _C .

(42)

Since ε − δ > 0 and C > 0 can exist as large as desired, ω_{ε , N} → ∞ as N → ∞ and (ix) holds.

Proof of Theorems 3 and 4

We show Theorem 4, since Theorem 3 can be considered to exist a special case of Theorem 4.

Case 1: θ_K < 0. For whatever 0 < δ < ε < 1, however small, and any N ≥ 1,

P θ ( Y Northward ∈ Fifty ε , N , 1000 ) ≥ P θ ( Y N ∈ L δ , N , K ) ≥ exp [ ∑ thousand = 1 K − 1 min y N ∈ 50 δ , N , G [ θ one thousand 1000 N , k ( y N ) ] + θ G δ U N , One thousand − ψ N ( θ ) ]

(43)

and

1 − P θ ( Y N ∈ L ε , North , Yard ) < exp [ N log M + ∑ k = 1 K − i max y N ∈ Y N ∖ L ε , Northward , Yard [ θ k g North , k ( y Due north ) ] + θ K ε U N , Chiliad − ψ N ( θ ) ] .

(44)

Thus, the log odds of P_θ ( Y _N ∈ ) is given past

ω ε , North , K = log P θ ( Y N ∈ L ε , N , One thousand ) 1 − P θ ( Y Due north ∈ 50 ε , N , K ) > − θ K ( ε − δ ) U N , Thousand − Due north log Grand − ∑ k = one One thousand − 1 ∣ θ k ∣ U N , k .

(45)

Since −θ_Chiliad > 0, ε − δ > 0, and the sufficient statistic thousand_{N , Chiliad} ( y _N ) is unstable, the term −θ_M (ε − δ) U_{North , 1000} on the right-hand side of (45) is positive and not bounded by North, while the stability of the sufficient statistics g_{N ,i}( y _{Due north} ), …, m_{Northward , K −one}( y _N ) implies that the other terms on the right-mitt side of (45) are bounded by N. Thus, for whatsoever θ_G < 0, ω_{ε , N , K} → ∞ as N → ∞ and (12) holds.

Case two: θ_Yard > 0. The case θ_K > 0 proceeds along the same lines as the case θ_K < 0, mutatis mutandis, to evidence that (13) holds.

Proof of Theorem 5

A proof of Theorem 5 proceeds along the same lines as the proof of Theorem 4, with the exception that the sufficient statistics g_{Northward ,one}( y _N ), …, g_{N , G −1}( y _N ) may be unstable but are dominated past the unstable sufficient statistic grand_{Due north , G} ( y _{Due north} ).

Proof of Corollaries ane and 2

We prove Corollary 2, since Corollary ane can be considered to be a special case of Corollary one.

Case 1: θ_Chiliad < 0. For any 0 < γ < ane, however modest, and any North ≥ 1, one can partition the sample space into the subsets and \ . Therefore,

(46)

By Theorem 4, for whatever 0 < δ < 1, withal small, and whatever θ_K < 0, there exists North_δ > 0 such that

(47)

Since γ and δ can be as pocket-sized every bit desired, (26) holds.

Case ii: θ_K > 0. The instance θ_K > 0 gain along the same lines every bit the case θ_K < 0, mutatis mutandi, to show that (27) holds.

Proof of Results 1–half-dozen

Let 0 _N exist the empty graph (0 _ij = 0, all i < j) and 1 _North be the complete graph (ane _ij = 1, all i < j) given n nodes and North = n(n − i)/2 degrees of liberty. For every ERGM of Results one–6, every θ ∈ Θ, and every n > 1, q_θ (0 _N ) = 0 and

Southward _Northward ( θ )−I _N ( θ ) ≥  ∣q _θ (ane _N )−q _θ (0 _{Due north} )∣ =  ∣q _θ (1 _N )∣.

(48)

Therefore, all θ ∈ Θ such that |q_θ (i _N )| is not divisional past Northward requite rise to unstable distributions P_θ , proving Results 1–6.

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Source: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3405854/

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