# EnsembleIV: Creating Instrumental Variables from Ensemble Learners for
Robust Statistical Inference

Gordon Burtch1, Edward McFowland III2, Mochen Yang3, Gediminas Adomavicius3
1 Questrom School of Business, Boston University
2 Harvard Business School, Harvard University
3 Carlson School of Management, University of Minnesota

(Current Draft: 3/5/2023)

##  1 Introduction

We study the measurement error problem that arises from the integration of
supervised machine learning and statistical inference in a hybrid, two-stage
process. In the first stage, a supervised machine learning model is trained to
predict a target outcome based on a set of features. In the second stage,
predicted values of the target are used as an independent variable, usually
within a regression model, for statistical inference purposes. However,
because predictions from the first-stage machine learning model are typically
imperfect, prediction errors will manifest as measurement error in the second-
stage regression model, leading to estimation biases and threatening the
validity of inferences.

This two-stage approach is an example of statistical inference with generated
regressors (Pagan,, 1984; Oxley and McAleer,, 1993). It has seen growing
adoption in social sciences, partly due to impressive advances in machine
learning techniques to efficiently extract useful information from large
amounts of both structured and unstructured data. To name just a few examples,
Cengiz et al., (2022) built a boosted tree model to identify minimum wage
workers based on their demographics, and then examined the effect of minimum
wage policies on labor market outcomes for these workers. Moreno and
Terwiesch, (2014) mined text sentiment from user-generated comments on an
online service platform, and subsequently estimated the impact of (predicted)
sentiment on buyers’ purchasing decisions. Zhang et al., (2021) used deep
learning techniques to measure the quality of property images on Airbnb, and
studied how image quality affects property demand.

While the measurement error problem has been largely ignored in early work
that adopted this two-stage approach, it has received increasing attention
from researchers in recent years (e.g., Yang et al.,, 2018, 2022; Qiao and
Huang,, 2021; Fong and Tyler,, 2021). Unlike a traditional measurement error
setting, where the errors are unobserved and the estimation biases are
therefore hard to address, the integration of machine learning offers a unique
opportunity for bias correction. Because the training and evaluation of a
machine learning model typically require a manually collected labeled dataset,
where the true values of the target outcome (e.g., text sentiment or image
quality) are known, one can observe the measurement error in the generated
variable on this labeled data and quantify its properties (e.g.,
distributional characteristics or correlations with other variables of
interest). This has enabled the use of some existing techniques to mitigate
estimation biases (e.g., Simulation-Extrapolation Stefanski and Cook,, 1995;
Küchenhoff et al.,, 2006; Yang et al.,, 2018), as well as the development of
new ones (Yang et al.,, 2022; Qiao and Huang,, 2021; Fong and Tyler,, 2021).

In this paper, we develop a novel method that leverages instrumental variables
to address the measurement error problem. The method consists of three key
ingredients, respectively focusing on the generation, transformation, and
selection of instruments. First, we propose to use ensemble learning
techniques (e.g., random forest) to build the first-stage machine learning
model. Doing so generates a set of weak learners (e.g., individual trees in a
random forest) whose predictions can serve as candidate instruments for each
other. However, these instruments are “imperfect” in that they are not
guaranteed to satisfy the exclusion condition. Second, we adapt a
transformation technique originally developed by Nevo and Rosen, (2012) to
transform candidate instruments to comply with the exclusion condition. Third,
to deal with the potential challenge of weak instruments (i.e., instruments
that only barely satisfy the relevance condition), we borrow a LASSO-based
selection technique proposed by Belloni et al., (2012). The selected
instruments are both valid and strong, and are subsequently used in
instrumental variable regressions to obtain estimates that are less affected
by the measurement error problem. We henceforth refer to this method as
EnsembleIV.

We carry out empirical evaluations of EnsembleIV to understand its properties
and performance on both synthetic and real-world data. Simulation studies with
synthetic data (Section 3.1) demonstrate EnsembleIV’s ability to substantially
mitigate estimation biases on machine learning generated variables in several
common regression models. We then apply EnsembleIV to a real-world dataset of
user-generated content on social media (Section 3.4), and illustrate how it
can be used together with modern deep learning techniques.

EnsembleIV represents a novel methodological contribution to the nascent
literature on robust statistical inference with machine learning generated
variables. It improves upon existing approaches in several important aspects.
First, EnsembleIV has a more general theoretical foundation, as the method can
accommodate (1) both continuous and binary generated variables and (2) both
independent and correlated measurement errors (also known as classical and
non-classical errors in the literature Carroll et al.,, 2006). In contrast,
many existing approaches are more limited, e.g., SIMEX (Stefanski and Cook,,
1995; Yang et al.,, 2018) only addresses continuous and classical measurement
error and MC-SIMEX (Küchenhoff et al.,, 2006) works for binary
misclassification. Second, prior approaches that use instrumental variables
for bias correction, such as ForestIV (Yang et al.,, 2022), typically need to
rely on the “diversity” of weak learners within an ensemble (namely, the
property that prediction errors from different weak learners should be weakly
correlated, Breiman,, 2001) to discover valid instruments. EnsembleIV is much
less dependent on this property. Even if the ensemble learning technique does
not automatically produce diverse weak learners, EnsembleIV can still generate
valid instrumental variables, owing to the IV transformation technique (Nevo
and Rosen,, 2012). This allows EnsembleIV to leverage ensemble learning
algorithms other than the random forest (e.g., boosting), and greatly enhances
its practical applicability. Third, as will be shown in Section 3.3, we find
EnsembleIV to outperform the benchmark (i.e., ForestIV) in terms of estimation
efficiency, producing estimates with smaller standard errors on the same
sample.

##  2 EnsembleIV: Theory and Algorithm

In this section, we develop the theoretical components needed to create valid
instrumental variables from ensemble learners, based on which we design the
EnsembleIV algorithm. We use the following notations throughout the paper.

Notation | Definition
---|---
$Y$ | The dependent variable in second-stage regression.
$X$ | The independent variable in second-stage regression generated via machine learning.
$\boldsymbol{Z}$ | The exogenous control variables in second-stage regression.
$\varepsilon$ | The exogenous error term in second-stage regression.
$\widehat{X}$ | Predicted values of $X$ (generated by a machine learning model).
$e$ | Measurement / Prediction error in $\widehat{X}$.
$W$ | A candidate instrumental variable.
$D_{\bullet}$ | A particular partition of the data, one of training, testing, labeled, or unlabeled.
$M$ | Total number of weak learners in an ensemble.
$\widehat{X}^{(i)}$ | Predicted values of $X$ produced by weak learner $i\in\\{1,\ldots,M\\}$.
Table 1: Glossary of Notations

###  2.1 Instrumental Variable Approach to the Measurement Error Problem

We first formulate the measurement error problem that arises when a machine
learning generated variable is incorporated into a regression as an
independent covariate, then describe the IV approach to address it.

Without loss of generality, consider the following regression equation:111For
generalized linear models (e.g., logistic regressions), the same measurement
error problem can be formulated based on the latent variable model
(Wooldridge,, 2002), where the latent outcome $Y^{*}$ is linearly related to
independent covariates in the same way as specified in Equation (1).

$Y=X\beta+\boldsymbol{Z\Pi}+\varepsilon$ (1)

Under the setting that we consider, $X$ can only be observed in a relatively
small set of labeled data ($D_{label}$) but is unobserved in the much larger
unlabeled data ($D_{unlabel}$, and $|D_{unlabel}|\gg|D_{label}|$). A typical
reason for the size disparity between labeled and unlabeled data is the cost
of labeling $X$. As an example, in studies that investigate the relationships
between textual sentiment and certain dependent variables of interest (e.g.,
Tirunillai and Tellis,, 2012; Goh et al.,, 2013; Moreno and Terwiesch,, 2014),
researchers often need to manually label the sentiment (e.g., by hiring crowd
workers, such as Amazon Mechanical Turkers, to manually read the text and
assign sentiment labels). Doing so for a large volume of text can be very
expensive. Of course, one can directly estimate regression equation (1) using
$D_{label}$, but the limited size of $D_{label}$ may result in imprecise
estimates that hinder statistical inference. To make use of the larger
$D_{unlabel}$, one can build machine learning models on $D_{label}$ to predict
the values of $X$ on $D_{unlabel}$.

However, the predicted values $\widehat{X}$ generally contain some degree of
prediction error, defined as $e:=\widehat{X}-X$. When $\widehat{X}$ is added
into regression equation (1) as a surrogate for $X$, the prediction errors
manifest as measurement errors, and the regression that is actually estimated
can be written as:

$Y=\widehat{X}\beta+\boldsymbol{Z\Pi}+(\varepsilon-e\beta)$ (2)

We denote $u:=\varepsilon-e\beta$ as the error term in the estimated
regression, because
$Cov(\widehat{X},u)=Cov(X+e,\varepsilon-e\beta)=-\beta(Cov(X,e)+Var(e))$.
Aside from a very unlikely scenario wherein $Cov(X,e)$ precisely cancels out
$Var(e)$, the regression will suffer from endogeneity and produce biased and
inconsistent estimates.

It is important to note that we do not make assumptions regarding the
distributions of $X$ or $\widehat{X}$ – they can be continuous or binary – nor
do we restrict the relationship between $e$ and $X$ (i.e., $e$ could be
independent of, or correlated with $X$). In other words, our proposed approach
is applicable for both continuous and binary mismeasured covariate, and under
both independent and correlated measurement errors.

A standard solution to the measurement error problem is the instrumental
variable approach (Buzas and Stefanski,, 1996; Greene,, 2003; Hu and
Schennach,, 2008). A valid instrumental variable (IV), $W$, needs to satisfy
two conditions, namely (1) relevance, $Cov(W,\widehat{X})\neq 0$, i.e., the IV
should be correlated with the mismeasured covariate; and (2) exclusion,
$Cov(W,u)=-\beta Cov(W,e)=0$, i.e., the IV should be uncorrelated with the
measurement error. If such an IV can be identified, one can obtain consistent
estimates via IV regression, e.g., two-stage least-squares (2SLS, Greene,,
2003; Wooldridge,, 2002). In a finite sample, even though the (asymptotic)
consistency may not be fully realized, IV regression can nevertheless mitigate
estimation biases from measurement error.

Despite attractive theoretical properties of the IV approach, it is often
challenging to identify IVs that satisfy the exclusion condition in practice,
because the measurement error (or more generally, the source of endogeneity)
is not directly observable. As such, the exclusion condition is usually argued
conceptually but rarely tested empirically (Conley et al.,, 2012). Again, in
our context, this is not the case; measurement error can be observed directly
in the sample of data that is used to train the machine learning model.

###  2.2 Ensemble Learners as Instrumental Variables

We now describe the key theoretical components of EnsembleIV. Suppose the
first-stage machine learning model is an ensemble model with $M$ weak learners
(e.g., a random forest with $M$ trees). Instead of the common practice, using
the aggregated predictions,
$\widehat{X}=\frac{1}{M}\sum_{i}\widehat{X}^{(i)}$, in the second-stage
regression, consider using the predictions from one weak learner,
$\widehat{X}^{(i)}$, as a mismeasured variable, and relying on the predictions
from other weak learners, $\widehat{X}^{(j)}(j\neq i)$, as candidate IVs.

Because the different weak learners are all predictive of the target ground
truth to some extent, their predictions will generally be correlated with one
another, i.e., $Cov(\widehat{X}^{(j)},\widehat{X}^{(i)})\neq 0$. This
satisfies the relevance condition. However, there is no theoretical guarantee
that these candidate IVs will satisfy the exclusion condition, which requires
that $Cov(\widehat{X}^{(j)},\widehat{X}^{(i)}-X)=0$. To address the exclusion
violation, we adopt an IV transformation technique proposed by Nevo and Rosen,
(2012).

Specifically, suppose a candidate IV, $W$, is “imperfect” in the sense that it
violates the exclusion condition, i.e., $Cov(W,u)=-\beta Cov(W,e)\neq 0$. Nevo
and Rosen, (2012) propose a procedure to transform the IV so that it satisfies
the exclusion restriction. Denote correlation coefficients
$\rho_{Wu}=\frac{Cov(W,u)}{\sigma_{W}\sigma_{u}}$ and
$\rho_{\widehat{X}u}=\frac{Cov(\widehat{X},u)}{\sigma_{\widehat{X}}\sigma_{u}}$
and consider the following quantity

$\lambda=\frac{\rho_{Wu}}{\rho_{\widehat{X}u}}=\frac{Cov(W,u)}{Cov(\widehat{X},u)}\cdot\frac{\sigma_{\widehat{X}}}{\sigma_{W}}=\frac{Cov(W,e)}{Cov(\widehat{X},e)}\cdot\frac{\sigma_{\widehat{X}}}{\sigma_{W}}$
(3)

Conceptually, $\lambda$ measures how $W$ compares to $\widehat{X}$ in terms of
the degree to which each variable violates the exclusion criterion, i.e., how
much “better” (or “worse”) $W$ is than $\widehat{X}$ in this regard. The
following Theorem 1 describes how $\lambda$ can be used to construct a valid
IV.

######  Lemma 1 (IV Transformation, Nevo and Rosen, (2012)).

Let $\widetilde{W}=\sigma_{\widehat{X}}W-\lambda\sigma_{W}\widehat{X}$, then
$Cov(\widetilde{W},u)=0$.

###### Proof.

By definition,
$\widetilde{W}=\sigma_{\widehat{X}}W-\lambda\sigma_{W}\widehat{X}=\sigma_{\widehat{X}}W-\frac{Cov(W,u)}{Cov(\widehat{X},u)}\cdot\sigma_{\widehat{X}}\widehat{X}$.
It follows that

$Cov(\widetilde{W},u)=\sigma_{\widehat{X}}Cov(W,u)-\frac{Cov(W,u)}{Cov(\widehat{X},u)}\cdot\sigma_{\widehat{X}}Cov(\widehat{X},u)=0$

∎

In a typical measurement error problem setting, $\lambda$ is unknown because
the measurement error component, $e$, is unobservable. However, when the
mismeasured variable is generated by a machine learning model, we can
empirically estimate $\lambda$ using the labeled data. Specifically, for a
pair of weak learners $i\neq j$, where $\widehat{X}^{(i)}$ serves as the
endogenous covariate and $\widehat{X}^{(j)}$ serves as the (imperfect) IV, the
following procedure first estimates $\lambda$ based on $D_{test}$ and then
employs it to transforms the IV into a valid instrument.

Data: Weak learners $i\neq j$, $D_{test}$, and $D_{unlabel}$

// Estimate $\lambda$ from testing data

Deploy weak learners on $D_{test}$ to get predictions
$\widehat{X}_{test}^{(i)}$, $\widehat{X}_{test}^{(j)}$;

Set $\widehat{X}\leftarrow\widehat{X}_{test}^{(i)}$ and
$W\leftarrow\widehat{X}_{test}^{(j)}$;

Compute prediction error $e\leftarrow\widehat{X}-X_{test}$;

Compute
$\widehat{\lambda}\leftarrow\frac{Cov(W,e)}{Cov(\widehat{X},e)}\cdot\frac{\sigma_{\widehat{X}}}{\sigma_{W}}$;

// Transform IV on unlabeled data

Deploy weak learners on $D_{unlabel}$ to get predictions
$\widehat{X}_{unlabel}^{(i)}$, $\widehat{X}_{unlabel}^{(j)}$;

Compute
$\widetilde{W}^{(j)}=\sigma_{\widehat{X}}\widehat{X}_{unlabel}^{(j)}-\widehat{\lambda}\sigma_{W}\widehat{X}_{unlabel}^{(i)}$;

Output: $\widehat{X}_{unlabel}^{(i)}$ as the mismeasured covariate and
$\widetilde{W}^{(j)}$ as transformed IV.

Algorithm 1 Instrumental Variable Transformation Procedure

Asymptotically, because $D_{test}$ is randomly drawn from the population,
$\widehat{\lambda}$ should tend to its true value in the limit. In a given
finite sample, $\widehat{\lambda}$ should still provide a reasonable
approximation of the true parameter value, as long as the size of $D_{test}$
is sufficiently large.

Taking $\widehat{X}^{(i)}$ as the mismeasured covariate, we apply Algorithm 1
to each $\widehat{X}^{(j)}$ where $j\in\\{1,\ldots,M\\}\setminus i$, which
produces $M-1$ transformed IVs denoted as $\widetilde{W}^{(j)}$. Following
this procedure will often yield numerous asymptotically valid instruments for
the IV estimation, but it may be sub-optimal to employ all of them in this
manner for two reasons. First, some IVs may only be weakly correlated with the
mismeasured variable, and having weak instruments are known to produce
inconsistent estimates (Bound et al.,, 1995; Stock and Yogo,, 2002; Andrews et
al.,, 2019). Second, with a finite dataset, the value of $\widehat{\lambda}$
estimated from $D_{test}$ (for the pair of weak learners $i\neq j$) may not
equal its true population value. As a result, although the transformed IVs
likely improve upon the original (non-transformed) ones in terms of exclusion,
they still may not be perfectly excluded. With these “almost valid”
instruments, it becomes even more important to select strong IVs to use in
estimation (Murray,, 2006). Finally, having too many instruments can overfit
the endogenous covariate and fail to isolate the exogenous variations
(Roodman,, 2009). Because of these reasons, we next carry out an “IV
selection” step where a subset of strong IVs are selected from the pool of
transformed IVs. We specifically consider three different approaches for IV
selection:

  1. 1.

Top-$k$: select the $k$ transformed IVs having the strongest correlation with
the mismeasured covariate $\widehat{X}^{(i)}$. This serves as a simple
heuristic.

  2. 2.

PCA: apply PCA on the transformed IVs
$\\{\widetilde{W}^{(j)}\\}_{j\in\\{1,\ldots,M\\}\setminus i}$, then select the
top $k$ components to use as “condensed” IVs for estimation. This follows the
“factorized IV” approach of Mehrhoff, (2009).

  3. 3.

LASSO: employ a LASSO-based selection method based on Belloni et al., (2012).
Roughly speaking, we run a LASSO regression
$\widehat{X}^{(i)}\sim\sum_{j\in\\{1,\ldots,M\\}\setminus
i}\gamma_{j}\widetilde{W}^{(j)}$ and retain the transformed IVs that exhibit
non-zero coefficients. The penalty term in this LASSO regression is selected
in a data-driven manner, through cross-validation. Belloni et al., (2012) show
that this approach will yield a subset of strong IVs that, when employed in an
IV regression, produces consistent estimates of effects.

Because this IV selection step relies on the transformed IVs
$\widetilde{W}^{(j)}$, it is carried out on $D_{unlabel}$.

###  2.3 EnsembleIV Algorithm

Combining the IV transformation and selection procedures, we are now ready to
describe the EnsembleIV estimation procedure in the following Algorithm 2.

Data: $D_{train}$, $D_{test}$, and $D_{unlabel}$

Train an ensemble model with $M$ weak learners on $D_{train}$;

foreach  _$i\in\\{1,\ldots M\\}$_ do

Designate $\widehat{X}^{(i)}$ as mismeasured covariate;

// IV Transformation

Transform each $\widehat{X}^{(j)},(j\neq i)$ using Algorithm 1 to obtain
$\widetilde{W}^{(j)}$;

// IV Selection

Select a subset of strong IVs based on the top-$k$, PCA, or LASSO approach;

// IV Estimation

Estimate IV regression with the selected $\widetilde{W}^{(j)}$;

Store estimates $\boldsymbol{\widehat{\beta}}_{IV}^{(i)}$;

end foreach

Output: Average IV estimates
$\boldsymbol{\widehat{\beta}}_{IV}=\frac{1}{M}\sum_{i}\boldsymbol{\widehat{\beta}}_{IV}^{(i)}$.

Algorithm 2 EnsembleIV

The “IV Estimation” step amounts to any standard IV regression approach
established in the econometrics literature, e.g., two-stage least-squares
(2SLS, Greene,, 2003; Wooldridge,, 2002) for linear models and two-stage
residual-inclusion (2SRI, also referred to as the control function approach,
Terza et al.,, 2008; Angrist and Pischke,, 2008) for generalized linear models
(GLMs).222We note that IV estimation for GLMs suffers from some known issues
and is still an active area of research. For example, due to the non-
collapsibility of logistic regressions (e.g., Daniel et al.,, 2021; Schuster
et al.,, 2021), the 2SRI approach is not guaranteed to recover consistent
estimates (Wan et al.,, 2018). While addressing this issue is beyond the scope
of the current work, we advise caution in applying IV estimation under GLMs.
Notably, instead of relying on the IV estimates associated with a single weak
learner (and its instruments), we average over the IV estimates obtained from
each weak learner. Doing so can potentially produce more precise
estimations.333For example, in the case where different
$\boldsymbol{\widehat{\beta}}_{IV}^{(i)}$ are independent of each other,
averaging can shrink the standard errors by a factor of $\sqrt{M}$. Even if
different $\boldsymbol{\widehat{\beta}}_{IV}^{(i)}$ are correlated, some
variance shrinkage can still be expected.

Deriving standard error estimations for EnsembleIV is a challenging task
because the algorithm involves variable selection (in the IV selection step),
and there lacks a standard approach for post-selection inference (e.g., Berk
et al.,, 2013; Taylor and Tibshirani,, 2015; Dezeure et al.,, 2015).
Therefore, we rely on bootstrapping to approximate the standard errors. In
particular, we consider both a “full bootstrap” and a “split-sample
bootstrap”. Under “full bootstrap”, we draw from the entire
$\\{D_{train},D_{test},D_{unlabel}\\}$ with replacement, and carry out
Algorithm 2 on each bootstrapped sample. This represents an end-to-end
approach to approximate the uncertainty in EnsembleIV estimates. Under “split-
sample bootstrap”, we borrow the idea of Rinaldo et al., (2019) to use a fixed
$\\{D_{train},D_{test}\\}$ for ensemble model training, IV transformation, and
IV selection (i.e., steps related to model selection), then bootstrap only
$D_{unlabel}$ during the IV estimation (i.e., the inference step) to obtain
the standard errors. Expectedly, the bootstrapped standard errors under “full
bootstrap” are larger than those under “split-sample bootstrap”. To be
conservative, we report the former in subsequent empirical evaluations.

To summarize, EnsembleIV offers a systematic approach to generate, transform,
select, and use weak learners within an ensemble machine learning model as
instrumental variables to address the measurement error problem in second-
stage regressions. The idea of instrumenting one weak learner’s predictions
with those of other weaker learners has been explored in the prior literature
(ForestIV, Yang et al.,, 2022). However, EnsembleIV offers several notable
improvements over prior approaches, particularly the ForestIV approach of Yang
et al., (2022). First, the validity of ForestIV depends on specific
theoretical properties of one machine learning technique, the random forest
algorithm, and that validity has only been formally established for scenarios
involving continuous machine learning-generated variables. In contrast,
EnsembleIV is a general-purpose approach that can accommodate the use of any
machine learning ensemble technique (see Section 3.2 for a demonstration of
how EnsembleIV can be used with gradient boosting), and it is provably valid
for both binary and continuous machine learning generated variables. Second,
whereas ForestIV depends on the chance discovery of weak learners from a
trained random forest, which happen to yield predictions that meet the
exclusion criterion, EnsembleIV purposefully transforms all candidate
instruments to guarantee their compliance with the exclusion criterion. For
this reason, EnsembleIV is more broadly applicable. Third, whereas ForestIV
identifies the single “best” combination of weak learners that would yield
estimates exhibiting the smallest deviation from the unbiased estimates
obtained on $D_{label}$, EnsembleIV averages over all combinations.
Accordingly, in a given dataset, EnsembleIV will generally be more efficient
than ForestIV (see Section 3.3 for empirical demonstrations).

##  3 Empirical Evaluations

We carry out comprehensive empirical evaluations to understand the
effectiveness of EnsembleIV in mitigating estimation biases. In the first set
of evaluations (Section 3.1), we rely on synthetic data and simulations to
assess EnsembleIV’s correction performance for several widely-used regression
specifications. In the second set of evaluations (Section 3.4), we apply
EnsembleIV on a real-world dataset consisting of unstructured textual data to
demonstrate how the estimator can be used in combination with deep learning
techniques.

###  3.1 Simulation Studies with Synthetic Data

Recall the two-stage approach of combining machine learning with statistical
inference. In the first, prediction stage, a supervised learning model is
trained using the labeled data that can predict the variable of interest in
the unlabeled data. In the second, statistical inference stage, the predicted
values are incorporated into a regression model as an independent covariate.
In our simulation studies, we leverage public datasets to implement the
machine learning stage, and we synthesize data atop those real-world samples
to inform our assessments of the statistical inference stage.

We employ two publicly available datasets as the basis for the first-stage
prediction, namely the “Bike Sharing” dataset (Fanaee-T and Gama,, 2014) and
the “Bank Marketing” dataset (Moro et al.,, 2014). Both datasets are available
from the UCI Machine Learning Repository444Bike Sharing data:
https://archive.ics.uci.edu/ml/datasets/bike+sharing+dataset; Bank Marketing
data: https://archive.ics.uci.edu/ml/datasets/bank+marketing. and are commonly
used as benchmarking datasets in the machine learning literature. The Bike
Sharing dataset contains 17,379 bike rental records. From this sample, we use
date and timestamps in tandem with weather-related features to predict the
logarithm of hourly rental volumes (denoted as $lnCnt$). The Bank Marketing
dataset contains 45,211 phone calls conducted as part of a Portuguese bank’s
telemarketing campaign. From this sample, we use client features (e.g.,
demographics and financial status) to predict conversion, i.e., subscription
to a term deposit account with the bank (denoted as $Deposit$). Notably, the
target variable to be predicted is continuous in the Bike Sharing dataset and
binary in the Bank Marketing dataset, allowing us to evaluate EnsembleIV’s
performance under both variable types.

We partition each dataset into $D_{train}$, $D_{test}$, and $D_{unlabel}$. We
use $D_{train}$ to train a random forest model, $D_{test}$ to perform IV
transformation (based on Algorithm 1), and $D_{unlabel}$ to carry out IV
selection and subsequent statistical estimation and inference. The following
Table 2 summarizes the key aspects of each dataset used in the machine
learning stage.

Dataset | Target Variable | $|D_{train}|$ | $|D_{test}|$ | $|D_{unlabel}|$ | Number of Weak Learners ($M$)
---|---|---|---|---|---
Bike Sharing | $lnCnt$ | 1000 | 2000 | 14379 | 100
Bank Marketing | $Deposit$ | 1500 | 3000 | 40711 | 100
Table 2: Settings in the Machine Learning Stage

Using both datasets, we first provide descriptive evidence to demonstrate that
the IV transformation step in Algorithm 1 and IV selection step (using LASSO
selection for illustration) can indeed create (approximately) valid
instruments and select strong ones. In particular, for a given mismeasured
covariate $\widehat{X}^{(i)}$, denote
$\Omega_{before}^{(i)}=\\{\widehat{X}^{(j)}\\}_{j\neq i}$ as the set of
candidate IVs before any transformation or selection, and denote
$\Omega_{after}^{(i)}\subseteq\\{\widetilde{W}^{(j)}\\}_{j\neq i}$ as the set
of transformed IVs selected by the LASSO step. We then compute
$\frac{1}{|\Omega_{\bullet}^{(i)}|}\sum_{w\in\Omega_{\bullet}^{(i)}}|Corr(\widehat{X}^{(i)},w)|$
to measure the average relevance of selected IVs, and
$\frac{1}{|\Omega_{\bullet}^{(i)}|}\sum_{w\in\Omega_{\bullet}^{(i)}}|Corr(\widehat{X}^{(i)}-X,w)|$
to measure the average exclusion of selected IVs. In Figure 1, we plot the
distribution of the average relevance and exclusion measures $\forall
i\in\\{1,\ldots,100\\}$ and across all simulation runs, both before and after
the transformation and selection steps.

Figure 1: Average Relevance/Exclusion Measures Before and After IV
Transformation and Selection

On the Bike Sharing dataset, the average relevance / exclusion measure clearly
increases / decreases after the IV transformation and selection steps. On the
Bank Marketing dataset, the average relevance increase is also clear; the
average exclusion measure is already small before IV transformation, and its
distribution becomes even narrower after the transformation. Overall, this
lends support to the good performance of IV transformation and selection
steps. Meanwhile, one can see that the average exclusion measure is not
exactly 0 on either dataset, indicating that even after transformation, the
IVs are almost but not perfectly valid. This highlights the practical
importance of selecting strong IVs for estimation (Murray,, 2006).

Next, we simulate data atop these samples, for use in the statistical
inference stage. We consider two widely-used regression specifications: a
linear regression and a logistic regression. The data generation process of
each specification is included as follows:

$Y=1+0.5MLV+2Z_{1}+Z_{2}+\varepsilon$ (Linear Regression)
$\ln\frac{\Pr(Y=1)}{\Pr(Y=0)}=1+0.5MLV+2Z_{1}+Z_{2}$ (Logistic Regression)

where $MLV$ refers to the machine learning generated variable (i.e., the
target variable in the machine learning stage, $MLV\in\\{lnCnt,Deposit\\}$);
$Z_{1}\sim Uniform(-10,10)$ and $Z_{2}\sim N(0,10^{2})$ are independent
covariates (i.e., control variables); $\varepsilon\sim N(0,2^{2})$ represents
the independent error term in linear specifications; and $Y$ is the dependent
variable.

For each specification, we run three sets of regressions: (1) a “Biased”
regression, using the aggregated prediction values produced by the random
forest for $MLV$ on $D_{unlabel}$; (2) an “Unbiased” regression, using the
true values for $MLV$ on $D_{train}\cup D_{test}$, and (3) the “EnsembleIV”
regression, where the proposed EnsembleIV method is applied. During the IV
selection step, we apply three different approaches, namely (i) selecting top
3 IVs with strongest relevance correlation, (ii) selecting the three largest
principle components, and (iii) using LASSO. The Biased regression produces
estimates that one obtains when combining machine learning with statistical
inference without any bias correction effort. The Unbiased regression, in
contrast, produces the most precise estimates one can get on the labeled data.
We expect the EnsembleIV estimates to be less biased than the Biased estimates
(due to the use of IV for correction), and more precise than the Unbiased
estimates (due to its ability to leverage the much larger unlabeled data for
estimation).

To obtain the empirical distributions of coefficient estimates, we repeat each
simulation experiment, end-to-end, 100 times (including data partitioning,
random forest training, regression data synthesis, and statistical
estimation). We report the mean and standard deviation of each coefficient
estimate across 100 simulation runs as the point estimate and standard error,
respectively. Further, to quantify the degree of estimation bias, as well as
the effectiveness of correction, we report the Mean-Squared Error (MSE) of
each regression, defined as

$MSE(\boldsymbol{\widehat{\beta}})=\lVert\boldsymbol{\widehat{\beta}}-\boldsymbol{\beta}\rVert_{2}+\sum
Var(\boldsymbol{\widehat{\beta}})$

where
$\boldsymbol{\beta}=(\beta_{0},\beta_{MLV},\beta_{Z_{1}},\beta_{Z_{2}})=(1,0.5,2,1)$
denote the true coefficient values. The MSE is computed as the sum of bias and
variance over all coefficients estimated in the regression, with smaller MSE
intuitively corresponding to “better” estimates, i.e., estimates that are
statistically closer to the true coefficients. We report the simulation
results in Tables 3-4.

| True | Linear Regression | Logistic Regression
---|---|---|---
|  | Biased | Unbiased | Ens.IV | Ens.IV | Ens.IV | Biased | Unbiased | Ens.IV | Ens.IV | Ens.IV
|  |  |  | (Top-3) | (PCA) | (LASSO) |  |  | (Top-3) | (PCA) | (LASSO)
$\beta_{0}$ | 1.0 | 0.715 | 0.998 | 1.016 | 1.008 | 1.027 | 0.705 | 1.051 | 1.007 | 1.001 | 1.017
|  | (0.081) | (0.112) | (0.066) | (0.063) | (0.063) | (0.191) | (0.336) | (0.163) | (0.165) | (0.162)
$\beta_{MLV}$ | 0.5 | 0.564 | 0.500 | 0.497 | 0.499 | 0.495 | 0.556 | 0.495 | 0.489 | 0.492 | 0.489
|  | (0.017) | (0.024) | (0.014) | (0.014) | (0.013) | (0.046) | (0.073) | (0.039) | (0.039) | (0.039)
$\beta_{Z_{1}}$ | 2.0 | 2.001 | 2.000 | 2.001 | 2.001 | 2.001 | 1.978 | 2.022 | 1.978 | 1.982 | 1.983
|  | (0.003) | (0.005) | (0.003) | (0.003) | (0.003) | (0.059) | (0.133) | (0.059) | (0.059) | (0.059)
$\beta_{Z_{2}}$ | 1.0 | 1.000 | 1.001 | 1.000 | 1.000 | 1.000 | 0.989 | 1.01 | 0.989 | 0.991 | 0.991
|  | (0.002) | (0.004) | (0.002) | (0.002) | (0.002) | (0.030) | (0.067) | (0.030) | (0.030) | (0.030)
MSE |  | 0.092 | 0.013 | 0.005 | 0.004 | 0.005 | 0.134 | 0.144 | 0.033 | 0.034 | 0.033
Table 3: Simulation Results on Bike Sharing Data | True | Linear Regression | Logistic Regression
---|---|---|---
|  | Biased | Unbiased | Ens.IV | Ens.IV | Ens.IV | Biased | Unbiased | Ens.IV | Ens.IV | Ens.IV
|  |  |  | (Top-3) | (PCA) | (LASSO) |  |  | (Top-3) | (PCA) | (LASSO)
$\beta_{0}$ | 1.0 | 1.041 | 1.004 | 1.002 | 0.996 | 1.006 | 1.035 | 1.004 | 0.994 | 0.988 | 0.998
|  | (0.011) | (0.034) | (0.013) | (0.013) | (0.013) | (0.039) | (0.117) | (0.043) | (0.045) | (0.042)
$\beta_{MLV}$ | 0.5 | 0.274 | 0.497 | 0.447 | 0.494 | 0.416 | 0.286 | 0.534 | 0.468 | 0.521 | 0.442
|  | (0.044) | (0.093) | (0.056) | (0.058) | (0.050) | (0.145) | (0.301) | (0.189) | (0.203) | (0.177)
$\beta_{Z_{1}}$ | 2.0 | 2.000 | 2.000 | 2.000 | 2.000 | 2.000 | 1.996 | 2.011 | 1.996 | 1.997 | 1.997
|  | (0.002) | (0.005) | (0.002) | (0.002) | (0.002) | (0.040) | (0.105) | (0.040) | (0.040) | (0.040)
$\beta_{Z_{2}}$ | 1.0 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 0.997 | 1.004 | 0.998 | 0.998 | 0.998
|  | (0.001) | (0.003) | (0.001) | (0.001) | (0.001) | (0.019) | (0.054) | (0.019) | (0.019) | (0.019)
MSE |  | 0.055 | 0.010 | 0.006 | 0.004 | 0.010 | 0.072 | 0.120 | 0.041 | 0.046 | 0.038
Table 4: Simulation Results on Bank Marketing Data

We observe that EnsembleIV is consistently able to reduce estimation bias
(compared to the Biased estimates) and improve estimation precision (compared
to the Unbiased estimates). Across all specifications on both datasets,
EnsembleIV produces (1) point estimates that are closer to the true
coefficient values than the Biased estimates, and (2) standard errors that are
smaller than those of the Unbiased estimates. EnsembleIV’s advantages are also
reflected in the MSE metric – EnsembleIV results in the lowest MSE in all
case. Moreover, among the three IV selection approaches (namely Top-3, PCA,
and LASSO), the PCA-selected IVs result in point estimates of $\beta_{MLV}$
that are closest to the true coefficient value whereas the LASSO-selected IVs
result in the smallest standard errors. The above simulation studies clearly
demonstrate EnsembleIV’s utility to mitigate estimation biases caused by the
measurement error problem, while also leveraging the larger $D_{unlabel}$ to
improve estimation precision.

###  3.2 Apply EnsembleIV with Boosting

An important advantage of EnsembleIV is that it can be applied for ensemble
learning techniques other than the random forest. Owing to the IV
transformation step, valid IVs can be created even if the weak learners are
not intentionally built to have low error correlations. In this section, we
demonstrate how EnsembleIV can be used with gradient boosting and evaluate its
performance using the two synthetic datasets.

We choose gradient boosting here because it represents a starkly different
type of ensemble learning technique than random forest. Under gradient
boosting, a collection of $M$ weak learners are trained in a sequential
manner, which we denote with a tuple $(1,\ldots,M)$. Each learner $i>1$ aims
to predict the errors (i.e., residuals) of the earlier learners $1\leq j<i$.
To apply EnsembleIV with gradient boosting, we first use the original $M$ weak
learners to construct $M$ “cumulative” learners, where the $i$-th cumulative
learner is the ensemble of weak learners in $(1,\ldots,i)$. Put differently,
the $i$-th cumulative learner is the aggregation of weak learners up to $i$.
We use the predictions from these $M$ cumulative learners as the endogenous
variable and its candidate instruments. Intuitively, different cumulative
learners are all somewhat predictive of the target ground truth, thereby
supporting the relevance condition of IVs. However, unlike in a random forest,
there is little reason to believe (a priori) that the prediction errors of
cumulative learner $i$ are only weakly correlated with the predictions of
cumulative learner $j$, because there the two cumulative learners have a non-
trivial overlap in their constituent weak learners. The likely violation of
exclusion condition makes this a challenging and meaningful test of
EnsembleIV’s ability to create valid IVs from data.

We apply EnsembleIV with XGBoost (Chen and Guestrin,, 2016), a highly
successful gradient boosting algorithm, on both the Bike Sharing dataset and
the Bank Marketing dataset with a linear or logit second-stage regression.
Simulation setups are kept the same as described in Section 3.1. The results
are presented in Tables 5-6.

| True | Linear Regression | Logistic Regression
---|---|---|---
|  | Biased | Unbiased | Ens.IV | Ens.IV | Ens.IV | Biased | Unbiased | Ens.IV | Ens.IV | Ens.IV
|  |  |  | (Top-3) | (PCA) | (LASSO) |  |  | (Top-3) | (PCA) | (LASSO)
$\beta_{0}$ | 1.0 | 1.023 | 0.992 | 1.05 | 1.047 | 1.051 | 1.147 | 0.953 | 1.063 | 1.068 | 1.154
|  | (0.066) | (0.115) | (0.06) | (0.060) | (0.058) | (0.161) | (0.428) | (0.201) | (0.201) | (0.164)
$\beta_{MLV}$ | 0.5 | 0.497 | 0.501 | 0.502 | 0.502 | 0.501 | 0.468 | 0.519 | 0.497 | 0.495 | 0.478
|  | (0.014) | (0.023) | (0.013) | (0.014) | (0.013) | (0.035) | (0.100) | (0.046) | (0.045) | (0.036)
$\beta_{Z_{1}}$ | 2.0 | 2.000 | 2.000 | 2.000 | 2.000 | 2.000 | 1.991 | 2.019 | 2.002 | 1.999 | 1.992
|  | (0.003) | (0.007) | (0.003) | (0.003) | (0.003) | (0.056) | (0.118) | (0.061) | (0.060) | (0.055)
$\beta_{Z_{2}}$ | 1.0 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 0.993 | 1.009 | 1.000 | 0.998 | 0.994
|  | (0.002) | (0.003) | (0.002) | (0.002) | (0.002) | (0.029) | (0.055) | (0.031) | (0.030) | (0.028)
MSE |  | 0.005 | 0.014 | 0.006 | 0.006 | 0.006 | 0.054 | 0.213 | 0.051 | 0.052 | 0.056
Table 5: Simulation Results on Bike Sharing Data with XGBoost | True | Linear Regression | Logistic Regression
---|---|---|---
|  | Biased | Unbiased | Ens.IV | Ens.IV | Ens.IV | Biased | Unbiased | Ens.IV | Ens.IV | Ens.IV
|  |  |  | (Top-3) | (PCA) | (LASSO) |  |  | (Top-3) | (PCA) | (LASSO)
$\beta_{0}$ | 1.0 | 1.038 | 1.004 | 1.023 | 1.020 | 1.027 | 1.036 | 1.016 | 1.022 | 1.019 | 1.026
|  | (0.009) | (0.032) | (0.010) | (0.011) | (0.010) | (0.037) | (0.106) | (0.039) | (0.038) | (0.037)
$\beta_{MLV}$ | 0.5 | 0.242 | 0.488 | 0.443 | 0.483 | 0.387 | 0.266 | 0.512 | 0.459 | 0.492 | 0.403
|  | (0.039) | (0.094) | (0.073) | (0.084) | (0.063) | (0.127) | (0.273) | (0.236) | (0.239) | (0.196)
$\beta_{Z_{1}}$ | 2.0 | 2.000 | 2.001 | 2.000 | 2.000 | 2.000 | 1.993 | 2.033 | 1.993 | 1.993 | 1.993
|  | (0.002) | (0.006) | (0.002) | (0.002) | (0.002) | (0.037) | (0.114) | (0.037) | (0.037) | (0.037)
$\beta_{Z_{2}}$ | 1.0 | 1.000 | 1.001 | 1.000 | 1.000 | 1.000 | 0.996 | 1.019 | 0.997 | 0.997 | 0.997
|  | (0.001) | (0.003) | (0.001) | (0.001) | (0.001) | (0.018) | (0.059) | (0.018) | (0.018) | (0.018)
MSE |  | 0.070 | 0.010 | 0.009 | 0.008 | 0.018 | 0.075 | 0.104 | 0.061 | 0.061 | 0.052
Table 6: Simulation Results on Bank Marketing Data with XGBoost

On the Bike Sharing dataset, the XGBoost model has achieved very high
predictive performance: given that the target ground truth ($lnCnt$) has mean
value 4.57 and standard deviation of 1.47, the prediction MSE on $D_{test}$ is
only 0.30. Accordingly, the relatively small measurement error in the Biased
regression does not result in substantial estimation biases (especially under
the linear regression specification). In this case, we observe that EnsembleIV
estimates are still very close to the true coefficient values and have smaller
standard errors than the Unbiased estimates. In other words, EnsembleIV does
not inadvertently create additional biases when measurement error is small to
begin with. On the Bank Marketing dataset, directly using the XGBoost model’s
predictions in the Biased regression has resulted in over 50% underestimation
of $\beta_{MLV}$. Importantly, EnsembleIV is again able to mitigate the
estimation biases and maintain a smaller standard errors than the Unbiased
estimates. We further note that, on the Bank Marketing data, EnsembleIV with
LASSO-selected IVs leads to point estimates on $\beta_{MLV}$ that are rather
far from the true coefficient value. Fortunately, in those cases, EnsembleIV
with PCA-based IVs produces much closer point estimates (despite small
increases in standard errors).555We find that the unsatisfactory point
estimates associated with LASSO-selected IVs are caused by relatively large
discrepancies between the $\lambda$ values estimated from $D_{test}$ and the
ideal values based on $D_{unlabel}$. As a result, the transformed IVs are not
as excluded, which violates the assumption behind the LASSO selection
procedure (Belloni et al.,, 2012). Overall, this set of evaluations clearly
demonstrate EnsembleIV’s ability to be used in combination with the gradient
boosting technique.

###  3.3 Comparisons with ForestIV

Using the same simulation setup as in the previous Section, we compare the
correction performance between EnsembleIV and ForestIV (see Yang et al.,,
2022, for implementation details of the ForestIV algorithm). On both the Bike
Sharing and Bank Marketing datasets, we separately apply EnsembleIV and
ForestIV under linear second-stage regressions, and report the mean and
standard error of each coefficient estimate across 100 simulation runs. The
results are included in Table 7

| True | Bike Sharing Data | Bank Marketing Data
---|---|---|---
|  | Biased | Unbiased | Ens.IV | ForestIV | Biased | Unbiased | Ens.IV | ForestIV
$\beta_{0}$ | 1.0 | 0.715 | 0.998 | 1.028 | 0.969 | 1.041 | 1.004 | 1.005 | 0.995
|  | (0.081) | (0.112) | (0.063) | (0.084) | (0.011) | (0.034) | (0.012) | (0.013)
$\beta_{MLV}$ | 0.5 | 0.564 | 0.500 | 0.495 | 0.509 | 0.274 | 0.497 | 0.419 | 0.502
|  | (0.017) | (0.024) | (0.013) | (0.017) | (0.044) | (0.093) | (0.048) | (0.069)
$\beta_{Z_{1}}$ | 2.0 | 2.001 | 2.000 | 2.001 | 2.001 | 2.000 | 2.000 | 2.000 | 2.000
|  | (0.003) | (0.005) | (0.003) | (0.003) | (0.002) | (0.005) | (0.002) | (0.002)
$\beta_{Z_{2}}$ | 1.0 | 1.000 | 1.001 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000
|  | (0.002) | (0.004) | (0.002) | (0.002) | (0.001) | (0.003) | (0.001) | (0.001)
MSE |  | 0.092 | 0.013 | 0.005 | 0.008 | 0.055 | 0.0099 | 0.0098 | 0.005
Table 7: Comparison with ForestIV

We first note that EnsembleIV estimates have smaller standard errors than
ForestIV estimates in both datasets. This offers empirical support to
EnsembleIV’s advantage in estimation efficiency. Point estimates from
EnsembleIV are not necessarily always better than those from ForestIV – on the
Bank Marketing data, ForestIV produces a point estimate of $\beta_{MLV}$ that
is closer to the true coefficient value. In other words, EnsembleIV does not
fully dominate ForestIV. Given a finite sample, EnsembleIV is generally more
efficient than ForestIV, but the correction performance in point estimates can
be data dependent.

###  3.4 Field Evaluation 1: Engagement with User-Generated Content on
Facebook

We now conduct an “in-vivo” evaluation of EnsembleIV using a real-world
dataset collected from Facebook businesses pages. Business pages on Facebook
are public spaces managed by firms to share marketing content (e.g., product
promotions), who seek to interact with Facebook users (e.g., answering their
questions or complaints). A number of prior studies have examined firm- and
user-generated content in this context, and their impact on user engagement or
firm performance (e.g., Goh et al.,, 2013; Lee et al.,, 2018; Yang et al.,,
2019). For this evaluation, we estimate how the sentiment (positive or
negative) of a user-generated post affects content engagement, measured as the
number of comments that the post receives from other users. This evaluation
serves two purposes: (1) it represents a realistic empirical context, where
neither the data-generation process nor the true coefficient values are known,
and (2) it demonstrates how EnsembleIV can be used in combination with modern
deep learning techniques for language tasks.

We use the dataset collected by Yang et al., (2019), which includes a total of
10,403 user-generated posts, created in 2012 on the business pages of 41
Fortune-500 companies. The sentiment of each post has been manually labeled by
5 independent Amazon Mechanical Turk workers, and the ground truth sentiment
label (positive or negative) is determined via majority voting. We partition
the data into 10% $D_{train}$ (1041 instances), 10% $D_{test}$ (1041
instances), and 80% $D_{unlabel}$ (8321 instances).

For the first-stage machine learning task, we train a binary classifier on
$D_{train}$ to predict the sentiment of each post. We consider two different
machine learning techniques when building the sentiment classifier. In the
first approach, which we refer to as “BoW + RF”, we adopt a traditional bag-
of-words representation for posts, namely a term-frequency inverse document-
frequency (TF-IDF) matrix. We train a random forest classifier comprised of
100 trees, taking the TF-IDF matrix as input. In the second approach, which we
refer to as “BERT + RF”, we encode each post as a 768-dimension embedding
vector using the BERT pre-trained model (Devlin et al.,, 2018). As before, we
train a random forest classifier comprised of 100 trees, taking post
embeddings as input features. When evaluated on $D_{test}$, the “BERT + RF”
approach outperforms the “BoW + RF” approach. We summarize the accuracy and
F-scores of each model in Table 8.

| BoW + RF | BERT + RF
---|---|---
Accuracy | 0.756 | 0.794
$F_{positive}$ | 0.244 | 0.502
$F_{negative}$ | 0.854 | 0.871
Table 8: Predictive Performance of Sentiment Classifiers

For the second-stage inference task, we estimate a linear regression on
$D_{unlabel}$ with the number of comments received by each post serving as the
dependent variable. The sentiment of each post, denoted as $sentiment$, takes
a value of 1 if the predicted sentiment is positive and 0 otherwise. This
variable serves as the mismeasured independent variable. In addition, we
control for several other covariates, including (1) the number of words in a
post ($wordcount$), (2) the content type of the post ($type$, one of photo,
status, video, or link), (3) activeness of the post author ($activity$),
reflecting the total number of posts created by the author on the business
page in 2012, and (4) the popularity of the business page ($pagepost$),
reflecting the total number of user- and firm-generated posts made to the page
in 2012. In Table 9, we report estimation results from the Biased regression,
Unbiased regression, and EnsembleIV, with bootstrapped standard errors. We use
the PCA-based instruments in EnsembleIV estimation, due to the superior
correction performance observed in the previous simulation studies. Note that
the unbiased estimates do not rely on machine learning predictions, and
therefore remain the same across results associated with each of the three
sentiment classifiers.

|  | BoW + RF | BERT + RF
---|---|---|---
| Unbiased | Biased | Ens.IV | Biased | Ens.IV
sentiment |  $-$0.620∗∗∗ |  $-$0.253 |  $-$0.346∗ |  $-$0.364∗∗∗ |  $-$0.437∗∗∗
| (0.120) | (0.147) | (0.141) | (0.081) | (0.132)
wordcount | 0.513∗∗∗ | 0.662∗∗∗ | 0.653∗∗∗ | 0.629∗∗∗ | 0.632∗∗∗
| (0.089) | (0.040) | (0.040) | (0.040) | (0.042)
activity | 0.017∗ | 0.009∗ | 0.009∗ | 0.009∗ | 0.009∗
| (0.007) | (0.004) | (0.004) | (0.004) | (0.004)
pagepost | 0.206∗∗ | 0.215∗∗∗ | 0.216∗∗∗ | 0.217∗∗∗ | 0.217∗∗∗
| (0.063) | (0.043) | (0.043) | (0.043) | (0.043)
type_photo | 0.816∗∗ | 0.631∗∗∗ | 0.634∗∗∗ | 0.689∗∗∗ | 0.666∗∗∗
| (0.316) | (0.135) | (0.135) | (0.139) | (0.136)
type_status | 1.056∗∗∗ | 1.277∗∗∗ | 1.270∗∗∗ | 1.280∗∗∗ | 1.261∗∗∗
| (0.206) | (0.070) | (0.070) | (0.072) | (0.073)
type_video | 0.399 |  $-$0.277 |  $-$0.273 |  $-$0.235 |  $-$0.251
| (0.374) | (0.214) | (0.213) | (0.222) | (0.215)
Constant |  $-$2.766∗∗∗ |  $-$3.657∗∗∗ |  $-$3.583∗∗∗ |  $-$3.524∗∗∗ |  $-$3.440∗∗∗
| (0.765) | (0.441) | (0.440) | (0.440) | (0.453)
Table 9: Evaluation on Facebook Data (${}^{***}\leavevmode\nobreak\ p<0.001$,
${}^{**}\leavevmode\nobreak\ p<0.01$, ${}^{*}\leavevmode\nobreak\ p<0.05$).
Bootstrapped standard errors of EnsembleIV estimates are reported.

Several findings are worth noting. First, directly using the aggregated
predictions from random forest in the second-stage regressions results in
nontrivial bias. The magnitude of the coefficient on $sentiment$ is
underestimated, and, under the “BoW + RF” classifier, such bias leads to a
Type II inferential error (i.e., we would fail to reject the null). Second,
EnsembleIV is able to mitigate the estimation bias on the sentiment variable
to some extent in all cases. The point estimate associated with $sentiment$
under the “BERT + RF” classifier is closest to that in the unbiased
regression. Third, EnsembleIV’s correction performance seems to align with the
predictive performance of the sentiment classifier. In particular, the “BERT +
RF” classifier achieves better predictive performance than the “BoW + RF”
classifier, and is also associated with more effective correction. Therefore,
EnsembleIV should not be relied upon as a substitute for predictive accuracy
in first-stage machine learning models. In practice, having a better
predictive model tends to also improve the performance of EnsembleIV.

##  4 Conclusion

Despite increasing popularity in empirical studies, the integration of machine
learning generated variables into regression models for statistical inference
suffers from the measurement error problem, which can bias estimation and
threaten the validity of inferences. In this paper, we develop a novel
approach to alleviate associated estimation biases. Our proposed approach,
EnsembleIV, creates valid and strong instrumental variables from weak learners
in an ensemble model, and uses them to obtain consistent estimates that are
robust against the measurement error problem. Our empirical evaluations, using
both synthetic and real-world datasets, show that EnsembleIV can effectively
reduce estimation biases across several common regression specifications, and
can provide an efficiency advantage over other existing methods in the
literature, namely ForestIV (Yang et al.,, 2022).

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# Decoherence of Dirac-particle quantumness for fermionic fields in a
dilatonic black hole

Chengjun Yao  Yating Shao  Kai Yan  Yinzhong Wu  School of Physical Science
and Technology, Suzhou University of Science and Technology, Suzhou,Jiangsu
215009, People’s Republic of China  Xiang Hao  Corresponding author
<EMAIL_ADDRESS>School of Physical Science and Technology, Suzhou
University of Science and Technology, Suzhou,Jiangsu 215009, People’s Republic
of China  Pacific Institute of Theoretical Physics, Department of Physics and
Astronomy,
University of British Columbia, 6224 Agriculture Rd., Vancouver B.C., Canada
V6T 1Z1.

###### Abstract

The quantumness of Dirac paticles for quantized fields in a dilatonic black
hole is estimated by means of quantum channel. We develop a general Bloch
vector representation of quantum channel in black hole spacetimes beyond
single mode approximation. The nonclassicality of Dirac particles can be
measured by the minimization of quantum coherence over all orthonormal basis
sets. The quantumness of the channel decreases as the dilaton parameter
increases. The interplay between the external reservoir noise and dilaton
black hole on the dynamical behavior of quantum coherence and steerability is
investigated in the Pauli basis. The external environment is modelled by a
random telegraph noise channel. The monotonous decay of quantum nonlocality
occurs in the weak coupling case. The degradation and revival of quantum
nonlocality are observed in the strong coupling condition. It is found that
quantum fluctuation effects of the external reservoir can protect quantum
coherence and steerability from the information loss of the black hole.

Keywords: dilatonic black hole, quantum channel, quantum coherence, quantum
steering

PACS: 03.67.Mn, 03.65.Yz, 04.62.+v, 04.70.Dy

##  I Introduction

The observation of gravitational waves and confirmation of black holes
Klaer2017  have received much attention. The quantum property of field modes
near the event horizon plays a prominent role in the study of black hole
thermodynamics Huang2021 . The relativistic effect on quantum dynamics in
accelation frames Giacomini2019  and vacuum fluctuation in black hole
spacetimes Wang2016  have been studied in the field of relativistic quantum
information Peres2004 . As is well known, Unruh acceleration effects Unruh1976
; Banerjee2017  and gravitational fields Hawking1975 ; He2016  lead to
information loss of particles or antiparticles in a relativistic framework.
Recent studies have demonstrated that the geometric inflation and shadows have
impacts on the nonclassicality of quasibound states of black holes
Edelstein2020 . From the viewpoint of quantum nonlocality Nielsen2011 ,
quantumness of scalar and Dirac fields in the black holes can be quantified by
quantum entanglement Martin2012  and quantum correlationWen2020 . The
relativistic metrology based on quantum Fisher information was also applied to
the analysis of vacuum structures in curved spacetimes Huang2019 ; Jin2020 .
The class of dilatonic black holes has become an interesting focus in
astrophysical black holes Anabalon2013 ; Amarilla2013 ; Badia2020 ; Kumar2020
. The superradiant instability and scattering properties of dilatonic black
holes are studied in the references of Bernard2017 ; Huang2020 . In the low
energy limit of string theory, the static spherical charged dilatonic black
hole solution was provided by Gibbons and Maeda Gibbons1988 , independently by
Garfinkle, Horowitz, and Strominger Garfinkle1991 ($\mathrm{GMGHS}$). The
study of dilatonic black holes contributes to a deeper understanding of
quantum gravity Howl2021 . It is of a great interest to explore quantumness of
dilatonic black holes in the combination of quantum information and general
relativity.

On the other hand, quantum fluctuations in some curved spacetimes can be
investigated in the context of open quantum system approach Yu2011 . As one
feasible method, quantum channel can be used to characterize dynamics of
quantum systems interacting with their reservoirs Scully1997 ; Barnett1997 .
The random telegraph noise channel Daffer2004 ; Benedetti2013 , for instance,
can describe quantum dissipation from a structured reservoir with a
fluctuation rate. Different types of environments bring out diverse
decoherence. In the background of black holes, we expect to find a unified
quantum channel which discovers the connection between gravitation and
decoherence. The quantum channel can be represented as the evolution of Bloch
states for quantized fields in relativistic spacetimes. The quantum resource
theory Baumgratz2014  provides an efficient way to measure quantumness of
channels Shahbeigi2018 . In fact, no feasible localized detector models can be
applied to the measurement of quantum effects of fields inside the event
horizon of a black hole. Therefore, it is advantageous to probe quantum
nonlocality near the outside region of the event horizon of black holes. We
assume that the two observers can detect local field modes. After both of them
share a quantum correlated initial state, one stays stationary at an
asymptotically flat region and experiences quantum dissipation from an
external reservoir, while the other moves with uniform acceleration and hovers
near the event horizon of the dilatonic black hole. In the frame of quantum
channel, quantum coherence and quantum steering Jevtic2014 ; Costa2016  are
used to estimate the dynamical behavior of open Dirac particles.

The paper is organized as follows. In the next section, the quantized fields
of Dirac particles and antiparticles are discussed beyond single mode
approximation. We suggest the generalized representation of quantum channel in
the form of Bloch vectors. The available estimation of quantumness of the
channel is obtained by means of quantum coherence. The relativistic effects
from the dilatonic parameter and Dirac mode approximation are considered. In
Sec. III, the interplay between the external environment and dilatonic black
hole spacetime is analyzed in terms of quantum coherence and quantum steering.
We put forward a unified method of quantum decoherence in the Pauli basis. We
will discuss our conclusion in the last section.

##  II Quantum channel in the dilatonic black hole

In accordance with string theory, the dilatonic black holes are formed by
gravitational systems coupled to Maxwell and dilaton fields. The action of the
four-dimensional dilaton theory is written as Gibbons1988 ; Garfinkle1991

$\mathcal{S}=\int\mathrm{d}^{4}x\sqrt{-g}[R-2(\nabla\phi)^{2}-e^{-2a\phi}F_{\mu\nu}F^{\mu\nu}],$
(1)

where $\phi$ represents the dilaton and
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
$\mathrm{Maxwell}$ tensor. The coupling parameter between the dilaton and
$\mathrm{Maxwell}$ field is given by $a=1$ which corresponds to the
$\mathrm{GMGHS}$ black hole. The line element of this spherical dilatonic
black hole is given by

$\mathrm{d}s^{2}=-\big{(}\frac{r-2M}{r-2D}\big{)}\mathrm{d}t^{2}+\big{(}\frac{r-2M}{r-2D}\big{)}^{-1}\mathrm{d}r^{2}+r(r-2D)(\mathrm{d}\theta^{2}+\sin^{2}\theta\;\mathrm{d}\phi^{2}).$
(2)

The natural units $\hbar=G=c=\kappa_{B}=1$ are used. The parameters $M$ and
$D$ are the mass of the black hole and dilaton respectively. For a charged
dilatonic black hole, $D=\frac{Q^{2}}{2M}$ where $Q$ denotes the charge. The
term $e^{-2\phi}=e^{-2\phi_{0}}(1-\frac{2D}{r})$. When the dilaton value at
spacelike infinity $\phi_{0}=0$, the case implies an asymptotic flat manifold.
The event horizon of $\mathrm{GMGHS}$ black hole is located at $r_{+}=2M$. The
area of the sphere goes to zero when $r=r_{-}=2D$ and the surface is singular.
The quantized Dirac fields in the dilatonic black hole satisfy the Dirac
equation which is given by
$\gamma^{\alpha}e_{a}^{\mu}(\partial_{\mu}+\Gamma_{\mu})\Psi=0$ Wang2010 .
$\gamma^{\alpha}$ is the Dirac matrix, $e_{a}^{\mu}$ corresponds to the
inverse of the tetrad $e^{\mu}_{a}$ and $\Gamma_{\mu}$ denotes the spin
connection coefficient. By solving the Dirac equation, we can obtain the
positive frequency outgoing solutions outside region and inside region of the
event horizon, $\Psi_{\iota,\bm{k}}^{+}=\Re e^{\mp i\omega\mathcal{U}}$ where
$\iota=(\mathrm{out},\mathrm{in})$ represents the two regions, $\bm{k}$
denotes a variable that labels the field mode, $\Re$ is a $4$-component Dirac
spinor and $\omega$ is a monochromatic frequency of the Dirac field. The
parameter $\mathcal{U}=t-2(M-D)\ln[(r-2M)/(2M-2D)]$ is the retarded time.

In terms of the complete orthogonal basis $\psi_{\iota,\bm{k}}^{\pm}$, the
Dirac field can be expanded as

$\Psi=\sum_{\iota=(\mathrm{out},\mathrm{in})}\int\mathrm{d}\bm{k}(\hat{a}_{\bm{k}}^{\iota}\Psi_{\iota,\bm{k}}^{+}+\hat{b}_{\bm{k}}^{\iota{\dagger}}\Psi_{\iota,\bm{k}}^{-}),$
(3)

where $\hat{a}_{\bm{k}}^{\iota}$ and $\hat{b}_{\bm{k}}^{\iota{\dagger}}$
correspond to the fermion annihilation and antifermion creation operators
acting on the state of the $\iota$ region. Making the analytic continuations
for $\Psi_{\iota,\bm{k}}^{+}$, we can quantize Dirac fields as the combination
of Kruskal modes according to the suggestion of Domour-Ruffini Damoar1976 .
The quantization of Dirac fields beyond single mode approximation JWang2010
can be taken into account. Using the Bogoliubov transformations between the
creation and annihilation operators in two kinds of coordinates, the operators
for mode $\bm{k}$ are expressed as Bruschi2010 ; Friis2011

$\displaystyle\hat{d}_{R,\bm{k}}$ $\displaystyle\;=\;\cos
r\hat{a}_{\bm{k}}^{\mathrm{out}}-\sin
r\hat{b}_{-\bm{k}}^{\mathrm{in}{\dagger}}$ $\displaystyle\hat{d}_{L,\bm{k}}$
$\displaystyle\;=\;\cos r\hat{a}_{\bm{k}}^{\mathrm{in}}-\sin
r\hat{b}_{-\bm{k}}^{\mathrm{out}{\dagger}},$ (4)

where $\cos r=c=[e^{-8\pi\omega(M-D)}+1]^{-\frac{1}{2}}$ and $\sin
r=s=[e^{8\pi\omega(M-D)}+1]^{-\frac{1}{2}}$. $\hat{d}_{R(L),\bm{k}}$ denotes
the annihilation operator for the right ($\mathrm{R}$) or left ($\mathrm{L}$)
mode. Assuming that $|0_{\bm{k}}\rangle_{R(L)}$ is annihilated by the operator
$\hat{d}_{R(L),\bm{k}}$, the Kruskal vacuum for mode $\bm{k}$ is defined as
$|0\rangle_{K}=|0_{\bm{k}}\rangle_{R}|0_{\bm{k}}\rangle_{L}=c^{2}|0000\rangle-
sc|0011\rangle-sc|1100\rangle-s^{2}|1111\rangle$. Here,
$|mn\tilde{m}\tilde{n}\rangle=|m_{\bm{k}}\rangle^{+}_{\mathrm{out}}|n_{-\bm{k}}\rangle^{-}_{\mathrm{in}}|\tilde{m}_{\bm{-}k}\rangle^{-}_{\mathrm{out}}|\tilde{n}_{\bm{k}}\rangle^{+}_{\mathrm{in}}$
and the superscripts $\\{+,-\\}$ represent the particle and antiparticle. When
the observer is located outside the event horizon, the Hawking radiation
spectrum with a thermal Fermi-Dirac distribution can be obtained by $\langle
0|\hat{a}_{\bm{k}}^{\mathrm{out}{\dagger}}\hat{a}_{\bm{k}}^{\mathrm{out}}|0\rangle=\frac{1}{e^{\omega/T}+1}$
where $T=\frac{1}{8\pi(M-D)}$ is the Hawking temperature Hawking1975 . Because
of the Pauli exclusion principle, only the first excited state for particle
mode is obtained by

$\displaystyle|1\rangle_{K}^{+}$
$\displaystyle=\;[q_{R}(\hat{d}_{R,\bm{k}}^{{\dagger}}\otimes
I_{L})+q_{L}(I_{R}\otimes\hat{d}_{L,\bm{k}}^{{\dagger}})]|0\rangle_{K}$
$\displaystyle=\;q_{R}(c|1000\rangle-s|1011\rangle)+q_{L}(s|1101\rangle+c|0001\rangle),$
(5)

with $|q_{R}|^{2}+|q_{L}|^{2}=1$. The physical interpretation on the operators
of Eq. (4) and states in Eq. (5) beyond single mode approximation was
presented Wang2014 . The creation operator $\hat{d}_{R,\bm{k}}^{{\dagger}}$
implies the creation of a fermion in the exterior vacuum and an antifermion in
the interior vacuum of the black hole. The spontaneous creation of particles
and antiparticles results in Hawking radiation with the total probability
$|q_{R}|^{2}+|q_{L}|^{2}=1$. When the parameter $|q_{R}|=1$, all the fermions
propagate toward the outside of event horizon and all the antifermions move to
the interior region. Similarly, the case of $|q_{L}|=1$ represents that all
the antifermions radiate toward the exterior region.

Since the exterior region is causally disconnected from the interior region in
the dilatonic black hole, we can obtain the physical accessible outside states
by tracing over the states of the interior region. The whole process of
information loss can be characterized by a quantum channel. In the open
quantum system approach, the dynamical map for the system of interest can be
obtained by tracing over all the degrees of freedom of the other environment.
With respect to the black hole spacetime, the outside states of all particles
or antiparticles can be considered as the open system of interest. Thus, the
quantum channel for Dirac particles in the outside region of the event horizon
can be given by

$\displaystyle\bm{\varepsilon}^{+}_{\mathrm{out}}(|0\rangle_{K}\langle 0|)$
$\displaystyle\;=\;c^{2}|0\rangle^{+}_{\mathrm{out}}\langle
0|+s^{2}|1\rangle^{+}_{\mathrm{out}}\langle 1|$
$\displaystyle\bm{\varepsilon}^{+}_{\mathrm{out}}(|0\rangle_{K}\langle 1|)$
$\displaystyle\;=\;q_{R}^{\ast}\cdot c|0\rangle^{+}_{\mathrm{out}}\langle 1|$
$\displaystyle\bm{\varepsilon}^{+}_{\mathrm{out}}(|1\rangle_{K}\langle 0|)$
$\displaystyle\;=\;q_{R}\cdot c|1\rangle^{+}_{\mathrm{out}}\langle 0|$
$\displaystyle\bm{\varepsilon}^{+}_{\mathrm{out}}(|1\rangle_{K}\langle 1|)$
$\displaystyle\;=\;|q_{L}|^{2}c^{2}|0\rangle^{+}_{\mathrm{out}}\langle
0|+(|q_{R}|^{2}+|q_{L}|^{2}s^{2})|1\rangle^{+}_{\mathrm{out}}\langle 1|,$ (6)

where the partial trace over all the states of the interior region and the
antiparticle states of the outside is carried out. For single particle state,
the density matrix of the state $\rho$ can be expressed by the Bloch vector
$\bm{\upsilon}$, i.e.,
$\rho=\frac{1}{2}(\mathbb{I}+\sum_{\mu=1}^{3}\upsilon_{\mu}\sigma_{\mu})$
where $\upsilon_{\mu}=\mathrm{Tr}(\rho\sigma_{\mu}),\quad(\mu=1,2,3)$ and
$\sigma_{\mu}(\mu=1,2,3)$ denote the three Pauli operators. In the pauli basis
of $\\{\sigma_{0}=\mathbb{I},\sigma_{\mu}(\mu=1,2,3)\\}$, the dynamical
process from an arbitrary initial particle vector
$\bm{X}=(1,\bm{\upsilon}^{(0)})$ to the evolved one $\bm{Y}=(1,\bm{\upsilon})$
is written as

$\bm{Y}=\bm{\hat{}}{\Phi}^{+}_{\mathrm{out}}\cdot\bm{X}=\Big{(}\begin{matrix}1&\bm{0}^{\mathrm{T}}\\\
\bm{\lambda}^{+}_{\mathrm{out}}&\bm{\Gamma}^{+}_{\mathrm{out}}\end{matrix}\Big{)}\Big{(}\begin{matrix}1\\\
\bm{\upsilon}^{(0)}\end{matrix}\Big{)},$ (7)

where $\bm{0}^{\mathrm{T}}=(0,0,0)$ and the superscript $\mathrm{T}$ denotes
the transpose. The quantum channel
$\bm{\varepsilon}^{+}_{\mathrm{out}}(\cdot)$ is equivalent to the action
operator $\bm{\hat{}}{\Phi}$ which is composed of the mapping vector and
mapping matrix. Here,
$\lambda_{\mu}(\mu=1,2,3)=\mathrm{Tr}[\frac{\sigma_{\mu}}{2}\cdot\bm{\varepsilon}^{+}_{\mathrm{out}}(\mathbb{I})]$.
The elements of the matrix $\bm{\Gamma}$ are obtained by
$\Gamma_{\mu,\nu}(\mu,\nu=1,2,3)=\mathrm{Tr}[\frac{\sigma_{\mu}}{2}\cdot\bm{\varepsilon}^{+}_{\mathrm{out}}(\sigma_{\nu})]$.
By using Eq. (6), we obtain the channel matrix and vector as

$\bm{\Gamma}^{+}_{\mathrm{out}}=\left(\begin{matrix}\mathrm{Re}(q_{R})\cdot
c&-\mathrm{Im}(q_{R})\cdot c&0\\\ \mathrm{Im}(q_{R})\cdot
c&\mathrm{Re}(q_{R})\cdot c&0\\\ 0&0&|q_{R}|^{2}\cdot
c^{2}\end{matrix}\right),\quad\bm{\lambda}^{+}_{\mathrm{out}}=\left(\begin{matrix}0\\\
0\\\ (1-|q_{R}|^{2})c^{2}-s^{2}\end{matrix}\right).$ (8)

We consider a real parameter, i.e., $\mathrm{Re}(q_{R})=q_{R}\in[0,1]$ and
$\mathrm{Im}(q_{R})=0$ in this case. The evolved particle state is expressed
as
$\bm{\upsilon}=\bm{\Gamma}^{+}_{\mathrm{out}}\cdot\bm{\upsilon}^{(0)}+\bm{\lambda}^{+}_{\mathrm{out}}$
in the form of Bloch vectors. It is known that a geometric picture of an
arbitrary pure state is a symmetric sphere which corresponds to a Bloch vector
$\bm{\upsilon}^{(0)}=(\cos\phi\sin\theta,\sin\phi\sin\theta,\cos\theta)^{\mathrm{T}},\;\theta\in[0,\pi),\phi\in[0,2\pi)$
without information loss. If the quantum channel of Eq. (8) acts on a Dirac
qubit, the sphere deformation is depicted in Figure 1(a). Under the condition
of the small values of $q_{R}$, the Bloch sphere for Dirac particle states
shrinks drastically. The geometrical contraction can verify the fact about the
loss of quantum information induced by quantum fluctuations near the event
horizon of the black hole.

Moreover, we try to quantify the quantumness of the channel which describes
the thermal radiation of particles near the event horizon of the dilatonic
black hole. The squared $l_{1}$ norm can be used to measure quantum coherence
of a particle state. The quantity of quantum coherence is given by
$C(\rho)=\sum_{i\neq j}|\varrho_{ij}|$ where $\varrho_{ij}$ is the element of
the state density matrix. When a particle state is subjected to the action
from quantum channel, the decay of quantum coherence depends both on the
channel and incoherent basis. The nonunitary evolution of states addresses the
question of how much quantumness a channel can preserve. By averaging on
quantum coherence of all states and minimizing over all orthonormal basis sets
Shahbeigi2018 , the measure of nonclassicality of the quantum channel
$\bm{\varepsilon}^{+}_{\mathrm{out}}$ can be obtained as

$Q(\bm{\varepsilon}^{+}_{\mathrm{out}})=\min_{\bm{n}}[\mathrm{Tr}(\mathcal{\bm{M}})-\bm{n}^{\mathrm{T}}\mathcal{\bm{M}}\bm{n}],$
(9)

where
$\mathcal{\bm{M}}=\frac{1}{2}(\bm{\Gamma}^{+}_{\mathrm{out}}\bm{\Gamma}^{+\mathrm{T}}_{\mathrm{out}}+5\bm{\lambda}^{+}_{\mathrm{out}}\bm{\lambda}^{+\mathrm{T}}_{\mathrm{out}})$
is determined by the mapping vector and matrix. The minimization is achieved
when $\bm{n}$ is chosen to be an eigenvector of the matrix $\mathcal{\bm{M}}$
with the largest eigenvalue. Therefore, the quantumness of the channel is
calculated by
$Q(\bm{\varepsilon}^{+}_{\mathrm{out}})=\mathrm{Tr}(\mathcal{\bm{M}})-\max(m_{1},m_{2},m_{3})$
where $m_{1},m_{2},m_{3}$ are the eigenvalues of the matrix. The channel
quantumness from quantum fluctuations near the event hozion is related to the
dilatonic parameter $D$ and the probability $q_{R}$ is illustrated in Figure
1(b). The increase of the dilationic parameter $D$ will diminish the
quantumness. When $q_{R}\rightarrow 1$, the quantity of the nonclassicality in
the black hole can keep a large value. If the dilatonic parameter $D>0.8$, the
quantumness declines rapidly. When an observer hovers near the event horizon
of the black hole, the detection of Dirac particle states from the outside
region can be applied to the measure of quantumness.

##  III Relativistic decoherence with external noise

The Dirac particles escaping from the outside region may experience external
noisy environments. When one observer detects states of open Dirac particles,
the effects from external noises need be considered. It is necessary to study
the impacts from external noises on quantum nonlocality in the background of
black hole spacetime. In a protocol, two observers can detect local field
modes of Dirac particles. Both of them share an initial state of two Dirac
particles. One observer stays stationary at an asymptotically flat region and
the local Dirac particle $A$ suffers from an external reservoir. While the
other detects the state of Dirac particle $B$ near the event horizon of the
black hole. In terms of quantum channels, the two Dirac particles experience
two independent channels of $\bm{\varepsilon}_{A}$ and $\bm{\varepsilon}_{B}$
respectively. In general, an arbitrary state for two Dirac particles $A$ and
$B$ can be written as
$\rho_{AB}^{(0)}=\frac{1}{4}\sum_{\mu,\nu=0}^{3}\Theta_{\mu\nu}^{(0)}(\sigma_{\mu,A}\otimes\sigma_{\nu,B})$.
Here, the matrix element
$\Theta_{\mu\nu}^{(0)}=\mathrm{Tr}(\rho_{AB}^{(0)}\sigma_{\mu,A}\otimes\sigma_{\nu,B})$
is obtained. In the pauli basis, we can obtain the equivalent form of the
initial state by using the matrix

$\displaystyle\bm{\Theta}^{(0)}=\left(\begin{matrix}1&\bm{\upsilon}_{B}^{(0)\mathrm{T}}\\\
\bm{\upsilon}_{A}^{(0)}&\bm{T}^{(0)}\end{matrix}\right),$ (10)

where $\bm{\upsilon}_{A(B)}^{(0)}$ denotes the initial Bloch vector of the
particle $A$ or $B$. The matrix $\bm{T}^{(0)}$ implies the correlation
information Horodecki1996 ; Luo2003 . Each quantum channel is determined by
the mapping vector $\bm{\lambda}$ and matrix $\bm{\Gamma}$. In the Bloch
vector representation, the transformation of quantum channel is given by
$\bm{\varepsilon}:\bm{\upsilon}=\bm{\Gamma}\cdot\bm{\upsilon}^{(0)}+\bm{\lambda}$.
In the context of open quantum systems, the evolved state is obtained by
$\rho_{AB}=\bm{\varepsilon}_{A}\otimes\bm{\varepsilon}_{B}(\rho_{AB}^{(0)})$.
We can expand $\rho_{AB}$ as

$\rho_{AB}=\frac{1}{4}\sum_{\mu,\nu=0}^{3}\Theta_{\mu\nu}^{(0)}[\bm{\varepsilon}_{A}(\sigma_{\mu,A})\otimes\bm{\varepsilon}_{B}(\sigma_{\nu,B})]=\frac{1}{4}\sum_{\mu,\nu=0}^{3}\Theta_{\mu\nu}(\sigma_{\mu,A}\otimes\sigma_{\nu,B}),$
(11)

where the part of
$\bm{\varepsilon}_{\alpha}(\sigma_{\mu,\alpha}),\;\alpha=A,B$ is given by

$\displaystyle\bm{\varepsilon}_{\alpha}(\mathbb{I}_{\alpha})$
$\displaystyle=\mathbb{I}_{\alpha}+\sum_{\mu=1}^{3}\lambda_{\mu,\alpha}\cdot\sigma_{i,\alpha}$
$\displaystyle\bm{\varepsilon}_{\alpha}(\sigma_{\nu,\alpha})$
$\displaystyle=\sum_{\mu=1}^{3}\Gamma_{\mu\nu,\alpha}\cdot\sigma_{\mu,\alpha}\quad\nu=1,2,3.$
(12)

The matrix $\bm{\Theta}$ for the evolved state is
$\bm{\Theta}=\Big{(}\begin{matrix}1&\bm{\upsilon}_{B}^{\mathrm{T}}\\\
\bm{\upsilon}_{A}&\bm{S}\end{matrix}\Big{)}$ and the vectors satisfy that
$\bm{\upsilon}_{\alpha}=\bm{\Gamma}_{\alpha}\cdot\bm{\upsilon}_{\alpha}^{(0)}+\bm{\lambda}_{\alpha},\;\alpha=A,B$.
According to Eqs. (11) and (12), the correlation matrix $\bm{S}$ for the
evolved state is obtained by

$\bm{S}=\bm{\lambda}_{A}\cdot\bm{\upsilon}_{B}^{\mathrm{T}}+(\bm{\upsilon}_{A}-\bm{\lambda}_{A})\cdot\bm{\lambda}_{B}^{\mathrm{T}}+\bm{\Gamma}_{A}\cdot\bm{T}^{(0)}\cdot\bm{\Gamma}_{B}^{\mathrm{T}}.$
(13)

Here, $\bm{\varepsilon}_{B}=\bm{\varepsilon}^{+}_{\mathrm{out}}$ is given by
the quantum channel for Dirac particles from the outside region near the event
horizon of the black hole.

To further explore the decoherence of Dirac particles, we study the dynamics
of quantum coherence and quantum steering. The Werner state is chosen to be an
initial one. In the Hilbert space spanned by
$\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$,
$\rho_{AB}^{(0)}=\frac{1-\eta}{4}\mathbb{I}+\eta|\psi^{-}\rangle\langle\psi^{-}|$
where $|\psi^{-}\rangle=\frac{1}{\sqrt{2}}(|01\rangle-|10\rangle)$ and the
parameter $\eta\in[0,1]$. In this case, the evolved state has a real
$\mathrm{X}$ shaped form. According to quantum resource theory, the measure of
quantum coherence based on $l_{1}$ norm is obtained by

$C(\rho_{AB})=\frac{1}{2}(|S_{11}-S_{22}|+|S_{11}+S_{22}|),$ (14)

where $S_{ij}$ are the elements of the correlation matrix for the evolved
state. Recently, the inequality has been developed to judge whether a
bipartite state is steerable when both of the observers are allowed to measure
observations in their local sites Costa2016 . This inequality is composed of a
finite sum of bilinear expectation values, which is given by
$F_{n}^{\mathrm{CJWR}}(\rho,\mu)=\frac{1}{\sqrt{n}}|\sum_{i=1}^{n}\langle
A_{i}\otimes B_{i}\rangle|\leq 1$. Here,
$A_{i}=\bm{u}_{i}\cdot\bm{\sigma},B_{i}=\bm{v}_{i}\cdot\bm{\sigma}$ where
$\bm{u}_{i}\in\mathbb{R}^{3}$ are unit vectors and
$\bm{v}_{i}\in\mathbb{R}^{3}$ are orthonormal vectors and
$\bm{\zeta}=\\{\bm{u}_{1},\dots,\bm{u}_{n},\bm{v}_{1},\dots,\bm{v}_{n}\\}$ is
the set of measurement directions. The $\mathrm{CJWR}$ inequality for $n=2$ is
used to quantify steerability
$Q(\rho_{AB})=\max[0,\frac{F_{2}(\rho_{AB})-1}{F_{2}^{max}-1}]$ where
$F_{2}(\rho_{AB})=\max_{\bm{\zeta}}F(\rho_{AB},\bm{\zeta})$ and
$F_{2}^{max}=\max_{\rho_{AB}}F_{2}(\rho_{AB})$. In this case, the steerability
is obtained by

$Q_{s}(\rho_{AB})=\max[0,\frac{\sqrt{\mathrm{Tr}(\bm{S}\bm{S}^{\mathrm{T}})-\min_{i}(S^{2}_{ii})}-1}{\sqrt{2}-1}].$
(15)

To clearly demonstrate the effects of external noises on the quantumness of
Dirac particles, we consider a general open system model which is applicable
to a qubit subjected to the fluctuating quantum fields with random phase
noises. This kind of reservoir is modelled by the random telegraph noise
channel. The noise is characterized by the autocorrelation function given as
$\langle\chi(\tau)\chi(s)\rangle=\delta^{2}e^{-\gamma|\tau-s|}$ Daffer2004 ;
Benedetti2013 . $\chi$ denotes the stochastic variable and $\delta$ represents
the coupling strength between the system an reservoir. And $\gamma$ signifies
the fluctuation rate of the noise. In the Kraus representation, the quantum
channel for the random telegraph noise is given by
$\bm{\varepsilon}(\rho)=\sum_{i}K_{i}\rho K_{i}^{{\dagger}}$ where the Kraus
operators are $K_{1}(\nu)=\sqrt{\frac{1+\kappa(\nu)}{2}}\mathbb{I}$ and
$K_{2}(\nu)=\sqrt{\frac{1-\kappa(\nu)}{2}}\sigma_{3}$ and
$\kappa(\nu)=e^{-\nu}[\cos(\mu\nu)+\frac{1}{\mu}\sin(\mu\nu)]$ with
$\mu=\sqrt{(\frac{2\delta}{\gamma})^{2}-1}$ and $\nu=\gamma\tau$. In the Bloch
representation, we can rewrite the random telegraph noise channel by using the
mapping matrix and vector,

$\bm{\Gamma}_{\mathrm{RTN}}=\mathrm{Diag}\left[\kappa(\nu),\kappa(\nu),1\right],\quad\bm{\lambda}_{\mathrm{RTN}}=\bm{0},$
(16)

where the mapping matrix $\bm{\Gamma}_{\mathrm{RTN}}$ has the diagonal form.
If the Dirac particle $A$ suffers from the dissipation of random telegraph
noise, the correlation matrix $\bm{S}$ in the pauli basis is obtained by

$\bm{S}=\mathrm{Diag}\left[-\eta\kappa(\nu)q_{R}\cos
r,-\eta\kappa(\nu)q_{R}\cos r,-\eta|q_{R}|^{2}\cos^{2}r\right].$ (17)

In accordance with the results of Eqs. (14) and (15), The measures of quantum
coherence and quantum steering are obtained by
$C(\rho_{AB})=|\eta\kappa(\nu)q_{R}|\cos r$ and
$Q_{s}(\rho_{AB})=\max\left[\frac{F_{2}-1}{\sqrt{2}-1},0\right]$ where the
quantity $F_{2}=\sqrt{2|\eta\kappa(\nu)q_{R}|}\cos r$. It is seen that both of
them satisfy the relation
$Q_{s}=\max\left[\frac{\sqrt{2}C(\rho_{AB})-1}{\sqrt{2}-1},0\right]$. The
interplay between the external noise and quantum fluctuations near the event
horizon of the black hole is numerically analyzed. It is seen that the maximum
of quantum nonlocality can be attainable in the case of $\eta=1,q_{R}=1$. The
dynamical behavior of quantum coherence with the reservoir evolution time
$\tau$ and dilatonic parameter $D$ is illustrated in Figure 2. In the
condition of weak couplings $(\frac{2\delta}{\gamma})^{2}<1$, the values of
quantum coherence are monotonously decreased with the increase of the
reservoir parameter in Fig. 2(a). The dilaton effect suppresses quantum
coherence when the dilatonic parameter is large. In the strong coupling of
$(\frac{2\delta}{\gamma})^{2}>1$, Figure 2(b) demonstrates that the revivals
of quantum coherence oscillate with time. The result shows that the quantum
fluctuation from the external reservoir leads to the oscillation of quantum
coherence. The similar behavior of quantum steerability can be shown in Figure
3. In the evolution process, the values of quantum steerability are diminished
to zero in the weak coupling case. While the death and revival of the
steerability are presented in the strong coupling condition. These results
demonstrate that the evolutions of quantum coherence and steerability are
dependent on the interplay between the external noise and quantum fluctuations
near the event horizon of the dilatonic black hole. The non-monotonous
decoherence effects from the external environment can protect quantum
nonlocality against the information loss from the relativistic effects of the
dilatonic black hole.

##  IV Discussion

The effects of black hole’s dilaton and external reservoir fluctuation on the
quantumness for Dirac particles are investigated in the unified quantum
channels. The process of information loss from quantum fluctuation near the
event horizon of the black hole is modelled by the quantum channel beyond
single mode approximation. The geometric feature and quantity of quantumness
for the channel are obtained. We find out that the quantumness for Dirac
particles escaping from the outside region can be preserved to a high extent
under the circumstance of $q_{R}\rightarrow 1$ and small dilaton. In order to
explore the decoherence of quantum nonlocality, we also put forward a general
method by using the Pauli basis. The interplay between the external
environmental noise and relativistic effects of the dilatonic black hole can
be further studied. For an instance of external reservoir, the random
telegraph noise channel is chosen. In the weak coupling condition, the
monotonous degradation of quantum coherence and quantum steering with time can
be observed when the dilaton parameter is increased. The oscillation revivals
of quantum coherence and steerability are also presented when the strong
coupling is considered. From the viewpoint of relativistic quantum
information, this kind of non-monotonous decoherence contributes to the
protection of quantum resources from the information loss near the event
horizon of the black hole.

###### Acknowledgements.

We would like to thank Professor Bill Unruh and Professor Philip C. E. Stamp
for the discussions on the related work. The authors were supported by
Postgraduate Research and Practice Innovation Program of Jiangsu Province.
This work is funded by $\mathrm{SCOAP}^{3}$.

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Figure Captions

Figure 1: The quantumness of Dirac particles in the outside region from the
event horizon is illustrated with the variation of the parameter $q_{R}$ and
dilaton value $D$. The sphere contraction in the Bloch representation is
pictured when $q_{R}=0.1$ and $D=0.95$. The contour plot of the quantumness
quantity based on quantum coherence is shown. Some parameters of the dilatonic
black hole $M=1,\;\omega=1$ are chosen.

Figure 2: The evolutions of quantum coherence are plotted as a function of the
dilaton $D$ and reservoir time $\tau$ for the parameters of $M=1,\;\omega=1$
and $\eta=1,q_{R}=1$. The left plot denotes the weak coupling case where the
reservoir parameters $\delta=0.05,\gamma=1$ are chosen. The right plot
represents the strong coupling case where $\delta=0.05,\gamma=0.001$ are
chosen.

Figure 3: The decoherence of quantum steering is illustrated for the
parameters of of $M=1,\;\omega=1$ and $\eta=1,q_{R}=1$. The black dash line
depicts the evolution of the steerability when the weak coupling
$\delta=0.05,\gamma=1$ is considered. The solid red line corresponds to the
case of the strong coupling when $\delta=0.05,\gamma=0.001$ are chosen.