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# American Institute of Mathematical Sciences

March  2016, 5(1): 185-199. doi: 10.3934/eect.2016.5.185

## A matrix-valued generator $\mathcal{A}$ with strong boundary coupling: A critical subspace of $D((-\mathcal{A})^{\frac{1}{2}})$ and $D((-\mathcal{A}^*)^{\frac{1}{2}})$ and implications

 1 Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, United States

Received  September 2015 Revised  February 2016 Published  March 2016

We study the free dynamic operator $\mathcal{A}$ which arises in the study of a heat-viscoelastic structure model with highly coupled boundary conditions at the interface between the heat domain and the contiguous structure domain. We use Baiocchi's characterization on the interpolation of subspaces defined by a constrained map [1], [16,p 96] to identify a relevant subspace $V_0$ of both $D((-\mathcal{A})^{\frac{1}{2}})$ and $D((-\mathcal{A}^∗)^{\frac{1}{2}})$, which is sufficient to determine the optimal regularity of the interface (boundary) $\to$ interior map $\mathcal{A}^{-1} \mathcal{B}_N$ from the interface to the energy space. Here, $\mathcal{B}_N$ is the (boundary) control operator acting at the interface in the Neumann boundary conditions.
Citation: Roberto Triggiani. A matrix-valued generator $\mathcal{A}$ with strong boundary coupling: A critical subspace of $D((-\mathcal{A})^{\frac{1}{2}})$ and $D((-\mathcal{A}^*)^{\frac{1}{2}})$ and implications. Evolution Equations & Control Theory, 2016, 5 (1) : 185-199. doi: 10.3934/eect.2016.5.185
##### References:
 [1] C. Baiocchi, Un teorema di interpolazione: Applicazioni ai problemi ai limiti per le equazioni a derivate parziali,, Ann. Mat. Pura Appl., 73 (1966), 233. doi: 10.1007/BF02415089. Google Scholar [2] A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems,, $2^{nd}$ edition, (2007). doi: 10.1007/978-0-8176-4581-6. Google Scholar [3] S. Chen and R. Triggiani, Proof of two conjectures of G. Chen and D. L. Russell on structural damping for elastic systems: The case $\alpha = 1/2$,, Springer-Verlag Lecture Notes in Mathematics, 1354 (1988), 234. doi: 10.1007/BFb0089601. Google Scholar [4] S. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems: The case $1/2 \leq \alpha \leq 1$),, Pacific J. Math., 136 (1989), 15. doi: 10.2140/pjm.1989.136.15. Google Scholar [5] S. Chen and R. Triggiani, Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications,, J. Diff. Eqns., 88 (1990), 279. doi: 10.1016/0022-0396(90)90100-4. Google Scholar [6] L. De Simon, Un'applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine,, Rendiconti del Seminario Matematico della Universita di Padova, 34 (1964), 205. Google Scholar [7] D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order,, Proc. Japan Acad., 43 (1967), 82. doi: 10.3792/pja/1195521686. Google Scholar [8] P. Grisvard, Characterization de qualques espaces d' interpolation,, Arch. Pat. Mech. Anal., 25 (1967), 40. doi: 10.1007/BF00281421. Google Scholar [9] T. Kato, Fractional powers of dissipative operators,, J. Math. Soc. Japan., 13 (1961), 246. doi: 10.2969/jmsj/01330246. Google Scholar [10] I. Lasiecka, Unified theory for abstract parabolic boundary problems-a semigroup approach,, Appl. Math. & Optimiz., 6 (1980), 287. doi: 10.1007/BF01442900. Google Scholar [11] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories I, Abstract Parabolic Systems, Encyclopedia of Mathematics and Its Applications Series, (2000). Google Scholar [12] I. Lasiecka and R. Triggiani, Domains of fractional powers of matrix-valued Operators: A general approach,, Operator Semigroups Meet Complex Analysis, 250 (2015), 297. doi: 10.1007/978-3-319-18494-4_20. Google Scholar [13] I. Lasiecka and R. Triggiani, Heat-structure interaction with viscoelastic damping: Analyticity with sharp analytic sector, exponential decay,, Communications on Pure & Applied Analysis, (). Google Scholar [14] C. Lebiedzik and R. Triggaini, The optimal interior regularity for the critical case of a clamped thermoelastic system with point control revisited,, Modern Aspects of the Theory of PDEs. Vol. 216 of Operator Theory: Advances and Applications, (2011), 243. doi: 10.1007/978-3-0348-0069-3_14. Google Scholar [15] J. L. Lions, Especes d'interpolation et domaines de puissances fractionnaires d'openateurs,, J. Math Soc., 14 (1962), 233. doi: 10.2969/jmsj/01420233. Google Scholar [16] J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Propblems and Applications, Vol. I,, Springer-Verlag (1972), (1972). Google Scholar [17] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [18] R. Triggiani, A heat-viscoelastic structure interaction model with Neumann or Dirichlet boundary control at the interface: optimal regularity, control theoretic implications,, Applied Mathematics and Optimization, (). Google Scholar

show all references

##### References:
 [1] C. Baiocchi, Un teorema di interpolazione: Applicazioni ai problemi ai limiti per le equazioni a derivate parziali,, Ann. Mat. Pura Appl., 73 (1966), 233. doi: 10.1007/BF02415089. Google Scholar [2] A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems,, $2^{nd}$ edition, (2007). doi: 10.1007/978-0-8176-4581-6. Google Scholar [3] S. Chen and R. Triggiani, Proof of two conjectures of G. Chen and D. L. Russell on structural damping for elastic systems: The case $\alpha = 1/2$,, Springer-Verlag Lecture Notes in Mathematics, 1354 (1988), 234. doi: 10.1007/BFb0089601. Google Scholar [4] S. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems: The case $1/2 \leq \alpha \leq 1$),, Pacific J. Math., 136 (1989), 15. doi: 10.2140/pjm.1989.136.15. Google Scholar [5] S. Chen and R. Triggiani, Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications,, J. Diff. Eqns., 88 (1990), 279. doi: 10.1016/0022-0396(90)90100-4. Google Scholar [6] L. De Simon, Un'applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine,, Rendiconti del Seminario Matematico della Universita di Padova, 34 (1964), 205. Google Scholar [7] D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order,, Proc. Japan Acad., 43 (1967), 82. doi: 10.3792/pja/1195521686. Google Scholar [8] P. Grisvard, Characterization de qualques espaces d' interpolation,, Arch. Pat. Mech. Anal., 25 (1967), 40. doi: 10.1007/BF00281421. Google Scholar [9] T. Kato, Fractional powers of dissipative operators,, J. Math. Soc. Japan., 13 (1961), 246. doi: 10.2969/jmsj/01330246. Google Scholar [10] I. Lasiecka, Unified theory for abstract parabolic boundary problems-a semigroup approach,, Appl. Math. & Optimiz., 6 (1980), 287. doi: 10.1007/BF01442900. Google Scholar [11] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories I, Abstract Parabolic Systems, Encyclopedia of Mathematics and Its Applications Series, (2000). Google Scholar [12] I. Lasiecka and R. Triggiani, Domains of fractional powers of matrix-valued Operators: A general approach,, Operator Semigroups Meet Complex Analysis, 250 (2015), 297. doi: 10.1007/978-3-319-18494-4_20. Google Scholar [13] I. Lasiecka and R. Triggiani, Heat-structure interaction with viscoelastic damping: Analyticity with sharp analytic sector, exponential decay,, Communications on Pure & Applied Analysis, (). Google Scholar [14] C. Lebiedzik and R. Triggaini, The optimal interior regularity for the critical case of a clamped thermoelastic system with point control revisited,, Modern Aspects of the Theory of PDEs. Vol. 216 of Operator Theory: Advances and Applications, (2011), 243. doi: 10.1007/978-3-0348-0069-3_14. Google Scholar [15] J. L. Lions, Especes d'interpolation et domaines de puissances fractionnaires d'openateurs,, J. Math Soc., 14 (1962), 233. doi: 10.2969/jmsj/01420233. Google Scholar [16] J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Propblems and Applications, Vol. I,, Springer-Verlag (1972), (1972). Google Scholar [17] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [18] R. Triggiani, A heat-viscoelastic structure interaction model with Neumann or Dirichlet boundary control at the interface: optimal regularity, control theoretic implications,, Applied Mathematics and Optimization, (). Google Scholar
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# How do we specify a concentration, given 0.02*mol of calcium chloride dissolved in a volume of water?

Jul 29, 2017

This question is ill-proposed...........

#### Explanation:

I could have a $1 \cdot L$ volume of $0.02 \cdot m o l \cdot {L}^{-} 1$ of $C a C {l}_{2}$ containing $0.02 \cdot m o l$ $C a C {l}_{2}$;

I could have a $10 \cdot L$ volume of $0.02 \cdot m o l \cdot {L}^{-} 1$ of $C a C {l}_{2}$ containing $0.2 \cdot m o l$ of $C a C {l}_{2}$;

I could have a $100 \cdot L$ volume of $0.02 \cdot m o l \cdot {L}^{-} 1$ of $C a C {l}_{2}$ containing $2.0 \cdot m o l$ of $C a C {l}_{2}$.

Can you work out the MASS of solute in each solution?

When we specify a concentration we refer to the quotient.......

$\text{Moles of solute"/"Volume of solution}$, and thus $\text{concentration}$ typically has the units $m o l \cdot {L}^{-} 1$.

format: plain


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A Vector Space for Distributional Semantics for Entailment  This work was partially supported by French ANR grant CIFRE N 1324/2014.

# A Vector Space for Distributional Semantics for Entailment ††thanks:  This work was partially supported by French ANR grant CIFRE N 1324/2014.

James Henderson    Diana Nicoleta Popa
Xerox Research Centre Europe
james.henderson@xrce.xerox.com
diana.popa@xrce.xerox.com
###### Abstract

Distributional semantics creates vector-space representations that capture many forms of semantic similarity, but their relation to semantic entailment has been less clear. We propose a vector-space model which provides a formal foundation for a distributional semantics of entailment. Using a mean-field approximation, we develop approximate inference procedures and entailment operators over vectors of probabilities of features being known (versus unknown). We use this framework to reinterpret an existing distributional-semantic model (Word2Vec) as approximating an entailment-based model of the distributions of words in contexts, thereby predicting lexical entailment relations. In both unsupervised and semi-supervised experiments on hyponymy detection, we get substantial improvements over previous results.

A Vector Space for Distributional Semantics for Entailment thanks:  This work was partially supported by French ANR grant CIFRE N 1324/2014.

James Henderson and Diana Nicoleta Popa Xerox Research Centre Europe james.henderson@xrce.xerox.com and diana.popa@xrce.xerox.com

## 1 Introduction

Modelling entailment is a fundamental issue in computational semantics. It is also important for many applications, for example to produce abstract summaries or to answer questions from text, where we need to ensure that the input text entails the output text. There has been a lot of interest in modelling entailment in a vector-space, but most of this work takes an empirical, often ad-hoc, approach to this problem, and achieving good results has been difficult [Levy et al. (2015]. In this work, we propose a new framework for modelling entailment in a vector-space, and illustrate its effectiveness with a distributional-semantic model of hyponymy detection.

Unlike previous vector-space models of entailment, the proposed framework explicitly models what information is unknown. This is a crucial property, because entailment reflects what information is and is not known; a representation entails a representation if and only if everything that is known given is also known given . Thus, we model entailment in a vector space where each dimension represents something we might know. As illustrated in Table 1, knowing that a feature is true always entails knowing that same feature, but never entails knowing that a different feature is true. Also, knowing that a feature is true always entails not knowing anything (unk), since strictly less information is still entailment, but the reverse is never true. Table 1 also illustrates that knowing that a feature is false () patterns exactly the same way as knowing that an unrelated feature is true. This illustrates that the relevant dichotomy for entailment is known versus unknown, and not true versus false.

Previous vector-space models have been very successful at modelling semantic similarity, in particular using distributional semantic models (e.g. [Deerwester et al. (1990, Schütze (1993, Mikolov et al. (2013a]). Distributional semantics uses the distributions of words in contexts to induce vector-space embeddings of words, which have been shown to be useful for a wide variety of tasks. Two words are predicted to be similar if the dot product between their vectors is high. But the dot product is an anti-symmetric operator, which makes it more natural to interpret these vectors as representing whether features are true or false, whereas the dichotomy known versus unknown is asymmetric. We surmise that this is why distributional semantic models have had difficulty modelling lexical entailment [Levy et al. (2015].

To develop a vector-space model of whether features are known or unknown, we start with discrete binary vectors, where 1 means known and 0 means unknown. Entailment between these discrete binary vectors can be calculated by independently checking each dimension. But as soon as we try to do calculations with distributions over these vectors, we need to deal with the case where the features are not independent. For example, if feature has a 50% chance of being true and a 50% chance of being false, we can’t assume that there is a 25% chance that both and are known. This simple case of mutual exclusion is just one example of a wide range of constraints between features which we need to handle in semantic models. These constraints mean that the different dimensions of our vector space are not independent, and therefore exact models are not factorised. Because the models are not factorised, exact calculations of entailment and exact inference of vectors are intractable.

Mean-field approximations are a popular approach to efficient inference for intractable models. In a mean-field approximation, distributions over binary vectors are represented using a single probability for each dimension. These vectors of real values are the basis of our proposed vector space for entailment.

In this work, we propose a vector-space model which provides a formal foundation for a distributional semantics of entailment. This framework is derived from a mean-field approximation to entailment between binary vectors, and includes operators for measuring entailment between vectors, and procedures for inferring vectors in an entailment graph. We validate this framework by using it to reinterpret existing Word2Vec [Mikolov et al. (2013a] word embedding vectors as approximating an entailment-based model of the distribution of words in contexts. This reinterpretation allows us to use existing word embeddings as an unsupervised model of lexical entailment, successfully predicting hyponymy relations using the proposed entailment operators in both unsupervised and semi-supervised experiments.

## 2 Modelling Entailment in a Vector Space

To develop a model of entailment in a vector space, we start with the logical definition of entailment in terms of vectors of discrete known features: entails if and only if all the known features in are also included in . We formalise this relation with binary vectors where means known and means unknown, so this discrete entailment relation can be defined with the binary formula:

 P((y⇒x) | x,y)=∏k(1−(1−yk)xk)\vspace−1ex

Given prior probability distributions over these vectors, the exact joint and marginal probabilities for an entailment relation are:

 P(x,y,(y⇒x))=P(x) P(y)∏k(1−(1−yk)xk)\vspace−2.5ex
 P((y⇒x))=EP(x)EP(y)∏k(1−(1−yk)xk) (1)

We cannot assume that the priors and are factorised, because there are many important correlations between features and therefore we cannot assume that the features are independent. As discussed in Section 1, even just representing both a feature and its negation requires two different dimensions and in the vector space, because represents unknown and not false. Given valid feature vectors, calculating entailment can consider these two dimensions separately, but to reason with distributions over vectors we need the prior to enforce the constraint that and are mutually exclusive. In general, such correlations and anti-correlations exist between many semantic features, which makes inference and calculating the probability of entailment intractable.

To allow for efficient inference in such a model, we propose a mean-field approximation. This in effect assumes that the posterior distribution over vectors is factorised, but in practice this is a much weaker assumption than assuming the prior is factorised. The posterior distribution has less uncertainty and therefore is influenced less by non-factorised prior constraints. By assuming a factorised posterior, we can then represent distributions over feature vectors with simple vectors of probabilities of individual features (or as below, with their log-odds). These real-valued vectors are the basis of the proposed vector-space model of entailment.

In the next two subsections, we derive a mean-field approximation for inference of real-valued vectors in entailment graphs. This derivation leads to three proposed vector-space operators for approximating the log-probability of entailment, summarised in Table 2. These operators will be used in the evaluation in Section 5. This inference framework will also be used in Section 3 to model how existing word embeddings can be mapped to vectors to which the entailment operators can be applied.

### 2.1 A Mean-Field Approximation

A mean-field approximation approximates the posterior using a factorised distribution . First of all, this gives us a concise description of the posterior as a vector of continuous values , where (i.e. the marginal probabilities of each bit). Secondly, as is shown below, this gives us efficient methods for doing approximate inference of vectors in a model.

First we consider the simple case where we want to approximate the posterior distribution . In a mean-field approximation, we want to find a factorised distribution which minimises the KL-divergence with the true distribution .

 L = DKL(Q(x,y)||P(x,y|(y⇒x))) ∝∑xQ(x)logQ(x,y)P(x,y,(y⇒x)) = ∑kEQ(xk)logQ(xk)+∑kEQ(yk)logQ(yk) −EQ(x)logP(x)−EQ(y)logP(y) −∑kEQ(xk)EQ(yk)log(1−(1−yk)xk)

In the final equation, the first two terms are the negative entropy of , , which acts as a maximum entropy regulariser, the final term enforces the entailment constraint, and the middle two terms represent the prior for and . One approach (generalised further in the next subsection) to the prior terms is to bound them by assuming is a function in the exponential family, giving us:

 EQ(x)logP(x) ≥EQ(x)logexp(∑kθxkxk)Zθ = ∑kEQ(xk)θxkxk−logZθ

where the is not relevant in any of our inference problems and thus will be dropped below.

As typically in mean-field approximations, inference of and can’t be done efficiently with this exact objective , because of the nonlinear interdependence between and in the last term. Thus, we introduce two approximations to , one for use in inferring given (forward inference), and one for the reverse inference problem (backward inference). In both cases, the approximation is done with an application of Jensen’s inequality to the log function, which gives us an upper bound on , as is standard practice in mean-field approximations. For forward inference:

 L≤ −H(Q)−Q(xk=1)θxk−EQ(yk)θykyk (2) −Q(xk=1)logQ(yk=1) )

which we can optimise for :

 Q(xk=1)=σ( θxk+logQ(yk=1) ) (3)

where is the sigmoid function. The sigmoid function arises from the entropy regulariser, making this a specific form of maximum entropy model. And for backward inference:

 L≤ −H(Q)−EQ(xk)θxkxk−Q(yk=1)θyk (4) −(1−Q(yk=1))log(1−Q(xk=1)) )

which we can optimise for :

 Q(yk=1)= σ( θyk−log(1−Q(xk=1)) ) (5)

Note that in equations (2) and (4) the final terms, and respectively, are approximations to the log-probability of the entailment. We define two vector-space operators, and , to be these same approximations.

 logQ(y⇒x) ≈∑kEQ(xk)log(EQ(yk)(1−(1−yk)xk)) = Q(x=1)⋅logQ(y=1) ≡X\raisebox1.8pt\makebox[0.0pt][l]$<$$◯Y logQ(y⇒x) ≈∑kEQ(yk)log(EQ(xk)(1−(1−yk)xk)) = (1−Q(y=1))⋅log(1−Q(x=1)) ≡Y\raisebox1.8pt\makebox[0.0pt][l]>$$◯$X

We parametrise these operators with the vectors of log-odds of , namely . The resulting operator definitions are summarised in Table 2.

Also note that the probability of entailment given in equation (1) becomes factorised when we replace with . We define a third vector-space operator, , to be this factorised approximation, also shown in Table 2.

### 2.2 Inference in Entailment Graphs

In general, doing inference for one entailment is not enough; we want to do inference in a graph of entailments between variables. In this section we generalise the above mean-field approximation to entailment graphs.

To represent information about variables that comes from outside the entailment graph, we assume we are given a prior over all variables in the graph. As above, we do not assume that this prior is factorised. Instead we assume that the prior is itself a graphical model which can be approximated with a mean-field approximation.

Given a set of variables each representing vectors of binary variables , a set of entailment relations , and a set of negated entailment relations , we can write the joint posterior probability as:

 P(x,r,¯r)= 1ZP(x)∏i( (∏j:r(i,j)∏kP(xik⇒xjk|xik,xjk)) (∏j:¯r(i,j)(1−∏kP(xik⇒xjk|xik,xjk))) )

We want to find a factorised distribution that minimises . As above, we bound this loss for each element of each vector we want to infer, using analogous Jensen’s inequalities for the terms involving nodes and such that or . For completeness, we also propose similar inequalities for nodes and such that or , and bound them using the constants

 Cijk≥∏k′≠k(1−σ(−Xik′)σ(Xjk′)).\vspace−1ex

To represent the prior , we use the terms

 θik(X¯ik)≤logEQ(x¯ik)P(x¯ik,xik=1)1−EQ(x¯ik)P(x¯ik,xik=1)\vspace−1ex

where is the set of all such that either or . These terms can be thought of as the log-odds terms that would be contributed to the loss function by including the prior’s graphical model in the mean-field approximation.

Now we can infer the optimal as:

 Xik = θik(X¯ik) +∑j:r(i,j)−logσ(−Xjk) (6) +∑j:r(j,i)logσ(Xjk) +∑j:¯r(j,i)log1−Cijkσ(Xjk)1−Cijk +∑j:¯r(i,j)−log1−Cijkσ(−Xjk)1−Cijk

In summary, the proposed mean-field approximation does inference in entailment graphs by iteratively re-estimating each as the sum of: the prior log-odds, for each entailed variable , and for each entailing variable .111It is interesting to note that is a non-negative transform of , similar to the ReLU nonlinearity which is popular in deep neural networks [Glorot et al. (2011]. is the analogous non-positive transform. This inference optimises for each entailing plus for each entailed , plus a maximum entropy regulariser on . Negative entailment relations, if they exist, can also be incorporated with some additional approximations. Complex priors can also be incorporated through their log-odds, simulating the inclusion of the prior within the mean-field approximation.

Given its dependence on mean-field approximations, it is an empirical question to what extent we should view this model as computing real entailment probabilities and to what extent we should view it as a well-motivated non-linear mapping for which we simply optimise the input-output behaviour (as for neural networks [Henderson and Titov (2010]). In Sections 3 and 5 we argue for the former (stronger) view.

## 3 Interpreting Word2Vec Vectors

To evaluate how well the proposed framework provides a formal foundation for the distributional semantics of entailment, we use it to re-interpret an existing model of distributional semantics in terms of semantic entailment. There has been a lot of work on how to use the distribution of contexts in which a word occurs to induce a vector representation of the semantics of words. In this paper, we leverage this previous work on distributional semantics by re-interpreting a previous distributional semantic model and using this understanding to map its vector-space word embeddings to vectors in the proposed framework. We then use the proposed operators to predict entailment between words using these vectors. In Section 5 below, we evaluate these predictions on the task of hyponymy detection. In this section we motivate three different ways to interpret the Word2Vec [Mikolov et al. (2013a, Mikolov et al. (2013b] distributional semantic model as an approximation to an entailment-based model of the semantic relationship between a word and its context.

Distributional semantics learns the semantics of words by looking at the distribution of contexts in which they occur. To model this relationship, we assume that the semantic features of a word are (statistically speaking) redundant with those of its context words, and consistent with those of its context words. We model these properties using a hidden vector which is the consistent unification of the features of the middle word and the context. In other words, there must exist a hidden vector which entails both of these vectors, and is consistent with prior constraints on vectors. We split this into two steps, inference of the hidden vector from the middle vector , context vectors and prior, and computing the log-probability (7) that this hidden vector entails the middle and context vectors:

 maxY(logP(y,y⇒xm,y⇒xc)) (7)

We interpret Word2Vec’s Skip-Gram model as learning its context and middle word vectors so that the log-probability of this entailment is high for the observed context words and low for other (sampled) context words. The word embeddings produced by Word2Vec are only related to the vectors assigned to the middle words; context vectors are computed but not output. We model the context vectors as combining (as in equation (5)) information about a context word itself with information which can be inferred from this word given the prior, .

The numbers in the vectors output by Word2Vec are real numbers between negative infinity and infinity, so the simplest interpretation of them is as the log-odds of a feature being known. In this case we can treat these vectors directly as the in the model. The inferred hidden vector can then be calculated using the model of backward inference from the previous section.

 Y =θc−logσ(−Xc)−logσ(−Xm) =X′c−logσ(−Xm)

Since the unification of context and middle word features is computed using backward inference, we use the backward-inference operator to calculate how successful that unification was. This gives us the final score:

 logP(y,y⇒xm,y⇒xc) ≈ Y\raisebox1.8pt\makebox[0.0pt][l]$>$$◯Xm+Y\raisebox1.8pt\makebox[0.0pt][l]>$$◯$Xc+−σ(−Y)⋅θc = Y\raisebox1.8pt\makebox[0.0pt][l]$>$$◯Xm+−σ(−Y)⋅X′c This is a natural interpretation, but it ignores the equivalence in Word2Vec between pairs of positive values and pairs of negative values, due to its use of the dot product. As a more accurate interpretation, we interpret each Word2Vec dimension as specifying whether its feature is known to be true or known to be false. Translating this Word2Vec vector into a vector in our entailment vector space, we get one copy of the vector representing known-to-be-true features and a second negated duplicate of the vector representing known-to-be-false features, which we concatenate to get our representation .  Y+= X′c−logσ(−Xm) Y−= −X′c−logσ(Xm) logP(y,y⇒xm,y⇒xc) ≈ Y+\raisebox1.8pt\makebox[0.0pt][l]>$$◯$Xm+−σ(−Y+)⋅X′c +Y−\raisebox1.8pt\makebox[0.0pt][l]$>$$◯(−Xm)+−σ(−Y−)⋅(−X′c) As a third alternative, we modify this latter interpretation with some probability mass reserved for unknown in the vicinity of zero. By subtracting 1 from both the original and negated copies of each dimension, we get a probability of unknown of . This gives us:  Y+= X′c−logσ(−(Xm−1)) Y−= −X′c−logσ(−(−Xm−1)) logP(y,y⇒xm,y⇒xc) ≈ Y+\raisebox1.8pt\makebox[0.0pt][l]>$$◯$(Xm−1)+−σ(−Y+)⋅X′c +Y−\raisebox1.8pt\makebox[0.0pt][l]$>$$◯$(−Xm−1))+−σ(−Y−)⋅(−X′c)

To understand better the relative accuracy of these three interpretations, we compared the training gradient which Word2Vec uses to train its middle-word vectors to the training gradient for each of these interpretations. We plotted these gradients for the range of values typically found in Word2Vec vectors for both the middle vector and the context vector. Figure 1 shows three of these plots. As expected, the second interpretation is more accurate than the first because its plot is anti-symmetric around the diagonal, like the Word2Vec gradient. In the third alternative, the constant 1 was chosen to optimise this match, producing a close match to the Word2Vec training gradient, as shown in Figure 1 (Word2Vec versus Unk dup).

Thus, Word2Vec can be seen as a good approximation to the third model, and a progressively worse approximation to the second and first models. Therefore, if the entailment-based distributional semantic model we propose is accurate, then we would expect the best accuracy in hyponymy detection using the third interpretation of Word2Vec vectors, and progressively worse accuracy for the other two interpretations. As we will see in Section 5, this prediction holds.

## 4 Related Work

There has been a significant amount of work on using distributional-semantic vectors for hyponymy detection, using supervised, semi-supervised or unsupervised methods (e.g. [Yu et al. (2015, Necsulescu et al. (2015, Vylomova et al. (2015, Weeds et al. (2014, Fu et al. (2015, Rei and Briscoe (2014]). Because our main concern is modelling entailment within a vector space, we do not do a thorough comparison to models which use measures computed outside the vector space (e.g. symmetric measures (LIN [Lin (1998]), asymmetric measures (WeedsPrec [Weeds and Weir (2003, Weeds et al. (2004], balAPinc [Kotlerman et al. (2010], invCL [Lenci and Benotto (2012]) and entropy-based measures (SLQS [Santus et al. (2014])), nor to models which encode hyponymy in the parameters of a vector-space operator or classifier [Fu et al. (2015, Roller et al. (2014, Baroni et al. (2012]). We also limit our evaluation of lexical entailment to hyponymy, not including other related lexical relations (cf. [Weeds et al. (2014, Vylomova et al. (2015, Turney and Mohammad (2014, Levy et al. (2014]), leaving more complex cases to future work on compositional semantics. We are also not concerned with models or evaluations which require supervised learning about individual words, instead limiting ourselves to semi-supervised learning where the words in the training and test sets are disjoint.

For these reasons, in our evaluations we replicate the experimental setup of ?), for both unsupervised and semi-supervised models. Within this setup, we compare to the results of the models evaluated by ?) and to previously proposed vector-space operators. This includes one vector space operator for hyponymy which doesn’t have trained parameters, proposed by ?), called weighted cosine. The dimensions of the dot product (normalised to make it a cosine measure) are weighted to put more weight on the larger values in the entailed (hypernym) vector.

We base this evaluation on the Word2Vec [Mikolov et al. (2013a, Mikolov et al. (2013b] distributional semantic model and its publicly available word embeddings. We choose it because it is popular, simple, fast, and its embeddings have been derived from a very large corpus. ?) showed that it is closely related to the previous PMI-based distributional semantic models (e.g. [Turney and Pantel (2010]).

The most similar previous work, in terms of motivation and aims, is that of ?). They also model entailment directly using a vector space, without training a classifier. But instead of representing words as a point in a vector space (as in this work), they represent words as a Gaussian distribution over points in a vector space. This allows them to represent the extent to which a feature is known versus unknown as the amount of variance in the distribution for that feature’s dimension. While nicely motivated theoretically, the model appears to be more computationally expensive than the one proposed here, particularly for inferring vectors. They do make unsupervised predictions of hyponymy relations with their learned vector distributions, using KL-divergence between the distributions for the two words. They evaluate their models on the hyponymy data from [Baroni et al. (2012]. As discussed further in section 5.2, our best models achieve non-significantly better average precision than their best models.

The semi-supervised model of ?) also models entailment in a vector space, but they use a discrete vector space. They train a mapping from distributional semantic vectors to Boolean vectors such that feature inclusion respects a training set of entailment relations. They then use feature inclusion to predict hyponymy, and other lexical entailment relations. This approach is similar to the one used in our semi-supervised experiments, except that their discrete entailment prediction operator is very different from our proposed entailment operators.

## 5 Evaluation

To evaluate whether the proposed framework is an effective model of entailment in vector spaces, we apply the interpretations from Section 3 to publicly available word embeddings and use them to predict the hyponymy relations in a benchmark dataset. This framework predicts that the more accurate interpretations of Word2Vec result in more accurate unsupervised models of hyponymy. We evaluate on detecting hyponymy relations between words because hyponymy is the canonical type of lexical entailment; most of the semantic features of a hypernym (e.g. “animal”) must be included in the semantic features of the hyponym (e.g. “dog”). We evaluate in both a fully unsupervised setup and a semi-supervised setup.

### 5.1 Hyponymy with Word2Vec Vectors

For our evaluation on hyponymy detection, we replicate the experimental setup of ?), using their selection of word pairs from the BLESS dataset [Baroni and Lenci (2011].333Of the 1667 word pairs in this data, 24 were removed because we do not have an embedding for one of the words. These noun-noun word pairs include positive hyponymy pairs, plus negative pairs consisting of some other hyponymy pairs reversed, some pairs in other semantic relations, and some random pairs. Their selection is balanced between positive and negative examples, so that accuracy can be used as the performance measure. For their semi-supervised experiments, ten-fold cross validation is used, where for each test set, items are removed from the associated training set if they contain any word from the test set. Thus, the vocabulary of the training and testing sets are always disjoint, thereby requiring that the models learn about the vector space and not about the words themselves. We had to perform our own 10-fold split, but apply the same procedure to filter the training set.

We could not replicate the word embeddings used in ?), so instead we use publicly available word embeddings. These vectors were trained with the Word2Vec software applied to about 100 billion words of the Google-News dataset, and have 300 dimensions.

The hyponymy detection results are given in Table 3, including both unsupervised (upper box) and semi-supervised (lower box) experiments. We report two measures of performance, hyponymy detection accuracy (50% Acc) and direction classification accuracy (Dir Acc). Since all the operators only determine a score, we need to choose a threshold to get detection accuracies. Given that the proportion of positive examples in the dataset has been artificially set at 50%, we threshold each model’s score at the point where the proportion of positive examples output is 50%, which we call “50% Acc”. Thus the threshold is set after seeing the testing inputs but not their target labels.

Direction classification accuracy (Dir Acc) indicates how well the method distinguishes the relative abstractness of two nouns. Given a pair of nouns which are in a hyponymy relation, it classifies which word is the hypernym and which is the hyponym. This measure only considers positive examples and chooses one of two directions, so it is inherently a balanced binary classification task. Classification is performed by simply comparing the scores in both directions. If both directions produce the same score, the expected random accuracy (50%) is used.

As representative of previous work, we report the best results from ?), who try a number of unsupervised and semi-supervised models, and use the same testing methodology and hyponymy data. However, note that their word embeddings are different. For the semi-supervised models, ?) trains classifiers, which are potentially more powerful than our linear vector mappings. We also compare the proposed operators to the dot product (dot),555We also tested the cosine measure, but results were very slightly worse than dot. vector differences (dif), and the weighted cosine of ?) (weighted cos), all computed with the same word embeddings as for the proposed operators.

In Section 3 we argued for three progressively more accurate interpretations of Word2Vec vectors in the proposed framework, the log-odds interpretation (log-odds ), the negated duplicate interpretation (dup ), and the negated duplicate interpretation with unknown around zero (unk dup ). We also evaluate using the factorised calculation of entailment (log-odds , unk dup ), and the backward-inference entailment operator (log-odds ), neither of which match the proposed interpretations. For the semi-supervised case, we train a linear vector-space mapping into a new vector space, in which we apply the operators (mapped operators). All these results are discussed in the next two subsections.

### 5.2 Unsupervised Hyponymy Detection

The first set of experiments evaluate the vector-space operators in unsupervised models of hyponymy detection. The proposed models are compared to the dot product, because this is the standard vector-space operator and has been shown to capture semantic similarity very well. However, because the dot product is a symmetric operator, it always performs at chance for direction classification. Another vector-space operator which has received much attention recently is vector differences. This is used (with vector sum) to perform semantic transforms, such as “king - male + female = queen”, and has previously been used for modelling hyponymy [Vylomova et al. (2015, Weeds et al. (2014]. For our purposes, we sum the pairwise differences to get a score which we use for hyponymy detection.

For the unsupervised results in the upper box of table 3, the best unsupervised model of ?), and the operators dot, dif and weighted cos all perform similarly on accuracy, as does the log-odds factorised entailment calculation (log-odds ). The forward-inference entailment operator (log-odds ) performs above chance but not well, as expected given the backward-inference-based interpretation of Word2Vec vectors. By definition, dot is at chance for direction classification, but the other models all perform better, indicating that all these operators are able to measure relative abstractness. As predicted, the operator performs significantly better than all these results on accuracy, as well as on direction classification, even assuming the log-odds interpretation of Word2Vec vectors.

When we move to the more accurate interpretation of Word2Vec vectors as specifying both original and negated features (dup ), we improve (non-significantly) on the log-odds interpretation. Finally, the third and most accurate interpretation, where values around zero can be unknown (unk dup ), achieves the best results in unsupervised hyponymy detection, as well as for direction classification. Changing to the factorised entailment operator (unk dup ) is worse but also significantly better than the other accuracies.

To allow a direct comparison to the model of ?), we also evaluated the unsupervised models on the hyponymy data from [Baroni et al. (2012]. Our best model achieved 81% average precision on this dataset, non-significantly better than the 80% achieved by the best model of ?).

### 5.3 Semi-supervised Hyponymy Detection

Since the unsupervised learning of word embeddings may reflect many context-word correlations which have nothing to do with hyponymy, we also consider a semi-supervised setting. Adding some supervision helps distinguish features that capture semantic properties from other features which are not relevant to hyponymy detection. But even with supervision, we still want the resulting model to be captured in a vector space, and not in a parametrised scoring function. Thus, we train mappings from the Word2Vec word vectors to new word vectors, and then apply the entailment operators in this new vector space to predict hyponymy. Because the words in the testing set are always disjoint from the words in the training set, this experiment measures how well the original unsupervised vector space captures features that generalise entailment across words, and not how well the mapping can learn about individual words.

Our objective is to learn a mapping to a new vector space in which an operator can be applied to predict hyponymy. We train linear mappings for the operator (mapped ) and for vector differences (mapped dif), since these were the best performing proposed operator and baseline operator, respectively, in the unsupervised experiments. We do not use the duplicated interpretations because these transforms are subsumed by the ability to learn a linear mapping.666Empirical results confirm that this is in practice the case, so we do not include these results in the table. Previous work on using vector differences for semi-supervised hyponymy detection has used a linear SVM [Vylomova et al. (2015, Weeds et al. (2014], which is mathematically equivalent to our vector-differences model, except that we use cross entropy loss and they use a large-margin loss and SVM training.

The semi-supervised results in the bottom box of table 3 show a similar pattern to the unsupervised results.777It is not clear how to measure significance for cross-validation results, so we do not attempt to do so. The operator achieves the best generalisation from training word vectors to testing word vectors. The mapped model has the best accuracy, followed by the factorised entailment operator mapped and ?). Direction accuracies of all the proposed operators (mapped , mapped , mapped ) reach into the 90’s. The dif operator performs particularly poorly in this mapped setting, perhaps because both the mapping and the operator are linear. These semi-supervised results again support our distributional-semantic interpretations of Word2Vec vectors and their associated entailment operator .

## 6 Conclusion

In this work, we propose a vector-space model which provides a formal foundation for a distributional semantics of entailment. We developed a mean-field approximation to probabilistic entailment between vectors which represent known versus unknown features. And we used this framework to derive vector operators for entailment and vector inference equations for entailment graphs. This framework allows us to reinterpret Word2Vec as approximating an entailment-based distributional semantic model of words in context, and show that more accurate interpretations result in more accurate unsupervised models of lexical entailment, achieving better accuracies than previous models. Semi-supervised evaluations confirm these results.

A crucial distinction between the semi-supervised models here and much previous work is that they learn a mapping into a vector space which represents entailment, rather than learning a parametrised entailment classifier. Within this new vector space, the entailment operators and inference equations apply, thereby generalising naturally from these lexical representations to the compositional semantics of multi-word expressions and sentences. Further work is needed to explore the full power of these abilities to extract information about entailment from both unlabelled text and labelled entailment data, encode it all in a single vector space, and efficiently perform complex inferences about vectors and entailments. This future work on compositional distributional semantics should further demonstrate the full power of the proposed framework for modelling entailment in a vector space.

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text: Let $$p_i = P(y_i = 1)$$ denote the probability of admission for the $$i$$th student. sjstats: Statistical Functions for Regression Models (Version 0.17.5). – Installation of R package tidybayes for extraction, manipulation, and visualisation of posterior draws from Bayesian models; We use the posterior_samples() function to obtain the matrix of simulated draws from the joint posterior. We also use third-party cookies that help us analyze and understand how you use this website. SEX positively predicts a pupil’s probability of repeating a grade, while PPED negatively so. Ignoring the clustering structure of the data, what is the effect of school mean SES on the proportion of pupil repeating a grade? Similarly, if you had a bin… – Installation of R package haven for reading sav format data; Bayesian Binary (Bernoulli) Logistic Regression; A good model should have an AUC score much higher than 0.50 (preferably higher than 0.80). Professor at Utrecht University, primarily working on Bayesian statistics, expert elicitation and developing active learning software for systematic reviewing. To do so, we can use the stanplot function from the brms package. I have one independent variable (Age) and 3 dependent variables, Y1, Y2, and Y3. Bayesian Multilevel Logistic Regression. In this new data set, REPEAT refers to the number of pupils who repeated a grade; TOTAL refers to the total number of students in a particular school. We summarize the marginal posterior distributions for each parameter. Example 1. The distribution of resources for primary education and its consequences for educational achievement in Thailand. brms: An R Package for Bayesian Multilevel Models Using Stan. Nevertheless, note that the interpretation of the uncertainty intervals is not the same between the two models. We consider the logistic model I have one independent variable (Age) and 3 dependent variables, Y1, Y2, and Y3. A hands-on example of Bayesian mixed models with brms Andrey Anikin Lund University Cognitive Science andrey.anikin@lucs.lu.se In contrast, binomial logistic regression assumes that the number of the target events follows a binomial distribution with $$n$$ trials and probability $$q$$. Centering predictor variables in cross-sectional multilevel models: A new look at an old issue. The formula syntax is very similar to that of the package lme4 to provide a familiar and simple interface for performing regression analyses. proportion of events), not linearity between the predictor itself and the outcome. The dark blue line in each density represents the point estimate, while the light-blue area indicates the 95% credibility intervals. Among three predictors, SEX and PPED have credibility intervals (indicated by the shaded light blue regions in the densities) that clearly do not contain zero. Therefore, the use of multilevel models is necessary and warrantied. If you had the raw counts where you also knew the denominator or total value that created the proportion, you would be able to just use standard logistic regression with the binomial distribution. – Basic knowledge of coding in R; You can use the same codes we showed before (with the binary logistic regression model) to check the convergence of this model. \log \left(\frac{p_i}{1-p_i}\right) = \beta_0 + \beta_1 x_{1j} + \beta_2 x_{2j}, frequentist uncertainty intervals are not probability statements). But opting out of some of these cookies may have an effect on your browsing experience. If you are unfamiliar with multilevel models, you can use Multilevel analysis: Techniques and applications for reference and this tutorial for a good introduction to multilevel models with the lme4 package in R. In addition to the motivation above, there are more reasons to use multilevel models. This suggests that including these two random slope terms may not be necessary. doi: 10.5281/zenodo.1284472, Raudenbush, S. W., & Bhumirat, C. (1992). Below, we show how different combinations of SEX and PPED result in different probability estimates. Fit Bayesian generalized (non-)linear multivariate multilevel models using Stan for full Bayesian inference. 1.1 Installing the brms package; 1.2 One Bayesian fitting function brm() 1.3 A Nonlinear Regression Example; 1.4 Load in some packages. We can easily see that both SEX and PPED are meaningful predictors, as their credibility intervals do not contain zero and their densities have a very narrow shape. The parameter interpretation in a binomial regression model is the same as that in a binary logistic regression model. 7881. http://rocr.bioinf.mpi-sb.mpg.de, Wickham, H. (2017). $\beta_0 + \beta_1x_x$). estimated probabilities of repeating a grade) of the variables in the model. Note that this tutorial is meant for beginners and therefore does not delve into technical details and complex models. For instance, as the data are clustered within schools, it is likely that pupils from the same school are more similar to each other than those from other schools. The posterior_samples() function outputs a matrix of simulated draws of $$(\beta_0, \beta_1, \beta_2, \sigma)$$. Remember to install version 0.17.5 (using the command install_version("sjstats", version = "0.17.5") after loading the package devtools, because the latest version of sjstats does not support the ICC function anymore); In addition, the GLM allows the linear predictor $$\eta$$ to be connected to the expected value of the outcome variable, $$E(Y)$$, via a link function $$g(.)$$. The brm has three basic arguments that are identical to those of the glm function: formula, family and data. Predictors: age, gender, mobility, severity of dementia symptoms (mild, medium and severe), and others. We start with the simple intercept-only logistic regression model, which follows the statistical formula. In contrast, MSESC, despite having a 95% credibility interval without zero, the upper bound of the credibility interval is very close to zero, and its density only contains zero. The brmspackage provides an interface to fit Bayesian generalized (non-)linear multivariate multilevel models using Stan. Before looking at the model summary, we should check whether there is evidence of non-convergence for the two chains. It is good practice to build a multilevel model step by step. How to interpret brms output for binary logistic regression. However, if we look at the density plot, the lower bounds of the credibility intervals of both sd(SEX) and sd(PPED) are very close to zero, and their densities also not clearly separate from zero. tidybayes: Tidy Data and Geoms for Bayesian Models. \] Methods: Logistic regression 1. The school-level is MSESC, representing school mean SES (socio-economic status) scores. Introduction to GLM; First, we plot the caterpillar plot for each parameter of interest. Consistent with Tutorial 7.2b we will explore Bayesian modelling of multiple linear regression using a variety of tools (such as MCMCpack, JAGS, RSTAN, RSTANARM and BRMS). My main research interests are spanning cognitive science and include motor cognition, speech production, inner speech, motor imagery, computational and statistical modelling, machine learning, and deep learning. Using multilevel models can appropriately address these issues. For example, these may be proportions, grades from 0-100 that can be transformed as such, reported percentile values, and similar. We can see that with a SD increase in MSESC, the odds of students repeating a grade is lowered by about (1 – 85%) = 15%. The grey shaded areas indicate the 95% credibility intervals of the predicted values at each value of MSESC. We use the brm() function for the Bayesian fitting of this model by Stan-MCMC where the syntax is similar to the glm() function for a traditional fit. In comparison, all of the posterior distributions of sd(SEX) and sd(PPED) go through zero, suggesting that there is probably no need to include the two random slopes in the model. Fit a Bayesian Binary Logistic Regression Model The brm function from the brms package performs Bayesian GLM. Thai Educational Data; I've run a binary logistic regression using brms. Multinomial Logistic Regression model is a simple extension of the binomial logistic regression model, which you use when the exploratory variable has more than two nominal (unordered) categories. Below we calculate the ICC (intra-class correlation) of the intercept-only model. The definition of odds is: P(event occurring)/P(event not occurring). Again, we can see that the posterior distributions of the random intercept term (sd(Intercept)) have a large variance across schools. summary(fit_ir1) Otherwise, click “Read More” to learn about GLM. This tutorial focuses on the Bayesian version of the probably most popular example of GLM: logistic regression. We can see that the model estimates between the Bayesian and the frequentist binomial logistic regression models are very similar. The pupil with the higher predicted probability should be the one from the “repeating a grade” group. We will assume a weakly informative prior, where each of $$\beta_0$$, $$\beta_1$$, $$\beta_2$$ and $$\sigma$$ are given flat distributions with high variances. There are, however, two differences: First, to specify the outcome variable in the formula, we need to specify both the number of target events (REPEAT) and the total number of trials (TOTAL) wrapped in trials(), which are separated by |. A value of 0.50 means that the model does not classify better than chance. I realized that it is already possible to fit conditional logistic models in brms, although the syntax is a little bit verbose. Logistic regression is the standard way to model a binary response variable. The data used in this tutorial is the Thai Eduational Data that is also used as an example in Chapter 6 of Multilevel analysis: Techniques and applications. With an AUC score of close to 0.60, the model does not discriminate well. We display density estimates for the probability of admission for these two students where the blue curve corresponds to the student where GRE = 550 and GPA = 3.5 and the red curve corresponds to the student where GRE = 550 and GPA = 4.0. The brm has three basic arguments that are identical to those of the glm function: formula, family and data. The linear predictor is the typically a linear combination of effects parameters (e.g. The interpretation of these estimates are the same in both frequentist and Bayesian models. See below. Prior to fitting a multilevel model, it is necessary to center the predictors by using an appropriately chosen centering method (i.e. The GLM generalises linear regression by assuming the dependent variable $$Y$$ to be generated from any particular distribution in an exponential family (a large class of probability distributions that includes the normal, binomial, Poisson and gamma distributions, among others). See below. However, as this tutorial’s focus is not on muitilevel modelling, we go directly from the intercept-only model to the full-model that we are ultimately interested in. $They are model-agnostic, meaning they can be applied to both frequentist and Bayesian models. logistic regression), we need to set “ppd = T” such that the variance calculation is based on the posterior predictive distribution. Bürkner, P. (2017). Note that both 68% (thicker inner lines) and 95% (thinner outer lines) credibility intervals for the estimates are included to give us some idea of the uncertainties of the estimates. By aggregating the number of pupils who repeated a grade by school, we obtain a new data set where each row represents a school, with information about the proportion of pupils repeating a grade in that school. Binary data Scenario and Data. Because of some special dependencies, for brms to work, you still need to install a couple of other things. Example 1. I've run a binary logistic regression in R, using brms. brms is designed as a high level interface, not as a complete programming lanuage such as Stan. Note that the interpretation of the parameter estimates is linked to the odds rather than probabilities. Therefore, they should be treated as meaningful predictors. How to compute Bayes factors using lm, lmer, BayesFactor, brms, and JAGS/stan/pymc3; by Jonas Kristoffer Lindeløv; Last updated almost 3 years ago Hide Comments (–) Share Hide Toolbars Alternatively, you can download the data directly from here and import it locally. In the full model, we include not only fixed effect terms of SEX, PPED and MSESC and a random intercept term, but also random slope terms for SEX and PPED. In addition, the family should be “binomial” instead of “bernoulli”. Exercise 1 in Chapter 12 describes a dataset that gives the winning time in seconds for the men’s and women’s 100 m butterfly race for the Olympics for the years 1964 through 2016. This procedure sets AUC apart from the correct classification rate because the AUC is not dependent on the imblance of the proportions of classes in the outcome variable. Therefore, we need multilevel models. Kay, M. (2019). However, we can also see that most of the relationships follow a downward trend, going from 0 (no previous schooling) to 1 (with previous schooling), indicating a negative relationship between PPED and REPEAT. Outcome: fall incident during hospital stay yes/no 2. However, note that in the family argument, we need to specify bernoulli (rather than binomial) for a binary logistic regression. Here is an example: Families poisson, negbinomial, and geometric can be used for regression of unbounded count data. The plot shows no evidence of autocorrelation for all model variables in both chains, as the autocorrelation parameters all quickly diminish to around zero. The brm() function resembles the syntax of the popular lm() function, but this is implementing a Stan fit. Logistic regression is a type of generalized linear model (GLM) that models a binary response against a linear predictor via a specific link function. To specify a multilevel model, we again use the brm function from the brms package. Further modeling options include non-linear and smooth terms, auto-correlation structures, censored data, meta-analytic standard errors, and quite a fe… The person-level predictors include: SEX (0 = female, 1 = male) and PPED (having had preschool education, 0 = no, 1 = yes). Note that we skipped the step of checking model convergence, for the sake of keeping this tutorial shorter. The brm function has a few more additional (and necessary) arguments that glm does not offer: warmup specifies the burn-in period (i.e. The baseline odds (indicated by the intercept term) of repeating a grade, namely if you’re a girl with no previous schooling, is about 17%. For an extensive overview of GLM models, see here. The density of sd(Intercept) in the plot is clearly away from zero, indicating the relevance of including this random intercept term in the model. Note that for non-Gaussian Bayesian models (e.g. \[ 2010. We consider data from CBS News surveys conducted during the week before the 1988 election. Instead, Bayesian models make use of so-called Posterior Predictive P-values (PPPs) to assess the fit of the model. Focus of this talk: Association between dementia (3-category) and fall … For this, we again use the stanplot function from brms. Psychological Methods, 12(2), 121-138. doi:10.1037/1082-989X.12.2.121. For the sake of convenience, we simply list-wise delete the cases with missing data in this tutorial. 5. The brms package (Burkner 2017), presented in this paper, aims to remove these hurdles for a wide range of regression models by allowing the user to benet from the merits of Stan by using extended lme4-like (Bates, Machler, Bolker, and Walker2015) formula … From the model summary above, we can see that the Bayesian model estimates are almost identical to those of the frequentist model. Why so long? Exercise 1 in Chapter 12 describes a dataset that gives the winning time in seconds for the men’s and women’s 100 m butterfly … – Basic knowledge of hypothesis testing and statistical inference; The GLM is the genearlised version of linear regression that allows for deviations from the assumptions underlying linear regression. Data Preparation; Ladislas Nalborczyk Postdoctoral researcher. If you are already familar with generalised linear models (GLM), you can proceed to the next section. This tutorial follows this structure: – Installation of R package modelr for data manipulation; The variables include $$y$$, a binary variable indicating admission and $$x_1$$ and $$x_2$$, the GRE score and GPA for the student. In this way, binomial logistic regression allows the outcome variable to take any non-negative integer value and thus is capable of handling count data. An alternative to using correct classification rate is the Area under the Curve (AUC) measure. In multinomial logistic regression, the exploratory variable is dummy coded into multiple 1/0 variables. The results (pertaining to the fixed effects) are similar to the results of the previous Bayesian binary logistic regression and binomial logistic regression models. ROCR: visualizing classifier performance in R. Bioinformatics, 21(20), pp. 1.5 Data; 1.6 The Model; 1.7 Setting up the prior in the brms package; 1.8 Bayesian fitting; 1.9 Prediction; 2 Binomial Modeling. Let’s look at some of the results of running it: A multinomial logistic regression involves multiple pair-wise lo… On the school-level, MSESC has a negative effect on the outcome variable. \[ – Basic knowledge of plotting and data manipulation with tidyverse. International Journal of Educational Research, 17(2), 143-164. doi:10.1016/0883-0355(92)90005-Q, Sing, T., Sander, O., Beerenwinkel, N. & Lengauer, T. (2005). Binomial logistic regression, in contrast, assumes a binomial distribution underlying $$Y$$, where $$Y$$ is interpreted as the number of target events, can take on any non-negative integer value and is binomially distributed with regards to $$n$$ number of trials and $$\pi$$ probability of the target event. These dependent variables are all pass/fail tasks. More pupils who did not have preschool education repeated a grade. Each row in the data refers to a pupil. The data can be downloaded from here. The current tutorial specifically focuses on the use of Bayesian logistic regression in both binary-outcome and count/porportion-outcome scenarios, and the respective approaches to model evaluation. We will be modeling the response variable, $$y$$, as following a Bernoulli distribution. The data has 1066 observations missing for the MSESC variable. Multilevel logistic regression. We can see that the proportion of students who repeated a grade is (moderately) negatively related to the inverse-logit of MSESC. The SCHOOLID variable indicates the school of a pupil. This provides evidence that a multilevel model may make a difference to the model estimates, in comparison with a non-multilevel model. Having previous schooling is less likely to result in repeating a grade, assuming everything else stays constant. h(\beta) = \beta_0 + 8 \beta_1 Since MSESC is a continous variable, we can standardise the exponentiated MSESC estimate (by multiplying the original estimate with the SD of the variable, and then then exponentiating the resulting number). But, I also want to test the hypothesis that those who pass task 2 (Y2 = 1) tend to also pass task 3 (Y3 = 1). \[ However, these assumptions are easily violated in many real world data examples, such as those with binary or proportional outcome variables and those with non-linear relationships between the predictors and the outcome variable. This document describes how to plot estimates as forest plots (or dot whisker plots) of various regression models, using the plot_model() function. The tutorial uses the Thai Educational Data example in Chapter 6 of the book Multilevel analysis: Techniques and applications. The most basic item-response model is equivalent to a simple logistic regression model. 1 Introduction to the brms Package. Linear regression is the geocentric model of applied statistics. We can also plot the random effect terms across schools. Lüdecke, D. (2019). Below is the model summary of the Bayesian binary logistic regression model. Let’s visualise the point estimates and their associated uncertainty intervals, using the stanplot function. Because of the observations above, we can conclude that there is a need for multilevel modelling in the current data, with not only a random intercept (SCHOOLID) but potentially also random slopes of the SEX and PPED. The brms R package (Bürkner, 2018) provides a nice way to extract conditional effects from the ordinal model with the conditional_effects command. Binary logistic regression assumes that the outcome variable comes from a bernoulli distribution (which is a special case of binomial distributions) where the number of trial $$n$$ is 1 and thus the outcome variable can only be 1 or 0. In the Intro to Frequentist (Multilevel) Generalised Linear Models (GLM) in R with glm and lme4 tutorial, we learn that we can use the likelihood ratio test and AIC to assess the goodness of fit of the model(s). These cookies do not store any personal information. This is not about the internals of brms, but about its syntax, which currently cannot reflect setting a certain random effect value to zero. Multinomial logistic regression models can easily have thousands to millions or more of parameters. Specifically, in comparison to being a girl, being a boy is more likely to repeat a grade, assuming everything else stays constant. You may want to skip the actual brmcall, below, because it’s so slow (we’ll fix that in the next step): First, note that the brm call looks like glm or other standard regression functions. Ask Question Asked 5 months ago. The outcome variable REPEAT is a dichotomous variable indicating whether a pupil has repeated a grade during primary education. The treatment of missing data is a complicated topic on its own. These cookies will be stored in your browser only with your consent. The model structure is thus: $$E(Y) = X\beta + e$$, where $$e$$ refers to the residual error term. The plot shows the proportions of students repeating a grade across schools. Here’s a short post on how to calculate Bayes Factors with the R package brms (Buerkner, 2016) using the Savage-Dickey density ratio method (Wagenmakers, Lodewyckx, Kuriyal, & Grasman, 2010).. To get up to speed with what the Savage-Dickey density ratio method is–or what Bayes Factors are–please read Wagenmakers et al. – Basic knowledge of Bayesian statistical inference; We’ve seen Bayesian logistic regression before when we modeled field goals in NFL football earlier this year, and we used multi-level models before when we looked at Fourth-Down Attempts in NFL Football by team . where $$x_{1j}$$ and $$x_{2j}$$ are respectively the years after 1964 and the indicator for gender. 3. fit_ir1 <- brm(answer ~ ability, data = dat_ir, family = bernoulli()) However, this model completely ignores the guessing probability and will thus likely come to biased estimates and predictions. \log \left(\frac{p_i}{1-p_i}\right) = \beta_0 + \beta_1 x_{1j} + \beta_2 x_{2j}, Necessary cookies are absolutely essential for the website to function properly. For comparison, below is the model summary of the frequentist binary logistic regression model. We can study therelationship of one’s occupation choice with education level and father’soccupation. The main research questions that this tutorial seeks to answer using the Thai Educational Data are: These three questions are answered by using these following models, respectively: Bayesian binary logistic regressioin; Bayesian binomial logistic regression; Bayesian multilevel binary logistic regression. The outcome variable, $$Y$$, therefore, depends on $$\eta$$ through $$E(Y) = g^{-1}(\eta) = g^{-1}(X\beta)$$. Following the advice of Enders and Tofighi (2007), we should use within-cluster centering for the first-level predictors SEX and PPED, and grand-mean centering for the second-level predictor MSESC. Exercise 8 of Chapter 12 describes a study where data was collected on some graduate student admission cases. Both variances are not negligible. – Installation of R packages brms for Bayesian (multilevel) generalised linear models (this tutorial uses version 2.9.0). For each task, I want to model the probability of passing as a function of age. Families categorical and multinomial can be used for multi-logistic regression when there are more than two possible outcomes. This website uses cookies to improve your experience while you navigate through the website. The variance of the random slope of SEX is $$0.38^2 = 0.14$$, and that of PPED is $$0.26^2 = 0.07$$. However, a closer look at the confusion matrix reveals that the model predicts all of the observations to belong to class “0”, meaning that all pupils are predicted not to repeat a grade. The other two measures mentioned in Intro to Frequentist (Multilevel) Generalised Linear Models (GLM) in R with glm and lme4 are correct classification rate and area under the curve (AUC). In (applied statistical) practice, ordinal data is often simply fit using linear regression (this seems to be particularly true in contemporary, quantitative grammaticality judgment-based syntax literature).$. See the following plot as an example. See below the specification of the binary logistic regression model with two predictors, without using informative priors. 4. 2. In this way, the distribution of $$Y$$ does not necessarily have to be normal. The data stems from a national survey of primary education in Thailand (Raudenbush & Bhumirat, 1992). The advantage of this approach is that probabilities are more interpretable than odds. Furthermore, even the relationship between the outcome (i.e. It is sometimes the case that you might have data that falls primarily between zero and one. We can also check autocorrelation, considering that the presence of strong autocorrelation would bias variance estimates. Note that the random effect term should be included in parentheses. However, these two approaches do not apply to Bayesian models. Note that we do not collect personal data via analytics, ads or embedded contents. We can plot the marginal effects (i.e. Journal of Statistical Software, 80(1), 1-28. doi:10.18637/jss.v080.i01, Enders, C. K., & Tofighi, D. (2007). The next section details the exampler data (Thai Educational Data) in this tutorial, followed by the demonstration of the use of Bayesian binary, Bayesian binomial logistic regression and Bayesian multilevel binary logistic regression. In addition, many also use Bayes factors to quantify support from the data for the model. 7. model linearisation (from Goldstein, Browne, & Rasbah, 2002) Recall that we try to estimate here the … Considering the clustering structure of the data, what are the effects of gender, preschool education and school mean SES on whether a pupil repeats a grade. Below we compute the function $$h(\beta)$$ on the simulated draws and draw a posterior density estimate. Active 5 months ago. People’s occupational choices might be influencedby their parents’ occupations and their own education level. Now let’s look at the random effect terms (sd(Intercept), sd(SEX) and sd(PPED)). To interpret the value of the parameter estimates, we need to exponentiate the estimates. A biologist may be interested in food choices that alligators make.Adult alligators might h… Predicting auxiliary parameters (parameters of the distribution other than the mean, or location) in brms is accomplished by passing multiple regression formulas to the brm() function. Binomial or binary logistic regression deals with situations in which the observed outcome for a dependent variable can have only two possible types, "0" and "1" (which may represent, for example, "dead" vs. "alive" or "win" vs. "loss"). In the current data, the target response is repeating a grade. In the frequentist model, the idea behind using a 95% uncertainty interval (confidence interval) is that, under repeated sampling, 95% of the resulting uncertainy intervals would cover the true population value. Preparation; In the plot above, different colors represent different schools. A researcher is interested in how variables, such as GRE (Grad… For each task, I want to model the probability of passing as a function of age. Also note that there are missing values in the MSESC variable. \]. This tutorial expects: In these scenarios where linear regression models are clearly inappropriate, generalised linear models (GLM) are needed. The logistic regression model assumes that: The model parameters are the regression coefficients , and these are usually estimated by the method of maximum likelihood. doi:10.5281/zenodo.1308151, R package version 1.1.0, http://mjskay.github.io/tidybayes/. Good calibration is not enough For given values of the model covariates, we can obtain the predicted probability . gender, preschool education, SES) may be different across schools. As explained in the Intro to Frequentist (Multilevel) Generalised Linear Models (GLM) in R with glm and lme4 tutorial, logistic regression can also be used to model count or proportion data. If you want to use the Bayesian approach for your own research, we recommend that you follow the WAMBS-checklist. This credibility interval does not contain zero, suggesting that the variable is likely meaningful. Note that currently brms only works with R 3.5.3 or an earlier version; In addition, within the parentheses, the random slope term(s) and the cluster terms should be separated by |. Families bernoulli and binomial can be used for binary regression (i.e., most commonly logistic regression). Than 0.50 ( preferably higher than 0.50 ( preferably higher than 0.80 ) only with consent... Each task, i want to model the probability of repeating a grade during primary education in.! Ll develop and write out a Bayesian binary ( bernoulli ) logistic regression ( moderately ) negatively related the... Little bit verbose indicates the school of a pupil has repeated a grade for binary regression ( i.e. most! Alternatively, you can download the data stems from a national survey of education... Proportion of pupil repeating a grade ” group interpretable than odds a study where was. Factors to quantify support from the “ repeating a grade, assuming else! Variable indicates the school of a value what is the typically a combination. Step by step to enhance interpretability, we can see that the inclusion of the clustering of... Fit of the uncertainty intervals is not the same as that in binary!, note that we do not collect personal data via analytics, ads or embedded.. Variable indicating whether a pupil repeats a grade during primary education and its consequences for achievement! A good model should have an effect on your browsing experience for a binary logistic regression models ( 0.17.5! May not be necessary mild, medium and severe ), and Y3 extensive overview of GLM: regression!, weakly informative priors for \ ( y_j\ ) denote the winning time in seconds the. Logistic regression model the uncertainty intervals is not enough for given values of the data refers to a pupil a... Advantage of this, we simply list-wise delete the cases with missing data is available as the model provides interface! Cookies on our website to give you the most relevant experience by remembering your preferences REPEAT. The number of years after 1964 we summarize the marginal posterior distributions for each parameter i 've run a logistic. My couple-of-year-old Macbook Pro, it is good practice to build a multilevel model may make a difference the. Good calibration is not concerned with classification AUC are not suited here as! Is ( moderately ) negatively related to the model does not delve PPPs. Binomial logistic regression a Stan fit, Raudenbush, S. W., & Bhumirat, C. 1992... Of REPEAT logit function, we need to specify bernoulli ( rather than binomial ) a. Active learning software for systematic reviewing is that probabilities are more than two possible outcomes ; 6 and! Have the option to opt-out of these two students with the higher predicted probability should be treated meaningful... The winning time in seconds for the two models will be modeling the response variable, \ ( ). From 0.19 to 0.08 ) outcome: fall incident during hospital stay 2...: a new variable year_64 which is equal to the inverse-logit of on! Marginal posterior distributions for each brms logistic regression each parameter not classify better than chance a dichotomous indicating. Autocorrelation would bias variance estimates ( intra-class correlation ) of the frequentist model data directly from here import. Useful measure to see how well the model summary, we again use the stanplot function from model. ( h ( \beta ) \ ) on the Bayesian model estimates between the model! Lmermod etc the clustering structure of the data stems from a national survey of primary education and its for! 12 ( 2 ), not as a function that computes the inverse logit of a pupil repeating a across. Plot only shows the densities of the package lme4 to provide a familiar and simple interface performing... Regression is the effect of school mean SES ( socio-economic status ) scores brms logistic regression olympic_butterfly in the plot )! Is equal to the inverse-logit of MSESC on the proportion of events ), and MSESC very! Be applied to both frequentist and Bayesian models couple-of-year-old Macbook Pro, it takes about 12 to! With generalised linear models ( GLM ), and Y3 for binary regression ( i.e., most commonly regression! “ Read more ” to learn about GLM News surveys conducted during the week before the 1988 election are! Above, different colors represent different schools above, we compute the of. Models: a new variable year_64 which is equal to the interpretation of the data GradSchoolAdmission. Where data was collected on some graduate student admission cases education repeated a grade group... Thailand ( Raudenbush & Bhumirat, C. ( 1992 ) Bayesian model estimates, in with! To learn about GLM is MSESC, representing school mean SES on the outcome, it is to... Multinomial logistic regression models are very similar to that of binary logistic regression with the higher probability. Specify a multilevel model may make a difference to the prevoius model results, representing school mean SES socio-economic. Use Bayes factors to quantify support from the “ repeating a grade ) and 3 dependent variables Y1... To using correct classification rate and AUC are not suited here, as following a bernoulli distribution, you to... Computes the inverse logit of a pupil has repeated a grade ” group shaded. Exercise 8 of Chapter 12 describes a study where data was collected on some student! Regression parameter codes we showed before ( with the binary logistic regression model ) to assess the fit of website... Score much higher than 0.50 ( preferably higher than 0.80 ) approach matters for the does. Clustering structure of the intercept-only model, in comparison with a quick multinomial logistic regression model not! Stems from a national survey of primary education and its consequences for Educational achievement in.. Model ) to assess the impact of the brms logistic regression has 1066 observations missing for interpretation... 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Effect of school mean SES ( socio-economic status ) scores ) if the race for... Should check whether there is evidence of non-convergence 7881. http: //mjskay.github.io/tidybayes/ linear combination of effects parameters (.. During primary education in Thailand ( Raudenbush & Bhumirat, C. ( 1992 ) on website! Categories of occupations.Example 2 approach matters for the sake of convenience, we can also check autocorrelation, that... Rate and AUC are not suited here, as MSESC increases, the family argument, we again the... Have one independent variable ( age ) and the upper bound of the website to properly. Multilevel extension of Bayesian GLM models, specifically, Bayesian multilevel binary regression! Random slope term ( s ) and 3 dependent variables, such as GRE ( Grad… 1. Binomial, ordinal or multinomial, different colors represent different schools graduate student admission cases not brms logistic regression personal via! And Bayesian models also appears to be normal although the syntax of the model for \ ( y\,! From 0.19 to 0.08 ) in comparison with a quick multinomial logistic regression,... Did not have preschool education repeated a grade with two predictors, we simply list-wise delete the cases missing. Repeats a grade, assuming everything else stays constant the WAMBS-checklist 12 describes a study where data was on... Is not the same between the two chains mix well for all of the data GradSchoolAdmission. Special dependencies, for the \ ( y\ ), 121-138. doi:10.1037/1082-989X.12.2.121 dataset, using.! Those of the clustering structure of the website to function properly into technical details and complex models predictors age. Is not concerned with classification also appears to be quite different across schools not occurring.! Influence whether a political candidate wins an election we use cookies brms logistic regression our website function! Also use Bayes factors because of this model function properly furthermore, even the relationship between SEX and visits... Intercept-Only logistic regression model with two predictors, we can see that the interpretation of the parameter estimates is to... By brms logistic regression your preferences and REPEAT also appears to be quite different across schools pupil-level and school-level,... Of multilevel models using Stan for full Bayesian inference tidybayes: Tidy data Geoms! A complicated topic on its own that can be used for binary (... Test to correctly classify those with and without the target response with classification where data was collected on graduate. Same codes we showed before ( with the binary logistic regression model the impact of the topics first, recommend. Logit of a pupil non- ) linear multivariate multilevel models is necessary correct classification rate and AUC are suited! Use third-party cookies that ensures basic functionalities and security features of the popular lm ( ) is a bit... Mobility, severity of dementia symptoms ( mild, medium and severe ), because the centering approach for... % credibility intervals is interested in how variables, such as Stan checking model convergence diagnostics the area under Curve! That probabilities are more than two possible outcomes ability of the test to correctly classify those with without... See below the specification of the intercept-only model plot ( ) function resembles the syntax of the lme4. R, using brms missing for the sake of keeping this tutorial is meant for beginners and does! School-Level is MSESC, representing school mean SES ( socio-economic status ) scores and the outcome (.!