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The Poisson transform for unnormalised statistical models

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Abstract

Contrary to standard statistical models, unnormalised statistical models only specify the likelihood function up to a constant. While such models are natural and popular, the lack of normalisation makes inference much more difficult. Extending classical results on the multinomial-Poisson transform (Baker In: J Royal Stat Soc 43(4):495–504, 1994), we show that inferring the parameters of a unnormalised model on a space \(\Omega \) can be mapped onto an equivalent problem of estimating the intensity of a Poisson point process on \(\Omega \). The unnormalised statistical model now specifies an intensity function that does not need to be normalised. Effectively, the normalisation constant may now be inferred as just another parameter, at no loss of information. The result can be extended to cover non-IID models, which includes for example unnormalised models for sequences of graphs (dynamical graphs), or for sequences of binary vectors. As a consequence, we prove that unnormalised parameteric inference in non-IID models can be turned into a semi-parametric estimation problem. Moreover, we show that the noise-contrastive estimation method of Gutmann and Hyvärinen (J Mach Learn Res 13(1):307–361, 2012) can be understood as an approximation of the Poisson transform, and extended to non-IID settings. We use our results to fit spatial Markov chain models of eye movements, where the Poisson transform allows us to turn a highly non-standard model into vanilla semi-parametric logistic regression.

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Correspondence to Nicolas Chopin.

Appendices

Derivatives of Poisson-transformed likelihoods

The first and second derivatives of \(\mathcal {L}\left( \varvec{\theta }\right) \) and \(\mathcal {M}\left( \varvec{\theta },\nu \right) \) are needed in the proofs and we collect them here.

Derivatives of \(\mathcal {L}\left( \varvec{\theta }\right) \):

$$\begin{aligned}&\mathcal {L}(\varvec{\theta }) = \sum _{i=1}^{n}f_{\varvec{\theta }}(\mathbf {y}_{i})-n\log \left( \int \text{ exp }\left\{ f_{\varvec{\theta }}\left( s\right) \right\} \text{ d }s\right) \\&\qquad :=\sum _{i=1}^{n}f_{\varvec{\theta }}(\mathbf {y}_{i})-n\phi \left( \varvec{\theta }\right) \\&\frac{\partial }{\partial \varvec{\theta }}\phi \left( \varvec{\theta }\right) = \!\int \!\frac{\partial }{\partial \varvec{\theta }}f_{\varvec{\theta }}(s)\text{ exp }\left\{ f_{\varvec{\theta }}\!\left( s\right) \!-\!\phi \left( \varvec{\theta }\right) \right\} \text{ d }s\!=\!E_{\varvec{\theta }}\!\left( \frac{\partial }{\partial \varvec{\theta }}f_{\varvec{\theta }}\right) \\&\frac{\partial ^{2}}{\partial \varvec{\theta }^{2}}\phi \left( \varvec{\theta }\right) = E_{\varvec{\theta }}\left( \frac{\partial ^{2}}{\partial \varvec{\theta }^{2}}f_{\varvec{\theta }}\right) \!+\!E_{\varvec{\theta }}\!\left( \left( \frac{\partial }{\partial \varvec{\theta }}f_{\varvec{\theta }}\right) \!\left( \frac{\partial }{\partial \varvec{\theta }}f_{\varvec{\theta }}\right) ^{t}\right) \\&\qquad \qquad \!\!\!\quad \,\,-E_{\varvec{\theta }}\left( \frac{\partial }{\partial \varvec{\theta }}f_{\varvec{\theta }}\right) E_{\varvec{\theta }}\left( \frac{\partial }{\partial \varvec{\theta }}f_{\varvec{\theta }}\right) ^{t}\\&\frac{\partial }{\partial \varvec{\theta }}\mathcal {L}= \sum _{i=1}^{n}\frac{\partial }{\partial \varvec{\theta }}f_{\varvec{\theta }}(\mathbf {y}_{i})-n\frac{d}{d\varvec{\theta }}\phi \left( \varvec{\theta }\right) \\&\frac{\partial ^{2}}{\partial \varvec{\theta }^{2}}\mathcal {L}\left( \varvec{\theta }\right) = \sum \frac{\partial ^{2}}{\partial \varvec{\theta }^{2}}f_{\varvec{\theta }}(\mathbf {y}_{i})-n\frac{d^{2}}{d\varvec{\theta }^{2}}\phi \left( \varvec{\theta }\right) \end{aligned}$$

where we have used \(E_{\varvec{\theta }}\) as shorthand for the expectation with respect to density \(\text{ exp }\left\{ f_{\varvec{\theta }}\left( s\right) -\phi \left( \varvec{\theta }\right) \right\} \).

Derivatives of \(\mathcal {M}\left( \varvec{\theta },\nu \right) \):

$$\begin{aligned}&\mathcal {M}\left( \varvec{\theta },\nu \right) = \sum _{i=1}^{n}\left\{ f_{\varvec{\theta }}(\mathbf {y}_{i})\!+\!\nu \right\} \!-\!n\int \text{ exp }\left\{ f_{\varvec{\theta }}\left( s\right) \!+\!\nu \right\} \text{ d }s\\&\frac{\partial }{\partial \varvec{\theta }}\mathcal {M}\left( \varvec{\theta },\nu \right) = \sum _{i=1}^{n}\frac{\partial }{\partial \varvec{\theta }}f_{\varvec{\theta }}(\mathbf {y}_{i})-nE_{\varvec{\theta },\nu }\left( \frac{\partial }{\partial \varvec{\theta }}f_{\varvec{\theta }}\right) \\&\frac{\partial }{\partial \nu }\mathcal {M}\left( \varvec{\theta },\nu \right) = n-n\int \text{ exp }\left\{ f_{\varvec{\theta }}\left( s\right) +\nu \right\} \text{ d }s\\&\frac{\partial ^{2}}{\partial \varvec{\theta }^{2}}\mathcal {M}\left( \varvec{\theta },\nu \right) = \sum \frac{\partial ^{2}}{\partial \varvec{\theta }^{2}}f_{\varvec{\theta }}\left( \mathbf {y}_{i}\right) -n\left( E_{\varvec{\theta },\nu }\left( \frac{\partial ^{2}}{\partial ^{2}\varvec{\theta }}f_{\varvec{\theta }}\right) \right. \\&\qquad \qquad \qquad \,\left. +\,E_{\varvec{\theta },\nu }\left( \left( \frac{\partial }{\partial \varvec{\theta }}f_{\varvec{\theta }}\right) \left( \frac{\partial }{\partial \varvec{\theta }}f_{\varvec{\theta }}\right) ^{t}\right) \right) \\&\frac{\partial ^{2}}{\partial \nu ^{2}}\mathcal {M}\left( \varvec{\theta },\nu \right) = -n\int \text{ exp }\left\{ f_{\varvec{\theta }}\left( s\right) +\nu \right\} \text{ d }s\\&\frac{\partial }{\partial \varvec{\theta }\partial \nu }\mathcal {M}\left( \varvec{\theta },\nu \right) = -nE_{\varvec{\theta },\nu }\left( \frac{\partial }{\partial \varvec{\theta }}f_{\varvec{\theta }}\right) \end{aligned}$$

where we have used \(E_{\varvec{\theta },\nu }\) as shorthand for the linear operator \(E_{\varvec{\theta },\nu }(\varphi )=\int \varphi (s)\text{ exp }\left\{ f_{\varvec{\theta }}\left( s\right) +\nu \right\} \,\text{ d }s\) (which is not an expectation in general).

Further properties of the Poisson transform

The Poisson transform preserves confidence intervals

The usual method for obtaining confidence intervals for \(\varvec{\theta }\) is to invert the Hessian matrix of \(\mathcal {L}\left( \varvec{\theta }\right) \) at the mode, \(\varvec{\theta }^{\star }\):

$$\begin{aligned} \mathbf {C}_{\mathcal {L}}=\left( -\frac{d^{2}}{d^{2}\varvec{\theta }}\mathcal {L}\left| _{\varvec{\theta }=\varvec{\theta }^{\star }}\right. \right) ^{-1} \end{aligned}$$

We can show that the same confidence intervals can be obtained from \(\mathcal {M}\left( \varvec{\theta },\nu \right) \) at the joint mode, \(\varvec{\theta }^{\star },\nu ^{\star }\).

At the joint maximum, \(\nu ^{\star }\) normalises the intensity function, and the Hessian of \(\mathcal {M}\) equals

$$\begin{aligned} H= & {} \left[ \begin{array}{cc} \mathbf {H}_{aa} &{} \mathbf {H}_{ba}\\ \mathbf {H}_{ab} &{} \mathbf {H}_{bb} \end{array}\right] =\left[ \begin{array}{cc} \frac{\partial ^{2}}{\partial ^{2}\varvec{\theta }}\mathcal {M}\left( \varvec{\theta },\nu \right) &{} \frac{\partial }{\partial \varvec{\theta }\partial \nu }\mathcal {M}\left( \varvec{\theta },\nu \right) \\ \frac{\partial }{\partial \nu \partial \varvec{\theta }}\mathcal {M}\left( \varvec{\theta },\nu \right) &{} \frac{\partial ^{2}}{\partial ^{2}\nu }\mathcal {M}\left( \varvec{\theta },\nu \right) \end{array}\right] \\= & {} \left[ \begin{array}{cc} \!\sum \!\frac{\partial ^{2}}{\partial \varvec{\theta }^{2}}f_{\varvec{\theta }}\left( \mathbf {y}_{i}\right) -nE_{\varvec{\theta }}\left( \frac{\partial ^{2}}{\partial ^{2}\varvec{\theta }}f\right) - nE_{\varvec{\theta }}\left( \left( \frac{\partial }{\partial \varvec{\theta }}f_{\varvec{\theta }}\right) \left( \frac{\partial }{\partial \varvec{\theta }}f_{\varvec{\theta }}\right) ^{t}\right) &{} -nE_{\varvec{\theta }}\left( \frac{\partial }{\partial \varvec{\theta }}f_{\varvec{\theta }}\right) ^{t}\\ -nE_{\varvec{\theta }}\left( \frac{\partial }{\partial \varvec{\theta }}f_{\varvec{\theta }}\right) &{} -n \end{array}\right] \end{aligned}$$

where again E denotes the expectation with respect to density \(\text{ exp }\left\{ f_{\varvec{\theta }}\left( s\right) -\phi \left( \varvec{\theta }\right) \right\} \).

Inverting \(-H\) also yields confidence intervals. By the inversion rule for block matrices, the approximate covariance for \(\varvec{\theta }\) using \(\mathcal {M}\left( \varvec{\theta },\nu \right) \) equals

$$\begin{aligned} \mathbf {C}_{\mathcal {M}}^{-1}= & {} -\left( \mathbf {H}_{aa}-\mathbf {H}_{ba}\mathbf {H}_{bb}^{-1}\mathbf {H}_{ab}\right) \\= & {} -\left( \mathbf {H}_{aa}+\frac{1}{n}n^{2}E_{\varvec{\theta }}\left( \frac{\partial }{\partial \varvec{\theta }}f_{\varvec{\theta }}\right) E_{\varvec{\theta }}\left( \frac{\partial }{\partial \varvec{\theta }}f_{\varvec{\theta }}\right) ^{t}\right) \\= & {} -\Bigg [\sum \frac{\partial ^{2}}{\partial \varvec{\theta }^{2}}f_{\varvec{\theta }}\left( \mathbf {y}_{i}\right) -nE_{\varvec{\theta }}\left( \frac{\partial ^{2}}{\partial ^{2}\varvec{\theta }}f\right) \\&-\,nE_{\varvec{\theta }}\left( \left( \frac{\partial }{\partial \varvec{\theta }}f_{\varvec{\theta }}\right) \left( \frac{\partial }{\partial \varvec{\theta }}f_{\varvec{\theta }}\right) ^{t}\right) \\&+\,nE_{\varvec{\theta }}\left( \frac{\partial }{\partial \varvec{\theta }}f_{\varvec{\theta }}\right) E\left( \frac{\partial }{\partial \varvec{\theta }}f_{\varvec{\theta }}\right) ^{t}\Bigg ]\\= & {} \mathbf {C}_{\mathcal {L}}^{-1} \end{aligned}$$

Preservation of log-concavity in exponential families

In exponential families, the log-likelihood is concave, which facilitates inference. The Poisson transform preserves this log-concavity.

In the natural parameterisation, exponential family models are given by

$$\begin{aligned} \mathcal {L}\left( \varvec{\theta }\right) =\exp \left\{ \sum _{i=1}^{n}s(\mathbf {y}_{i})^{t}\varvec{\theta }-\phi \left( \varvec{\theta }\right) \right\} \end{aligned}$$

with \(s(\mathbf {y})\) a vector of sufficient statistics. The second derivative of \(\mathcal {L}\left( \varvec{\theta }\right) \) simplifies to

$$\begin{aligned} -\frac{1}{n}\frac{\partial }{\partial ^{2}\varvec{\theta }}\mathcal {L}\left( \varvec{\theta }\right)= & {} \int s(\mathbf {y})s(\mathbf {y})^{t}\exp \left( s(\mathbf {y})^{t}\varvec{\theta }-\phi \left( \varvec{\theta }\right) \right) \\= & {} E_{\varvec{\theta }}\left\{ s(\mathbf {y})s(\mathbf {y})^{t}\right\} \end{aligned}$$

a p.s.d. matrix, which establishes concavity.

The second derivatives of \(\mathcal {M}\left( \varvec{\theta },\nu \right) \) (Section 1) also simplify

$$\begin{aligned} -\frac{1}{n}\frac{\partial ^{2}}{\partial \varvec{\theta }^{2}}\mathcal {M}\left( \varvec{\theta },\nu \right)= & {} \exp \left\{ \nu -\nu ^{\star }\left( \varvec{\theta }\right) \right\} \\&\times \int s(\mathbf {y})s(\mathbf {y})^{t}\exp \left( s(\mathbf {y})^{t}\varvec{\theta }-\phi \left( \varvec{\theta }\right) \right) \\ -\frac{1}{n}\frac{\partial ^{2}}{\partial \nu \partial \varvec{\theta }}\mathcal {M}\left( \varvec{\theta },\nu \right)= & {} \exp \left\{ \nu -\nu ^{\star }\left( \varvec{\theta }\right) \right\} \\&\times \int s(\mathbf {y})\exp \left( s(\mathbf {y})^{t}\varvec{\theta }-\phi \left( \varvec{\theta }\right) \right) \\ -\frac{1}{n}\frac{\partial ^{2}}{\partial \nu {}^{2}}\mathcal {M}\left( \varvec{\theta },\nu \right)= & {} \exp \left\{ \nu -\nu ^{\star }\left( \varvec{\theta }\right) \right\} \end{aligned}$$

so that the full Hessian \(\mathbf {H}\) can be written in block form as follows:

$$\begin{aligned} -\frac{1}{n}\exp \left\{ \nu ^{\star }\left( \varvec{\theta }\right) -\nu \right\} \mathbf {H}=\left[ \begin{array}{cc} E_{\varvec{\theta }}\left\{ s\left( \mathbf {y}\right) s\left( \mathbf {y}\right) ^{t}\right\} &{} E\left( s\left( \mathbf {y}\right) \right) \\ E_{\varvec{\theta }}\left\{ s\left( \mathbf {y}\right) ^{t}\right\} &{} 1 \end{array}\right] \!=\!\mathbf {A} \end{aligned}$$

and \(\mathbf {H}\) is n.s.d if and only if for all \(\mathbf {x},c\) such that \(({\mathbf {x}},c)\ne {\mathbf {0}}\):

$$\begin{aligned} \left[ \begin{array}{cc} \mathbf {x}^{t}&c\end{array}\right] \mathbf {A}\left[ \begin{array}{c} \mathbf {x}\\ c \end{array}\right] >0 \end{aligned}$$

which the following establishes:

$$\begin{aligned}&\left[ \begin{array}{cc} \mathbf {x}^{t}&c\end{array}\right] \left[ \begin{array}{cc} E_{\varvec{\theta }}\left\{ s\left( \mathbf {y}\right) s\left( \mathbf {y}\right) ^{t}\right\} &{} E\left\{ s\left( \mathbf {y}\right) \right\} \\ E_{\varvec{\theta }}\left\{ s\left( \mathbf {y}\right) ^{t}\right\} &{} 1 \end{array}\right] \left[ \begin{array}{c} \mathbf {x}\\ c \end{array}\right] \\&\quad = \left[ \begin{array}{cc} \mathbf {x}^{t}&c\end{array}\right] \left[ \begin{array}{c} E_{\varvec{\theta }}\left\{ s\left( \mathbf {y}\right) s\left( \mathbf {y}\right) ^{t}\right\} \mathbf {x}+cE_{\varvec{\theta }}\left\{ s\left( \mathbf {y}\right) \right\} \\ E_{\varvec{\theta }}\left\{ s\left( \mathbf {y}\right) ^{t}\right\} \mathbf {x}+c \end{array}\right] \\&\quad = E_{\varvec{\theta }}\left\{ \mathbf {x}^{t}s\left( \mathbf {y}\right) s\left( \mathbf {y}\right) ^{t}\mathbf {x}\right\} +2E_{\varvec{\theta }}\left\{ \mathbf {x}^{t}s\left( \mathbf {y}\right) \right\} c+c^{2}\\&\quad = E_{\varvec{\theta }}\left[ \left( s\left( \mathbf {y}\right) ^{t}\mathbf {x}+c\right) ^{2}\right] >0 \end{aligned}$$

assuming \(E_{\varvec{\theta }}\left\{ s(\mathbf {y})s(\mathbf {y})^{t}\right\} \) is p.s.d. for all \(\varvec{\theta }\).

Noise-constrative divergence approximates the Poisson transform (Theorem 7)

We have assumed that

$$\begin{aligned} f_{\varvec{\theta }}(\mathbf {y})-\log q(\mathbf {y})\le C(\varvec{\theta }) \end{aligned}$$

for a certain constant \(C(\varvec{\theta })\) that may depend on \(\varvec{\theta }\), and all \(\mathbf {y}\in \Omega \). We rewrite the log odds ratio as \(h(\mathbf {y})-\log (m)\) where

$$\begin{aligned} h(\mathbf {y}):=f_{\varvec{\theta }}(\mathbf {y})+\nu -\log q(\mathbf {y})+\log (n) \end{aligned}$$

does not depend on m; note \(h(\mathbf {y})\le \bar{h}:=C(\varvec{\theta })+\nu +\log (n)\). One has

$$\begin{aligned}&{\mathcal {R}}^{m}(\varvec{\theta },\nu )+\log (m/n)= \sum _{i=1}^{n}\log \left[ \frac{m\exp \left\{ f_{\varvec{\theta }}(\mathbf {y}_{i})+\nu \right\} }{n\exp \left\{ f_{\varvec{\theta }}(\mathbf {y}_{i})+\nu \right\} +mq(\mathbf {y}_{i})}\right] \\&\quad +\sum _{j=1}^{m}\log \left[ \frac{mq({\mathbf {r}}_{j})}{n\exp \left\{ f_{\varvec{\theta }}({\mathbf {r}}_{j})+\nu \right\} +mq({\mathbf {r}}_{j})}\right] \end{aligned}$$

where the first term trivially converges (as \(m\rightarrow +\infty \)) to

$$\begin{aligned} \sum _{i=1}^{n}\left\{ f_{\varvec{\theta }}(\mathbf {y}_{i})+\nu -\log q(\mathbf {y}_{i})\right\} . \end{aligned}$$

Regarding the second term, one has

$$\begin{aligned}&\log \left[ \frac{mq({\mathbf {r}}_{j})}{n\exp \left\{ f_{\varvec{\theta }}({\mathbf {r}}_{j})+\nu \right\} +mq({\mathbf {r}}_{j})}\right] \\&\quad =\log \left[ 1-\frac{1}{1+m\exp \left\{ -h({\mathbf {r}}_{j})\right\} }\right] \end{aligned}$$

where

$$\begin{aligned} 0\le \frac{1}{1+m\exp \left\{ -h({\mathbf {r}}_{j})\right\} }\le \frac{1}{m}\exp (\bar{h}). \end{aligned}$$

Since \(\left| \log (1-x)+x\right| \le x^{2}\) for \(x\in [0,1/2]\), we have, for m large enough, that

$$\begin{aligned}&\left| \log \left[ \frac{mq({\mathbf {r}}_{j})}{n\exp \left\{ f_{\varvec{\theta }}({\mathbf {r}}_{j})+\nu \right\} +mq({\mathbf {r}}_{j})}\right] \right. \nonumber \\&\quad \left. +\frac{1}{1+m\exp \left\{ -h({\mathbf {r}}_{j})\right\} }\right| \le \frac{\exp (2\bar{h})}{m^{2}} \end{aligned}$$
(5.1)

and

$$\begin{aligned} \left| \frac{1}{1+m\exp \left\{ -h({\mathbf {r}}_{j})\right\} }-\frac{1}{m}\exp \left\{ h({\mathbf {r}}_{j})\right\} \right| \le \frac{\exp (2\bar{h})}{m^{2}} \end{aligned}$$

and since, by the law of large numbers,

$$\begin{aligned}&\frac{1}{m}\sum _{j=1}^{m}\exp \left\{ h({\mathbf {r}}_{i})\right\} \rightarrow \mathbb {E}_{q}[\exp \left\{ h({\mathbf {r}}_{i})\right\} ]\nonumber \\&\quad =n\int \exp \left\{ f_{\varvec{\theta }}(\mathbf {y})+\nu \right\} \, d\mathbf {y}<+\infty \end{aligned}$$
(5.2)

almost surely as \(m\rightarrow +\infty \), one also has

$$\begin{aligned}&\sum _{j=1}^{m}\log \left[ \frac{mq(\mathbf {y}_{i})}{n\exp \left\{ f_{\varvec{\theta }}(\mathbf {y}_{i})+\eta \right\} +mq(\mathbf {y}_{i})}\right] \rightarrow \\&\quad -n\int \exp \left\{ f_{\varvec{\theta }}(\mathbf {y})+\nu \right\} \, d\mathbf {y}\end{aligned}$$

almost surely, since the difference between the two sums is bounded deterministically by \(\exp (2\bar{h})/m\).

Uniform convergence of the noise-constrative divergence (Theorem 8)

We first prove two intermediate results.

Lemma 9

Assuming that \(\left| f_{\varvec{\theta }}(\mathbf {y})-\log q(\mathbf {y})\right| \le C\) for all \(\mathbf {y}\in \Omega \), then there exists a bounded interval I such that, for any \(\varvec{\theta }\), the maximum of both functions \(\nu \rightarrow \mathcal {M}(\varvec{\theta },\nu )\) and \(\nu \rightarrow {\mathcal {R}}^{m}(\varvec{\theta },\nu )\) is attained in I.

Proof

Let \(\varvec{\theta }\) some fixed value. \(\mathcal {M}(\varvec{\theta },\nu )\) is maximised at \(\nu ^{\star }(\varvec{\theta })=-\log \int _{\Omega }\exp \left\{ f_{\varvec{\theta }}(\mathbf {y})\right\} \, d\mathbf {y}\in \left[ -C,C\right] \), since \(e^{-C}q\le f_{\varvec{\theta }}\le e^{C}q\). For \({\mathcal {R}}^{m}(\varvec{\theta },\nu )\), using again \(e^{-C}q\le f_{\varvec{\theta }}\le e^{C}q\), one sees that \(l(\nu )\le {\mathcal {R}}^{m}(\varvec{\theta },\nu )\le u(\nu )\), where l and u are functions of \(\nu \) that diverges at \(-\infty \) for both \(\nu \rightarrow +\infty \) and \(\nu \rightarrow -\infty \); i.e.

$$\begin{aligned}&{\mathcal {R}}^{m}(\varvec{\theta },\nu )+\log (m/n)\le u(\nu ):= \sum _{i=1}^{n}\log \left[ \frac{m\exp \left\{ C+\nu \right\} }{n\exp (-C+\nu )+m}\right] \\&\quad +\sum _{j=1}^{m}\log \left[ \frac{m}{n\exp \left\{ -C+\nu \right\} +m}\right] \end{aligned}$$

and the lower bound \(l(\nu )\) has a similar expression. Thus, one may construct an interval J such that the maximum of function \(\nu \rightarrow {\mathcal {R}}^{m}(\varvec{\theta },\nu )\) is attained in J for all \(\varvec{\theta }\) (e.g. take J such that for \(\nu \in J^{c}, u(\nu )\le M_{l}/2, l(\nu )\le M_{l}/2\), with \(M_{l}=\sup _{\nu }l\)) . To conclude, take \(I=J\cup [-C,C]\).

We now establish uniform convergence, but, in light of the previous result, we restrict \(\nu \) to the interval I defined in Lemma 9.

Lemma 10

Under the Assumptions that (i) \(\Theta \) is bounded, that (ii) \(\left| f_{\varvec{\theta }}(\mathbf {y})-\log q(\mathbf {y})\right| \le C\) for all \(\mathbf {y}\in \Omega \), that (iii) \(\left| f_{\varvec{\theta }}(\mathbf {y})-f_{\varvec{\theta }'}(\mathbf {y})\right| \le \kappa (\mathbf {y})\left\| \varvec{\theta }-\varvec{\theta }'\right\| \) with \(\kappa \) such that \(\mathbb {E}_{q}[\kappa ]<\infty \), one has, for fixed \({\mathcal {S}}=\left\{ \mathbf {y}_{1},\ldots ,\mathbf {y}_{n}\right\} \):

$$\begin{aligned}&\sup _{\left( \varvec{\theta },\nu \right) \in \Theta \times I}\left| {\mathcal {R}}^{m}(\varvec{\theta },\nu )+\log (m/n)+\sum _{i=1}^{n}\log q({\mathbf {y}}_{i})\right. \nonumber \\&\left. \quad -\mathcal {M}(\varvec{\theta },\nu )\right| \rightarrow 0 \end{aligned}$$
(5.3)

almost surely, relative to the randomness induced by \({\mathcal {R}}=\left\{ {\mathbf {r}}_{1},\ldots ,{\mathbf {r}}_{m}\right\} .\)

Proof

Recall that the absolute difference above was bounded by the sum of three terms in the previous Appendix. The first term was

$$\begin{aligned}&\sum _{i=1}^{n}\left[ \log \left[ \frac{m\exp \left\{ f_{\varvec{\theta }}(\mathbf {y}_{i})+\nu \right\} }{n\exp \left\{ f_{\varvec{\theta }}(\mathbf {y}_{i})+\nu \right\} +mq(\mathbf {y}_{i})}\right] \right. \\&\quad \left. -\left\{ f_{\varvec{\theta }}(\mathbf {y}_{i})+\nu -\log q(\mathbf {y}_{i})\right\} \right] \end{aligned}$$

which clearly converges deterministically to 0 as \(m\rightarrow +\infty .\) In addition, this convergence is uniform with respect to \(\left( \varvec{\theta },\nu \right) \in \Theta \times I\), since \(\left| \log x-\log y\right| \le c\left| x-y\right| \) for \(x,y\ge 1/c\), and here, by Assumption (ii),

$$\begin{aligned} x:= & {} \frac{m\exp \left\{ f_{\varvec{\theta }}(\mathbf {y}_{i})+\nu \right\} }{n\exp \left\{ f_{\varvec{\theta }}(\mathbf {y}_{i})+\nu \right\} +mq(\mathbf {y}_{i})}\ge \frac{m\exp \left\{ -C+\nu \right\} }{n\exp \left\{ C+\nu \right\} +m}\\\ge & {} \exp \left\{ -C+\nu \right\} \end{aligned}$$

and \(y=\exp \left\{ f_{\varvec{\theta }}(\mathbf {y}_{i})+\nu -\log q(\mathbf {y}_{i})\right\} \ge \exp \left\{ -C+\nu \right\} \), so both x and y are lower bounded since \(\nu \in I\). Similarly \((x-y)\) is bounded by \(C'/m\), where \(C'\) is some constant independent of \(\varvec{\theta }\).

The second term, see (5.1), was bounded by \(\exp (2\bar{h})/m^{2}\), where \(\bar{h}\), an upper bound of h, may now be replaced by a constant, since \(h(\mathbf {y}):=f_{\varvec{\theta }}(\mathbf {y})+\nu -\log q(\mathbf {y})+\log (n)\le C+\nu +\log (n)\) and \(\nu \in I\), again by Assumption (ii).

The third term is related to the law of large numbers (5.2) for random variable \(H_{(\varvec{\theta },\eta )}({\mathbf {r}}_{i}):=\exp \left\{ h({\mathbf {r}}_{i})\right\} \), which depended implicitly on \(\left( \varvec{\theta },\eta \right) \):

$$\begin{aligned} H_{(\varvec{\theta },\eta )}({\mathbf {r}}_{i})=\frac{n\exp \left\{ f_{\varvec{\theta }}({\mathbf {r}}_{i})+\nu \right\} }{q({\mathbf {r}}_{i})}. \end{aligned}$$

To obtain (almost surely) uniform convergence, we use the generalised version of the Glivenko-Cantelli theorem; e.g. Theorem 19.4 p. 270 in Van der Vaart (2000). From Example 19.7 of the same book, one sees that a sufficient condition in our case is that \(\Theta \) is bounded [Assumption (i)], and that

$$\begin{aligned} \left| H_{(\varvec{\theta },\eta )}({\mathbf {r}})-H_{(\varvec{\theta }',\eta ')}({\mathbf {r}})\right| \le m({\mathbf {r}})\left\| \varvec{\xi }-\varvec{\xi }'\right\| \end{aligned}$$

for \(\varvec{\xi }=(\varvec{\theta },\eta ), \varvec{\xi }'=(\varvec{\theta }',\eta ')\), and m a function such that \(\mathbb {E}_{q}[m]<\infty \). But

$$\begin{aligned}&\left| H_{(\varvec{\theta },\eta )}({\mathbf {r}})-H_{(\varvec{\theta }',\eta ')}({\mathbf {r}})\right| =\frac{n\exp \left\{ f_{\varvec{\theta }}({\mathbf {r}})+\nu \right\} }{q({\mathbf {r}})}\\&\quad \left| 1-\exp \left\{ f_{\varvec{\theta }}({\mathbf {r}})+\nu -f_{\varvec{\theta }'}({\mathbf {r}})-\nu '\right\} \right| \\&\quad \le ne^{C+\nu }\left| 1-\exp \left\{ f_{\varvec{\theta }}({\mathbf {r}})+\nu -f_{\varvec{\theta }'}({\mathbf {r}})-\nu '\right\} \right| \\&\quad \le C'\left\{ \kappa ({\mathbf {r}})\left\| \varvec{\theta }-\varvec{\theta }'\right\| +\left| \nu -\nu '\right| \right\} \\&\quad \le C'\left\{ \kappa ({\mathbf {r}})+1\right\} \left\| \varvec{\xi }-\varvec{\xi }'\right\| \end{aligned}$$

by Assumption (ii), and for some constant \(C'\) independent of \(\varvec{\theta }\), since \(\left| 1-e^{x}\right| \le Kx\) for xy in a bounded set. One may conclude, since, by Assumption (ii), \(\mathbb {E}_{q}[\kappa ]<\infty \).

We are now able to prove Theorem 8. Again, let \(\varvec{\xi }=(\varvec{\theta },\nu )\), and rewrite any function of \((\varvec{\theta },\nu )\) as a function of \(\varvec{\xi }\), i.e. \(\mathcal {M}(\varvec{\xi }), {\mathcal {R}}^{m}(\varvec{\xi })\). By e.g. Theorem 5.7 p. 45 of Van der Vaart (2000), the uniform convergence 5.3 implies that that the maximiser \(\hat{\varvec{\xi }}^{m}\) of \({\mathcal {R}}^{m}(\varvec{\theta },\nu )\) converges to the maximiser \(\hat{\varvec{\xi }}\) of \(\mathcal {M}(\varvec{\theta },\nu )\), provided that (a) the maximisation is with respect to \(\left( \varvec{\theta },\nu \right) \in \Theta \times I\); and (b) that \(\sup _{d(\varvec{\xi },\hat{\varvec{\xi }})\ge \epsilon }\mathcal {M}(\varvec{\xi })<\mathcal {M}(\hat{\varvec{\xi }})\). However, by Lemma 9 one sees that in (a) the same estimators would be obtained by maximising instead with respect to \(\left( \varvec{\theta },\nu \right) \in \Theta \times \mathbb {R}\), and (b) is a direct consequence of Assumption (iv) of the theorem, if one takes for \(d(\varvec{\xi },\hat{\varvec{\xi }})\) the supremum norm of \(\varvec{\xi }-\hat{\varvec{\xi }}\).

Additional information on the application

In our application we fit a spatial Markov chain model using logistic regression. Since the procedure involves the generation of a random set of reference points, we incur some Monte Carlo error in the estimates. Estimating the magnitude of the Monte Carlo error is just a matter of running the procedure several times to look at variability in the estimates. We did so over five repetitions and report the results in Fig. 5. For each repetition we plot the estimated smooth effect of saccade angle \(r_{\mathrm{ang}}\), along with a 95% confidence band. Since smoothing splines are used, smoothing hyperparameters had to be inferred from the data (using REML, Wood 2011), and the reported confidence band is conditional on the estimated value of the smoothing hyperparameters. The fits and confidence bands are extremely stable over independent repetitions. The R command we used was

figurea
Fig. 5
figure5

Eye movement model: 5 independent replications of the estimates under different sets of random reference points. We show here the estimated effect of saccade angle with an associated 95%  pointwise confidence interval. The 5 replicates are in different colours and overlap each other almost completely, showing that 20 reference points per true datapoint are more than enough to produce stable estimates. (Color figure online)

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Barthelmé, S., Chopin, N. The Poisson transform for unnormalised statistical models. Stat Comput 25, 767–780 (2015). https://doi.org/10.1007/s11222-015-9559-4

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Keywords

  • Logistic Regression
  • Point Process
  • Normalisation Constant
  • Markov Chain Model
  • Bregman Divergence