# Step by Step to Fairness: Attributing Societal Bias in
Task-oriented Dialogue Systems

Hsuan Su$\heartsuit$, Rebecca Qian♢, Chinnadhurai Sankar♢, Shahin Shayandeh♢,
Shang-Tse Chen$\heartsuit$, Hung-yi Lee$\heartsuit$, Daniel M. Bikel♢ Work
done when interning at Meta AI.

###### Abstract

Recent works have shown considerable improvements in task-oriented dialogue
(TOD) systems by utilizing pretrained large language models (LLMs) in an end-
to-end manner. However, the biased behavior of each component in a TOD system
and the error propagation issue in the end-to-end framework can lead to
seriously biased TOD responses. Existing works of fairness only focus on the
total bias of a system. In this paper, we propose a diagnosis method to
attribute bias to each component of a TOD system. With the proposed
attribution method, we can gain a deeper understanding of the sources of bias.
Additionally, researchers can mitigate biased model behavior at a more
granular level. We conduct experiments to attribute the TOD system's bias
toward three demographic axes: gender, age, and race. Experimental results
show that the bias of a TOD system usually comes from the response generation
model.

## Introduction

Recent advances in machine learning have led conversational assistants such as
Google Home, Alexa (Ram et al. 2018), and Siri to become popular and essential
in our daily life. However, models of conversational assistants can learn
harmful societal biases from datasets, leading to biased model behavior.

Various existing works studied bias in open-domain dialogue systems. Dinan et
al. (2020) investigated multiplayer text-based fantasy adventure dataset LIGHT
(Urbanek et al. 2019) and proposed a bias mitigation method by adding a
control bin as a prompt during training. Liu et al. (2020a) mitigated gender
bias with a regularization method. Sun et al. (2022) studied geographical
fairness in TOD systems and proposed a new dataset to evaluate TOD's
helpfulness toward different developing countries.

Figure 1: Language models can be biased toward specific demographic
attributes. We propose a diagnosis method to attribute bias to the components
of task-oriented dialogue systems.

Figure 1 demonstrates the composition of a TOD system. A TOD system can be
decomposed into three components: API call model, database, and response
generation model. The example user utterances in Figure 1 show that the TOD
system is biased toward gender. In this case, all three components of the TOD
system can contribute to the final biased behavior. However, existing works
only studied the accumulated bias. To get a more in-depth understanding of the
sources of bias, we propose a diagnosis method to attribute bias to each
component of TOD systems. We believe the proposed attribution method can
facilitate research on fairness mitigation.

In this work, we first study fairness on well-known TOD datasets. Then, we
attribute the societal bias of models trained with these well-known datasets
to the TOD system's components. Unlike the hand-crafted bias measuring methods
in previous works, our attribution is conducted by applying the perturber
(Qian et al. 2022) to the whole dialogue. We propose a newly designed fairness
metric to evaluate the bias by comparing original and perturbed user
utterances. After measuring the accumulated bias, we update each component's
setting to eliminate the resulting bias. The bias change after the component
setting is updated can be regarded as the bias contribution of the component.
By repeating this procedure, we can attribute the accumulated bias to each
component of the TOD system. We conduct experiments on several transformer-
based models with different pretrained methods. Our experiments cover three
essential demographic axes: gender, age, and race. The experimental results
show that the bias of a TOD system mainly comes from the response generation
model.

## Related Works

##### Fairness in Language Generation

Societal bias issues in natural language generation have drawn a lot of
attention recently (Sheng et al. 2021). We introduce two of them, bias
evaluation and bias mitigation, in this section.

Czarnowska, Vyas, and Shah (2021) showed that traditional bias evaluation
metrics, such as demographic parity (Dwork et al. 2012) and equalized odds
(Hardt et al. 2016), are not applicable to text results. Therefore, discrete
language generation results need dedicated evaluation methods. Researchers
have proposed several methods to measure the fairness of language generation
results. Liang et al. (2021) and Nadeem, Bethke, and Reddy (2021) divided the
bias evaluation methods into two categories: local bias-based and global bias-
based. Local bias-based methods use hand-crafted templates to evaluate the
fairness. For example, the template can be a sentence with some masked words.
We can then evaluate the fairness by comparing the model's token probability
of the masked words (Zhao et al. 2017; Kurita et al. 2019; Bordia and Bowman
2019). Global bias-based methods use multiple classifiers to evaluate the
fairness by comparing the classification results of generated texts from
various different perspectives. Previous works used regard ratio (Sheng et al.
2019, 2020; Dhamala et al. 2021), sentiment (Liu et al. 2020b; Groenwold et
al. 2020; Huang et al. 2020; Sheng et al. 2019; Dhamala et al. 2021; Liu et
al. 2019), offensive (Liu et al. 2020b), and toxicity (Dhamala et al. 2021) as
classifiers. In addition to off-the-shelf classifiers, existing works also
analyzed word usage on generated texts. Dinan et al. (2020) measured the word
count of the gender group (i.e., male, female) in the model's generated text;
Liu et al. (2020b, 2019) count the word frequency of ``career'' and ``family''
given male and female inputs.

For bias mitigation, an intuitive method is called Counterfactual Data
Augmentation (CDA). People applied word-based augmentation that augmented all
the demographics mentioned in texts, and trained models on the augmented data
to equalize the word usage imbalance problem in the dataset to mitigate bias
(Maudslay et al. 2019; Liu et al. 2019; Zmigrod et al. 2019). In addition to
CDA, there are various training-based methods to mitigate bias, for example,
Dinan et al. (2020) finetune models with controllable prefixes to equalize the
word usage for male and female. Liu et al. (2020b) proposed a novel
adversarial method that disentangles text into semantic and gender features to
mitigate bias. Liang et al. (2021) employed the concept of Null space
projection (May et al. 2019) to eliminate gender feature in models. More
recently, Sharma, Dey, and Sinha (2022) used hand-crafted prompts (Li and
Liang 2021) to mitigate bias in machine translation. Sheng et al. (2020) also
proposed to generate prompts to equalize gender and race's disparity in the
dialogue generation task.

##### End-to End Task-oriented Dialogue

Task-oriented dialogue (TOD) systems help people resolve specific problems in
domains such as restaurant booking and flight booking. TOD systems
traditionally can be decomposed into several components: Natural Language
Understanding, Dialogue State Tracking, Policy Learning, and Natural Language
Generation. (Ni et al. 2021; Liu and Lane 2018). Historically, these modules
were trained individually (Asri, He, and Suleman 2016; Bordes and Weston 2016;
Liu and Lane 2018; Lin et al. 2021). However, individual components have the
drawback of being non-differentiable. Pretrained large language models (LLMs)
provide an end-to-end solution, and have been shown to not only resolve the
aforementioned problems, but also improve the performance of TOD
systems(Hosseini-Asl et al. 2020a). In this training manner, the user
utterance, API call, and DB result are concatenated together and the model is
trained autoregressively to generate the final response (Hosseini-Asl et al.
2020a; Ham et al. 2020; Peng et al. 2020; Lin et al. 2020). In this work, we
followed the latter approach and trained end-to-end TOD systems with LLMs
(GPT-2 (Radford et al. 2019), BART (Lewis et al. 2019), and T5 (Raffel et al.
2020)).

##### Evaluation with Perturbations

When we deploy LLMs to products, it is expected that lots of real and out-of-
domain (OOD) issues exist, which can manifest in bias toward certain user
groups (Ribeiro et al. 2020). Previous approaches for evaluating the fairness
of ML systems relied on hand-crafted perturbation examples, counterfactuals,
or creating adversarial examples from scratch (Gardner et al. 2020; Teney,
Abbasnedjad, and Hengel 2020; Kaushik, Hovy, and Lipton 2019; Xu et al. 2021;
Ribeiro et al. 2020). Recently, researchers have applied model-based methods
to generate texts to influence model behavior (Wu et al. 2021; Qian et al.
2022; Wallace et al. 2019; Cho et al. 2021; Li et al. 2020; Qian et al. 2021).
These prior works showed that model-based methods not only enhance the

((a)) Percentage of gendered words in different datasets

((b)) Percentage of aged words in different datasets

Figure 2: Demographic distributions across datasets, the x-axis is the
dataset, and the y-axis is the proportion of the word usage.

efficiency of test case generation, but also model-generated texts have higher
performance in influencing and testing models' behavior.

## Attributing Bias in TOD Systems

### Problem Definition

This section will begin by providing a background on TOD systems and their
growing importance in various industries. It will then introduce the problem
of bias in TOD systems and highlight the need for fair and unbiased TOD
systems that provide equitable outcomes for all users.

#### Task-Oriented Dialogue (TOD) System

Figure 3: The proposed framework, we input both original and perturbed user
utterance and measure the bias attribution of each module by changing the
module step by step (The numbers in the figures are made up.).

As shown in Figure 1, a TOD system consists of three components: API call
model, database, and response generation model. Once a user utterance is
provided, the API call model generates an API call according to the user
utterance. Then, the TOD system converts the API call to a database search
command and searches the necessary data from the database. At last, the
response generation model generates the response according to the user
utterance, the API call, and the database results.

#### Bias in TOD Datasets

Machine learning models often replicate or amplify demographic bias present in
training data (Dinan et al. 2020). Therefore, measuring bias in datasets is
critical to understanding biased model responses. We start by analyzing the
percentage of demographic words in six well-known TOD datasets 111All six
datasets are public and can be found on ParlAI:
https://parl.ai/docs/tasks.html“#goal-tasks: Google SGD (Lee et al. 2022),
Multiwoz V22 (Zang et al. 2020), Taskmaster-(1, 2, 3) (Byrne et al. 2019),
MetaLWoZ (Lee et al. 2019), and MultiDoGo (Peskov et al. 2019), MSR E2E (Li et
al. 2018). The word list follows Qian et al. (2022)'s work. We count the
number of words in each demographic axis. Figure 2(a) and Figure 2(b) show
that words indicating gender and age are extremely imbalanced in the datasets.
In particular, words indicating man dominate other words in most of the
datasets. Such an imbalanced distribution of words could lead to heavily
biased model responses toward specific demographic attributes.

#### Definition of Fairness

To measure the bias of TOD systems, we formally define the fairness metric. We
utilize the concepts of helpfulness (Sun et al. 2022) and perturbation (Qian
et al. 2022) to define the metric. That is, the model responses should have
similar helpfulness for original and demographic-perturbed user utterances. We
use BLEU (Papineni et al. 2002) score of the model response and the ground
truth to evaluate helpfulness. For example, the response of the user
utterance: ``I would like to find a movie for my mom.'' should have similar
helpfulness to the response of the user utterance: ``I would like to find a
movie for my dad.'' According to this, we propose a newly designed fairness
metric (Equation 1). Note that fairscore refers to our newly designed
fairscore metric in the rest of the paper, not the fairscore proposed by
existing works.

$F_{S}=\frac{|BLEU(f_{t}(u_{t}),g_{t})-BLEU(f_{t}(\widetilde{u_{t}}),\widetilde{g_{t}})|}{BLEU(f_{t}(u_{t}),g_{t})}$
(1)

We denote the TOD system as $f_{t}$, original user utterances as $u_{t}$,
original responses as $f_{t}(u_{t})$, and original ground truths as $g_{t}$.
Perturbed user utterances and ground truths are denoted as $\widetilde{u_{t}}$
and $\widetilde{g_{t}}$, respectively. As aforementioned, we expect a fair
model to generate responses with similar helpfulness for original and
perturbed user utterances. More specifically, we calculate the absolute value
of the difference between original helpfulness $BLEU(f_{t}(u_{t},g_{t}))$ and
perturbed helpfulness $BLEU(f_{t}(\widetilde{u_{t}}),\widetilde{g_{t}})$, and
normalize the helpfulness change to original helpfulness to evaluate the bias
of a TOD system.

In this section, we propose a diagnosis method to attribute the bias to the
components of TOD systems. We first introduce the architecture of common TOD
systems in Section Task-Oriented Dialogue (TOD) System. We then formally
define the fairness metric in Section Definition of Fairness. The proposed
diagnosis method is described in Section Proposed Method.

1 Input: dataset $\SS$, set of attribute pairs $\mathcal{P}_{\text{d}}$,
dictionary of demographic words $\mathcal{D}_{\text{d}}$

2

3Initialize: new dataset $\widetilde{\mathcal{S}}\leftarrow\emptyset$

4 for  _dialog: $dialog\in\SS$_ do

5 new dialog $\widetilde{dialog}\leftarrow\emptyset$

6 for  _turn: $turn\in dialog$_ do

7 new turn $\widetilde{turn}\leftarrow turn$

8 new $K\leftarrow\emptyset$

9 for  _word $w\in turn\cap\mathcal{D}_{\text{d}}$_ do

10 for  _
$(\cdot,t)\in\left\\{(a_{s},a_{t})\in\mathcal{P}_{\text{d}}|a_{s}=a_{w},a_{s}\neq
a_{t}\right\\}$_ do

11 $K\leftarrow K\cup\left\\{w,t\right\\}$

12

13

14

15 $(w,t)\sim\mathcal{U}(K)$

16 for  _turn $turn\in dialog$_ do

17 $\widetilde{turn}\leftarrow$ perturber(turn, w, t)

18

19 $\widetilde{dialog}\leftarrow\widetilde{dialog}\cup\widetilde{turn}$

20
$\widetilde{\mathcal{S}}\leftarrow\widetilde{\mathcal{S}}\cup\left\\{\widetilde{s}\right\\}$

21

Output: $\widetilde{\mathcal{S}}$

Algorithm 1 Data Augmentation via Demographic Perturbation

### Proposed Method

To get a more in-depth understanding of the biased responses, we propose a
diagnosis method to attribute the bias to the potentially biased TOD
components. We introduce the perturbation details in Section Perturbation and
describe the attribution procedure in Section Step by step to fairness via
perturbating the Bias Attribution.

#### Perturbation

The proposed method requires perturbing the original input user utterances to
measure the bias. We modified Qian et al. (2022)'s perturbation algorithm to
suit our scenario.

Let $\SS$ be the original input dialogue dataset consisting of variable-length
turn of dialogues, where $turn\in dialog$ is a turn in dialogue and $w$ is a
word in $turn$ with demographic attribute $a_{w}$. We denote the attribute
pair (source, target) as $(a_{s},a_{t})\in\mathcal{P}$, where
$\mathcal{P}\subseteq\mathcal{A}\times\mathcal{A}$. $\mathcal{A}$ is the
demographic attribute set.

Algorithm 1 describes the dialogue augmentation procedure. We first identify
the set of perturbable words for each dialogue turn with the word lists. Note
that each perturbable word's source and target attributes are also identified.
We then sample a word from the collected perturbable words with target
attributes $K$ and apply the perturbation to every dialogue turn. To ensure
dialogue consistency, we perform the identical perturbation on dialogue states
and database search results. Table 2 shows the perturbed user utterances.

#### Step by step to fairness via perturbating the Bias Attribution

As mentioned in Section Task-Oriented Dialogue (TOD) System, a TOD system
usually consists of the following components: API call model, database (DB),
and response generation model. All these components can cause biased responses
toward different demographic attributes. When attributing the bias, we
eliminate each component's influence on the bias by changing the component
setting. After the setting of the component is updated, the change of the
fairscore is attributed to the component. The attribution consists of three
steps, we describe the details below.

  * •

Step 1 - Fairscore Evaluation: In the beginning, we evaluate the fairscore by
inputting a perturbed user utterance to the TOD system and get a fairscore of
uttr' based on generated responses. As shown in Figure 3, the fairscore of
$\tilde{uttr}$ is $0.23$.

  * •

Step 2 - Resolving DB Mismatch: We then exclude DB search's influence on bias
by resolving DB's mismatch issue. Though we can calculate the change of the
fairscore and attribute the difference to the DB, it can be meaningless as we
do not have the original dataset. That is, we do not know whether the
perturbed API call can find necessary data from the original dataset. In
Figure 3, the fairscore of $\tilde{uttr}$ after this step is $0.18$.

  * •

Step 3 - API Call Adjustment: In this step, we perturb the API call of uttr
and use the perturbed API call for the downstream components. The content of
the perturbed API call should be semantically identical to the API call of
uttr but with different demographic attributes. In Figure 3's example, the
change of the fairscore after the API call perturbation is $0.18-0.14=0.04$.
After solving the mismatch problem in the previous step, the bias ($0.18$)
comes from the API call model and the response generation model. Therefore, we
can attribute the bias contribution $0.04$ and the bias contribution $0.14$ to
the API call model and the response generation model, respectively.

|  | DST | Response Gen
---|---|---|---
Model | Dataset | JGA | BLEU
GPT-2 | Google SGD | 0.7920 | 0.1604
Taskmaster 2 | 0.5890 | 0.2188
BART | Google SGD | 0.8219 | 0.1880
Taskmaster 2 | 0.6212 | 0.2338
T5 | Google SGD | 0.8058 | 0.1672
Taskmaster 2 | 0.5948 | 0.2390
Table 1:  Automatic metrics of different modeling approaches tested on the
original Google SGD and Taskmaster 2 datasets. We used JGA for dialogue state
tracking (DST) and BLEU score to measure response generation

## Experiments

### Experimental Setup

#### Datasets

((a)) Google SGD

((b)) Taskmaster 2

Figure 4: The biased modules results on Google SGD and Taskmaster 2. The
y-axis indicates the Fairscore while the x-axis is the models with different
initial methods. In this figure, we stacked the API call module (light blue)
with the response generation module (deep blue). The whole bar indicates the
overall bias in the systems.

We chose two datasets and investigated the biased module when we trained
models on different datasets. We selected two well-known datasets in our
experimental setups, Google Schema Guided Dialogue (Lee et al. 2022) (Google
SGD) and Taskmaster-2 (Byrne et al. 2019). Google SGD is a huge dataset known
for its various schemas. Instead of the synthetic dataset, we also selected
Taskmaster 2, collected from real human dialogues. Since the perturbation
process is not deterministic. For each dataset, we applied perturber three
times and took the average of these three times fairness.

##### Google SGD

Google SGD dataset consists of over 20k annotated task-oriented, multi-domain,
human-virtual assistant conversations. Models may have a chance to access API
Schemas, which contain intents and services of conversation. We didn't
consider the API Schemas as the model's input in our experiments.

##### Taskmaster 2

The Taskmaster 2 dataset consists of 17,289 dialogues in the seven domains,
including restaurant suggestions, movie recommendations, and flight
reservations. Taskmaster 2 consists entirely of spoken two-person dialogues
containing a more diverse set of scenarios.

Axes Utterance API Call DB Response Generation Gender Original I want to find a female psychiatrist in Fremont. api_name: find_provider city: Fremont type: psychiatrist address = 39650 Liberty Street #310 city = Fremont phone_number = 510-498-2890 therapist_name = Charles Dennis Barton, Jr type = Psychiatrist Charles Dennis Barton, Jr is a great one in Fremont. Perturbed I want to find a male psychiatrist in Fremont. api_name: find_provider city: Fremont type: male therapist <BLANK> I couldn't find any results. Do you need help with anything else? Age Original Do I have to pay for a ticket to visit this? Do you think children may like it? <BLANK> <BLANK> Yes, I can book a ticket for you. Perturbed Do I have to pay for a ticket to visit this? Do you think adults may like it? <BLANK> <BLANK> Yes, I can book a ticket for you. You will be visiting historical landmark in a moment. Race Original I'm looking for American food. api_name = FindRestaurants city = Vacaville cuisine = American city = Vacaville cuisine = American has_live_music = False phone_number = 707-359-2200 price_range = moderate restaurant_name = Bj's Restaurant & Brewhouse serves_alcohol = True street_address = 190 Nut Tree Parkway Okay, I found 1 restaurant that matches your request. Bj's Restaurant & Brewhouse is a nice restaurant in Vacaville. Perturbed I'm looking for Hispanic food. api_name = FindRestaurants city = Vacaville cuisine = Mexican city = Vacaville cuisine = Mexican has_live_music = False phone_number = 707-448-4002 price_range = moderate restaurant_name = Hisui | Mexican Restaurant serves_alcohol = False street_address = 1350 East Monte Vista Avenue Okay, I found 1 restaurant that matches your request. How about Hisui | Mexican Restaurant in Vacaville?

Table 2: Generated examples in three demographic axes.

#### Model Settings

##### Model Framework

One of our experiments aimed to investigate bias phenomena in different model
architectures. We followed (Chen, Crook, and Roller 2021) that trained GPT-2
(Radford et al. 2019), BART (Lewis et al. 2019), and T5 (Raffel et al. 2020).
The implementation of all three models resembles the setup of SimpleTOD
(Hosseini-Asl et al. 2020b) as we introduced in Section Attributing Bias in
TOD Systems. Table 1 shows the automatic evaluation results on models across
three datasets.

##### Model Training Procedure

In addition to the original pretrained models' checkpoints, we used different
initiated checkpoints to investigate the bias transfer phenomenon. In our
experiments, we tried two settings: Prefinetunning and Cross training.

Cross Training To measure the bias transfer phenomenon, we tried cross
training that first trains models on Taskmaster 2 and transfer train on Google
SGD and vice versa. Under this setting, we hope to find the correlation
between these two training tasks and the effect of bias transfer given
different downstream training orders.

Prefinetunning As shown in Figure 2(a), there are heavy disparities in TOD
datasets. With that said, we want to know the bias transfer from TOD datasets
to downstream tasks. Before we finetuned models on our target dataset (e.g.,
Google SGD and Taskmaster 2), we previously prefinetuned the pretrained models
on a bunch of TOD datasets, we prefinetuned pretrained GPT-2, BART, and T5 on
Multiwoz V22 (Zang et al. 2020), Taskmaster 1, 3 (Byrne et al. 2019), MetaLWoZ
(Lee et al. 2019), and MultiDoGo (Peskov et al. 2019), MSR E2E (Li et al.
2018).

## Results

The experimental results are shown in Figure 4(a) and Figure 4(b). We perform
our bias attribution method on taskmaster-2 (Figure 4(b)) and Google SGD
(Figure 4(a)) datasets. The results of Bart, T5, and GPT-2 models with
different initiated methods are presented in each figure. The bias attribution
for the API call model and the response generation model is stacked in the
figures, so the whole bar represents the bias in the TOD systems.

### API Call versus Response Generation

In the first section, we would like to discuss the bias attribution of each
model inside TOD frameworks: the API call model and the response generation
model. In our experiments, as shown in Figure 4(a) and Figure 4(b). We found
that Google SGD (Figure 4(a)), in most cases, the response generation models
take more responsibility for biased results than the API call model. On the
other hand, we observed that the phenomenon is different in Taskmaster-2
(Figure 4(b)). Response generation models take less responsibility than the
API call model in terms of bias.

### Bias in Different Demographic Axes

In this subsection, we discussed the bias degree across different
demographics. We observed that on both Google SGD and taskmaster-2 dataset,
models are less biased on Gender and are most biased on race-perturbed data.
We attribute this because the demographics word percentage in the datasets.
Figure 2(a) and Figure 2(b) show that the datasets usually have most
percentage of gender words than other axes. With that said, although the
gender word distribution is imbalanced, models are still less biased compared
to other axes.

### Bias across Models

In this section, we want to compare the differences across the three models we
used in our work. We tried two sequence-to-sequence-based models (BART and T5)
and one decoder-only model (GPT-2). We found that the models trained with BART
were usually less biased. Although T5 models have better performance than
GPT-2 model in 1, the models trained with T5 were more biased than GPT-2. This
shows that model performance and bias are not correlated.

### Bias across Different Initial Methods

In the beginning, we expect that pre-finetunning and cross-training will
amplify models' bias in all axes since there's heavy bias in these datasets.
However, we didn't observe this phenomenon in our results. This means that the
upstream training might not have a significant influence on the downstream
tasks.

### Negative API Attribution

We observed that there are very few negative API Call model attributions in
the taskmaster-2 dataset (Figure 4(b)). After observing the generation
results, we found some examples that, API call generated some biased results
and didn't affect the model's generated response. However, we observed that
the response became biased when we changed the API call model and successfully
made parallel input for response generation models. This causes API call
attribution becomes negative in figure 4(b).

### Example Analysis

In table 2, we include the biased examples for each demographic detected by
the proposed method. In demographic 'gender' , we found that the model's
responses disparity is due to the model couldn't generate the correct type for
male psychiatrist in perturbed user utterances. In demographic 'age', since
models can decide whether generate a API call to search from the database,
it's '<BLANK>' in this case. We can see a bias in the response generation
module in that the model recommends historical landmarks for adults instead of
children. As for the demographic 'race', we found that the proposition of API
Call module's bias attribution is slightly higher than other demographics in
Figure 4. Among the generated results, we observed that there the cuisine in
API call is categorical but not a string class. Took Hispanic food as an
example. It only has Mexican category correlated to Hispanic, so models always
generate Mexican when the input is Hispanic. Qian et al. (2021) also show that
TOD datasets contain lots of locational bias that cause models to overfit on
certain locations, which matches our observations.

### Gender Disparities Analysis

In this subsection, we did a deeper analysis to check the model's disparity
across genders. We categorized original and perturbed utterances into three
dimensions (male, female, non-binary). With that said, the y-axis and x-axis
in Figure 5 mean the source and target pair $(a_{s},a_{t})$ attribute
mentioned in Perturbation. We refer results to Figure 5 and since the selected
target attribute can not be equal to the source attribute. We set the value in
diagonal to 0 in the figure. To be clear, we further read the heatmaps in two
ways: vertical and horizontal. Vertical observation means the effect on the
target attribute, and horizontal observation focuses on the source attribute
we choose to be perturbed.

In both vertical and horizontal observation, we first noticed that models
mostly have higher fairscores when we perturbed 'female' to 'non-binary' or
vice versa. In this situation, we would like to think it's the bias come from
the dataset. As shown in Figure 2(a), the proportion for female and non-binary
is less than male. The models that trained on very low frequency gender
attributes (i.e., female and non-biary) might become less fair. On the other
side, we also observed that fairscore raised when we selected 'male' as a
source attribute (the first row). In these cases, heavy unbalance distribution
of gender attributes across datasets enhances model's disparity between
frequent attributes and nonfrequent attributes.

((a)) API Call module's gender disparities under Google SGD

((b)) Response module 's gender disparities under Google SGD

((c)) API Call module's gender disparities under taskmaster 2

((d)) Response module 's gender disparities under taskmaster 2

Figure 5: These heatmaps indicate Fairscore on different source and target
attributes pair in gender demographic. The y-axis is the source attributes,
while the x-axis is the target attributes.

## Conclusion

In this paper, we present a first-of-its-kind study on the locations of bias
and fairness in task-oriented dialogue systems. The proposed method help us
understand

We conduct a complete bias investigation on TOD systems and found heavy biases
in well-known TOD datasets. Moreover, models show different degrees of
helpfulness in responses given different demographic attributes.

To dig into this problem and have a deeper understanding of the behavior
behind TOD systems in terms of bias, we proposed a generation version of
Fairscore that evaluates bias in the TOD framework based on the responses. We
further proposed a new method that studies the bias attributions of each
module in TOD frameworks. In our experiments, we found that models show the
most biased tendency on Age and Race demography but are relatively fair on
Gender and also observed that biased modules vary across different datasets.

We envision our method as a potential approach to studying bias interpretation
in systems composed of cascaded modules in the future.

## Broader Impacts

##### Fairness in TOD

To the best of our knowledge, very few works conduct fairness investigations
on TOD system. We hope our work could bring awareness to the community that
instead of the open domain dialogue systems, TOD also has a high probability
to generate biased responses. With that said, we encourage people to keep
exploring and digging into this area.

##### Module Polluting

In our works, we demonstrate the biased degree of each module in TOD systems.
There are two directions that the proposed method might be used by malicious
people. First, the method exposes the fairer modules in TOD systems which
might be an attack weakness for people to pollute. Second, malicious people
can take advantage of the most biased modules detected by our methods and use
them to pollute the whole system.

As developers of emerging technologies, we also take responsibility for
defining the boundaries of these technologies. We will continue to refine the
aforementioned method to ensure that the proposed methodology improves public
welfare as we intend it to

## Limitations

We proposed a diagnosis method to investigate fairness and attribute the
biases in TOD systems. There are some limitations that we want to discuss and
include in this section.

##### Database Limitation:

Since we didn't have access to the database that the authors used to create
the dataset. So in this paper, we implemented the database with a simple
lookup table. However, this may cause many data mismatches and non-found
problems in our diagnosis problem. To solve this problem, as mentioned in
previous sections, we perturbed the entities in the original database to
create a new 'simulated' database.

##### Demographic Catagorization:

The another limitation in our paper is that we categorized each demographic
axis into several attributes(e.g., we categorize gender into male, female, and
non-binary). However, this categorization behavior makes a distinction in
people and might not make sense to everyone.

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# GRAVITATIONAL BINARY-LENS EVENTS WITH PROMINENT EFFECTS OF LENS ORBITAL
MOTION

H. Park1,34, A. Udalski2,33, C. Han1,34,37, A. Gould3,34, J.-P. Beaulieu4,35,
Y. Tsapras5,6,36,
and
M. K. Szymański2, M. Kubiak2, I. Soszyński2, G. Pietrzyński2,7, R. Poleski2,3,
K. Ulaczyk2, P. Pietrukowicz2,
S. Kozłowski2, J. Skowron2, Ł. Wyrzykowski2,8
(The OGLE Collaboration),
J.-Y. Choi1, D. L. Depoy9, Subo Dong10, B. S. Gaudi3, K.-H. Hwang1, Y. K.
Jung1, A. Kavka3, C.-U. Lee11,
L. A. G. Monard12, B,-G. Park11, R. W. Pogge3, I. Porritt13, I.-G. Shin1, J.
C. Yee3
(The $\mu$FUN Collaboration),
M. D. Albrow14, D. P. Bennett15, J. A. R. Caldwell16, A. Cassan4, C.
Coutures4, D. Dominis17, J. Donatowicz18,
P. Fouqué19, J. Greenhill20, M. Huber21, U. G. Jørgensen22, S. Kane23, D.
Kubas4, J. -B. Marquette4, J. Menzies24,
C. Pitrou4, K. R. Pollard14, K. C. Sahu25, J. Wambsganss26, A. Williams27, M.
Zub26
(The PLANET Collaboration),
A. Allan28, D. M. Bramich29, P. Browne30, M. Dominik30, K. Horne30, M.
Hundertmark30, N. Kains29, C. Snodgrass31,
I. A. Steele32, R. A. Street5
(The RoboNet Collaboration)  1Department of Physics, Institute for
Astrophysics, Chungbuk National University, Cheongju 371-763, Korea  2Warsaw
University Observatory, Al. Ujazdowskie 4, 00-478 Warszawa, Poland
3Department of Astronomy, The Ohio State University, 140 West 18th Avenue,
Columbus, OH 43210, USA  4Institut d’Astrophysique de Paris, UMR 7095 CNRS -
Université Pierre & Marie Curie, 98bis Bd Arago, 75014 Paris, France  5Las
Cumbres Observatory Global Telescope Network, 6740 Cortona Drive, Suite 102,
Goleta, CA 93117, USA  6School of Mathematical Sciences, Queen Mary,
University of London, Mile End Road, London E1 4NS, UK  7Universidad de
Concepción, Departamento de Astronomia, Casilla 160-C, Concepción, Chile
8Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge
CB3 0HA, UK  9Department of Physics and Astronomy, Texas A&M University,
College Station, TX 77843, USA  10Institute for Advanced Study, Einstein
Drive, Princeton, NJ 08540, USA  11Korea Astronomy and Space Science
Institute, 776 Daedukdae-ro, Yuseong-gu, Daejeon 305-348, Korea  12Klein Karoo
Observatory, Calitzdorp, and Bronberg Observatory, Pretoria, South Africa
13Turitea Observatory, Palmerston North, New Zealand  14Department of Physics
and Astronomy, University of Canterbury, Private Bag 4800, Christchurch 8020,
New Zealand  15Department of Physics, University of Notre Dame, 225 Nieuwland
Science Hall, Notre Dame, IN 46556-5670, USA  16McDonald Observatory,
University of Texas, Fort Davis, TX 79734, USA  17Astrophysikalisches Institut
Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany  18Department of
Computing, Technical University of Vienna, Wiedner Hauptstrasse 10, A-1040
Vienna, Austria  19Observatoire Midi-Pyrénées, Laboratoire d’Astrophysique,
UMR 5572, Université Paul Sabatier - Toulouse 3, 14 avenue Edouard Belin,
31400 Toulouse, France  20School of Mathematics and Physics, University of
Tasmania, Private Bag 37, Hobart, TAS 7001, Australia  21Institute for
Astronomy, University of Hawaii, 2680 Woodlawn Drive Honolulu, HI 96822-1839,
USA  22Niels Bohr Institute, Astronomical Observatory, Juliane Maries vej 30,
2100 Copenhagen, Denmark  23NASA Exoplanet Science Institute, Caltech, MS
100-22, 770 South Wilson Avenue, Pasadena, CA 91125, USA  24South African
Astronomical Observatory, PO Box 9 Observatory 7935, South Africa  25Space
Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
26Astronomisches Rechen-Institut (ARI), Zentrum für Astronomie der Universität
Heidelberg (ZAH), Mönchhofstr. 12-14, 69120, Heidelberg, Germany  27Perth
Observatory, Walnut Road, Bickley, Perth 6076, Australia  28School of Physics,
University of Exeter Stocker Road, Exeter, Devon, EX4 4QL, UK  29European
Southern Observatory, Karl-Schwarzschild-Straße 2, 85748 Garching bei München,
Germany  30SUPA, University of St. Andrews, School of Physics and Astronomy,
North Haugh, St. Andrews, KY16 9SS, UK  31Max Planck Institute for Solar
System Research, Max-Planck-Str. 2, 37191 Katlenburg-Lindau, Germany
32Astrophysics Research Institute, Liverpool John Moores University, Twelve
Quays House, Egerton Wharf, Birkenhead, Wirral., CH41 1LD, UK  33The OGLE
Collaboration  34The $\mu$FUN Collaboration  35The PLANET Collaboration  36The
RoboNet Collaboration  37Corresponding author

###### Abstract

Gravitational microlensing events produced by lenses composed of binary masses
are important because they provide a major channel to determine physical
parameters of lenses. In this work, we analyze the light curves of two binary-
lens events OGLE-2006-BLG-277 and OGLE-2012-BLG-0031 for which the light
curves exhibit strong deviations from standard models. From modeling
considering various second-order effects, we find that the deviations are
mostly explained by the effect of the lens orbital motion. We also find that
lens parallax effects can mimic orbital effects to some extent. This implies
that modeling light curves of binary-lens events not considering orbital
effects can result in lens parallaxes that are substantially different from
actual values and thus wrong determinations of physical lens parameters. This
demonstrates the importance of routine consideration of orbital effects in
interpreting light curves of binary-lens events. It is found that the lens of
OGLE-2006-BLG-277 is a binary composed of a low-mass star and a brown dwarf
companion.

###### Subject headings:

gravitational lensing: micro – orbital motion – binaries: general

##  1\. INTRODUCTION

Progress in gravitational microlensing experiments for the last two decades
has enabled a great increase in the number of event detections from tens of
events per year at the early stage to several thousands per year in current
experiments. Among discovered lensing events, an important portion are
produced by lenses composed of two masses (Mao & Paczyński, 1991).

One reason why binary-lens events are important is that these events provide a
major channel to determine the physical parameters of lenses. For the
determination of the lens parameters from observed lensing light curves, it is
required to simultaneously measure the lens parallax $\pi_{\rm E}$ and the
angular Einstein radius $\theta_{\rm E}$. The lens parallax is measured from
long-term deviations in lensing light curves caused by the positional change
of an observer induced by the orbital motion of the Earth around the Sun:
parallax effect (Gould, 1992). On the other hand, the Einstein radius is
measured from deviations in lensing light curves affected by the finite size
of a source star: finite-source effect (Gould, 1994; Witt & Mao, 1994). With
the measured values of $\pi_{\rm E}$ and $\theta_{\rm E}$, the mass and
distance to the lens are determined respectively by $M_{\rm tot}=\theta_{\rm
E}/(\kappa\pi_{\rm E})$ and $D_{\rm L}={\rm AU}/(\pi_{\rm E}\theta_{\rm
E}+\pi_{\rm S})$, where $\kappa=4G/(c^{2}{\rm AU})$, ${\rm AU}$ is an
Astronomical Unit, $\pi_{\rm S}={\rm AU}/D_{\rm S}$, and $D_{\rm S}$ is the
distance to the lensed star (Gould, 1992; Gould et al., 2006). For single-lens
events, the chance to measure $\theta_{\rm E}$ is very low because finite-
source effects occur only for very rare events with extremely high
magnifications in which the lens passes over the surface of the source star,
e.g., Choi et al. (2012). By contrast, the chance to measure $\theta_{\rm E}$
is high for binary-lens events because most of these events involve source
stars’ caustic crossings or approaches during which finite-source effects are
important. As a result, the majority of gravitational lenses with measured
physical parameters are binaries.

It is known that changes of lens positions caused by the orbital motion of a
binary lens can induce long-term deviations in lensing light curves, similar
to deviations induced by parallax effects. Since this was first detected for
the event MACHO-97-BLG-41 (Bennett et al., 1999; Albrow et al., 2000; Jung et
al., 2013), orbital effects have been considered for more binary-lens events,
e.g., An et al. (2002); Jaroszyński et al. (2005); Skowron et al. (2011); Shin
et al. (2011). However, analyses have been carried out only for a limited
number of events. An important obstacle of orbital analyses is the heavy
computation required to consider the time variation of the caustic morphology
caused by the orbit-induced changes of the binary separation and orientation.
As a result, routine orbital analyses for general binary-lens events became
possible very recently after being able to utilize efficient modeling software
and powerful computing resources.

Considering orbital effects is important for accurate determinations of
physical lens parameters. Since orbital and parallax effects induce similar
long-term deviations, it might be that orbital effects can be mimicked by
parallax effects. Then, if only parallax effects are considered for events
affected by orbital effects, the determined physical parameters would be
different from their true values. In this work, we demonstrate the importance
of considering orbital effects by presenting analyses of two binary-lens
events.

Table 1Telescopes

Event | Telescopes
---|---
OGLE-2006-BLG-277 | OGLE, 1.3 m Warsaw, LCO, Chile
|  $\mu$FUN, 1.3 m SMARTS, CTIO, Chile
| PLANET, 1.5 m Boyden, South Africa
| PLANET, 1.0 m Canopus, Australia
| PLANET, 0.6 m Perth, Australia
| PLANET, 1.54 m Danish, Chile
| RoboNet, 2.0 m LT, La Palma, Spain
OGLE-2012-BLG-0031 | OGLE, 1.3 m Warsaw, LCO, Chile
|  $\mu$FUN, 1.3 m SMARTS, CTIO, Chile
|  $\mu$FUN, 0.36 m Turitea, New Zealand
|  $\mu$FUN, 0.36 m KKO, South Africa
| RoboNet, 2.0 m FTS, Australia
| Robonet, 2.0 m LT, La Palma, Spain

Note. — LCO: Las Campanas Observatory, CTIO: Cerro Tololo Inter-American
Observatory, KKO: Klein Karoo Observatory, LT: Liverpool Telescope, FTS:
Faulkes Telescope South

##  2\. Observation

The events analyzed in this work are OGLE-2006-BLG-277 and OGLE-2012-BLG-0031.
Both events occurred on stars toward the Galactic bulge field with equatorial
coordinates $(\alpha,\delta)_{\rm J2000}=(18^{\rm h}01^{\rm m}14^{\rm
s}\hskip-2.0pt.84,-27^{\circ}48^{\prime}36^{\prime\prime}\hskip-2.0pt.2)$,
corresponding to the Galactic coordinates
$(l,b)=(2.71^{\circ},-2.39^{\circ})$, for OGLE-2006-BLG-277 and
$(\alpha,\delta)_{\rm J2000}=(17^{\rm h}50^{\rm m}50^{\rm
s}\hskip-2.0pt.53,-29^{\circ}10^{\prime}48^{\prime\prime}\hskip-2.0pt.8)$,
corresponding to $(l,b)=(0.38^{\circ},-1.10^{\circ})$, for OGLE-2012-BLG-0031.
The events were discovered from survey observations conducted by the Optical
Gravitational Lensing Experiment (OGLE: Udalski, 2003). In addition to the
survey observation, the events were additionally observed by follow-up groups
including the Probing Lensing Anomalies NETwork (PLANET: Albrow et al., 1998),
the Microlensing Follow-Up Network ($\mu$FUN: Gould et al., 2006), and the
RoboNet (Tsapras et al., 2009) groups. In Table 1, we list survey and follow-
up groups who participated in observations of the individual events along with
the telescopes they employed as well as their locations. We note that the
event OGLE-2006-BLG-277 was previously analyzed by Jaroszyński et al. (2010),
but the analysis was based on only OGLE data. We, therefore, reanalyze the
event based on all combined data considering higher-order effects.

Reductions of data were done using photometry codes developed by the
individual groups, mostly based on difference image analysis (Alard & Lupton,
1998; Woźniak et al., 2001; Bramich, 2008; Albrow et al., 2009). In order to
use data sets acquired from different observatories, we readjust the error
bars. For this, we first add a quadratic error term so that the cumulative
distribution of $\chi^{2}$ ordered by magnifications is approximately linear
in data counts, and then rescale errors so that $\chi^{2}$ per degree of
freedom $(\chi^{2}/{\rm dof})$ becomes unity.

Figure 1.—  Light curve of OGLE-2006-BLG-277. In the legends indicating
observatories, the subscript of each observatory denotes the passband. The
subscript “$N$” denotes that no filter is used. The insets in the upper panel
show the enlargement of the caustic-crossing parts of the light curve. The
lower four panels show the residuals of data from four different models.

In Figures 1 and 2, we present the light curves of the individual events. Both
light curves exhibit sharp spikes that are characteristic features of caustic-
crossing binary-lens events. The spikes occur in pairs because the caustic
forms a closed curve. Usually, the inner region between two spikes has a
“U”-shape trough as is in OGLE-2006-BLG-277. For OGLE-2012-BLG-0031, the inner
region exhibits a complex pattern. Such a pattern can be produced when the
source trajectory runs approximately tangent to the fold of a caustic.

##  3\. Modeling

###  3.1. Standard Model

Figure 2.—  Light curve of OGLE-2012-BLG-0031. Notations are the same as those
in Fig. 1.

Knowing that the events were produced by binary lenses, we conduct modeling of
the observed light curves. Basic description of a binary-lens event requires 7
lensing parameters. Among them, the first three parameters describe the lens-
source approach. These parameters include the time of the closest approach of
the source to a reference position383838 For a binary lens with a projected
separation less than the Einstein radius, $s<1$, we set the reference position
of the lens as the center of mass of the binary lens. For a binary with a
separation greater than the Einstein radius, $s>1$, on the other hand, we set
the reference as the photocenter that is located at a position with an offset
$q/[s(1+q)]$ from the middle position between the two lens components. of the
binary lens, $t_{0}$, the lens-source separation at $t_{0}$ in units of the
Einstein radius, $u_{0}$, and the time required for the source to cross the
Einstein radius, $t_{\rm E}$ (Einstein time scale). The Einstein ring
represents the source image for an exact lens-source alignment and its radius
$\theta_{\rm E}$ is usually used as the length scale of lensing phenomena. The
Einstein radius is related to the physical lens parameters by $\theta_{\rm
E}=(\kappa{M}\pi_{\rm rel})^{1/2}$, where $M$ is the mass of the lens and
$\pi_{\rm rel}={\rm AU}(D^{-1}_{\rm L}-D^{-1}_{\rm S})$ is the relative lens-
source parallax. Another three lensing parameters describe the binary lens.
These parameters include the projected separation, $s$ (in units of
$\theta_{\rm E}$), the mass ratio between the binary lens components, $q$, and
the angle between the source trajectory and the binary axis, $\alpha$ (source
trajectory angle). The last parameter is the normalized source radius
$\rho_{\ast}=\theta_{\ast}/\theta_{\rm E}$, where $\theta_{\ast}$ is the
angular source radius. This parameter is needed to describe the parts of light
curves affected by finite-source effects, which are important when a source
star crosses over or approaches close to caustics formed by a binary lens.

In modeling the light curves based on the standard lensing parameters
(standard model), searches for best-fit solutions have been done in two steps.
In the first step, we identify local solutions by inspecting $\chi^{2}$
distributions in the parameter space. For this, we use both grid search and
downhill approach. We choose ($s,q,\alpha$) as grid parameters because lensing
magnifications can vary dramatically with small changes of these parameters.
By contrast, lensing magnifications vary smoothly with changes of the other
parameters, and thus we search for the solutions of these parameters by
minimizing $\chi^{2}$ using a downhill approach. We use the Markov Chain Monte
Carlo (MCMC) method for the $\chi^{2}$ minimization. In the second step, we
refine the lensing parameters for the individual local solutions by allowing
all parameters to vary. Then, the best-fit solution is obtained by comparing
$\chi^{2}$ values of the individual local solutions. We estimate the
uncertainties of the lensing parameters based on the distributions of the
parameters obtained from the MCMC chain of solutions.

Table 2Model Parameters

Model | $\chi^{2}/{\rm dof}$ | $t_{0}$ | $u_{0}$ | $t_{\rm E}$ | $s$ | $q$ | $\alpha$ | $\rho_{\ast}$ | $\pi_{{\rm E},N}$ | $\pi_{{\rm E},E}$ | $ds/dt$ | $d\alpha/dt$
---|---|---|---|---|---|---|---|---|---|---|---|---
|  | (HJD’) |  | (days) |  |  |  | ($10^{-3}$) |  |  | (${\rm yr}^{-1}$) | (${\rm yr}^{-1}$)
OGLE-2006-BLG-277 |  |  |  |  |  |  |  |  |  |  |  |
Standard | 2652.6 | 3941.620 | 0.157 | 39.13 | 1.374 | 2.600 | 1.477 | 5.83 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
| /1499 |  $\pm$0.020 |  $\pm$0.002 |  $\pm$0.08 |  $\pm$0.001 |  $\pm$0.037 |  $\pm$0.003 |  $\pm$0.02
Parallax only | 1811.0 | 3941.723 | 0.169 | 39.30 | 1.371 | 2.512 | 1.485 | 5.90 | 0.45 | 0.54 | $\cdots$ | $\cdots$
| /1497 |  $\pm$0.025 |  $\pm$0.003 |  $\pm$0.08 |  $\pm$0.001 |  $\pm$0.035 |  $\pm$0.003 |  $\pm$0.02 |  $\pm$0.07 |  $\pm$0.01
Orbit only | 1528.2 | 3943.066 | 0.170 | 38.78 | 1.347 | 2.033 | -1.485 | 5.98 | $\cdots$ | $\cdots$ | 0.73 | -0.33
| /1497 |  $\pm$0.031 |  $\pm$0.005 |  $\pm$0.07 |  $\pm$0.001 |  $\pm$0.030 |  $\pm$0.005 |  $\pm$0.02 |  $\pm$0.02 |  $\pm$0.11
Orbit+Parallax | 1511.9 | 3943.071 | -0.168 | 37.90 | 1.348 | 1.981 | 1.457 | 6.03 | 1.13 | -0.05 | 0.69 | 1.21
| /1495 |  $\pm$0.031 |  $\pm$0.005 |  $\pm$0.13 |  $\pm$0.001 |  $\pm$0.030 |  $\pm$0.006 |  $\pm$0.02 |  $\pm$0.16 |  $\pm$0.04 |  $\pm$0.03 |  $\pm$0.22
OGLE-2012-BLG-0031 |  |  |  |  |  |  |  |  |  |  |  |
Standard | 2580.5 | 6022.532 | 0.046 | 59.17 | 0.477 | 0.294 | 0.800 | 5.48 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
| /2403 |  $\pm$0.042 |  $\pm$0.001 |  $\pm$0.59 |  $\pm$0.003 |  $\pm$0.010 |  $\pm$0.009 |  $\pm$0.11
Parallax only | 2430.4 | 6022.233 | -0.047 | 56.47 | 0.510 | 0.223 | -0.739 | 5.63 | -0.29 | 0.10 | $\cdots$ | $\cdots$
| /2405 |  $\pm$0.043 |  $\pm$0.001 |  $\pm$0.66 |  $\pm$0.003 |  $\pm$0.008 |  $\pm$0.009 |  $\pm$0.11 |  $\pm$0.08 |  $\pm$0.02
Orbit only | 2422.1 | 6022.364 | 0.051 | 54.88 | 0.511 | 0.264 | 0.774 | 6.80 | $\cdots$ | $\cdots$ | 0.43 | 3.63
| /2405 |  $\pm$0.042 |  $\pm$0.001 |  $\pm$0.68 |  $\pm$0.003 |  $\pm$0.011 |  $\pm$0.009 |  $\pm$0.19 |  $\pm$0.08 |  $\pm$0.20
Orbit+Parallax | 2419.6 | 6022.350 | -0.051 | 54.58 | 0.511 | 0.268 | -0.773 | 6.81 | -0.09 | 0.03 | 0.47 | -2.98
| /2407 |  $\pm$0.042 |  $\pm$0.001 |  $\pm$0.77 |  $\pm$0.003 |  $\pm$0.010 |  $\pm$0.009 |  $\pm$0.21 |  $\pm$0.13 |  $\pm$0.02 |  $\pm$0.07 |  $\pm$0.39

Note. — HJD’=HJD-2,450,000

For magnification computations affected by finite-source effects, we use the
“map-making method” developed by Dong et al. (2006). In this method, a map of
rays for a given binary lens with a separation $s$ and a mass ratio $q$ is
constructed by using the inverse ray-shooting technique (Schneider & Weiss,
1986; Kayser et al., 1986; Wambsganss, 1997). In this technique, rays are
uniformly shot from the image plane, bent according to the lens equation, and
land on the source plane. The lens equation for a binary lens is represented
by

$\zeta=z-{m_{1}\over\overline{z}-\overline{z}_{\rm
L,1}}-{m_{2}\over\overline{z}-\overline{z}_{\rm L,2}},$ (1)

where $m_{1}={1}/(1+q)$ and $m_{2}=qm_{1}$ are the mass fractions of the
individual binary lens components, $\zeta=\xi+i\eta$, $z=x+iy$, and $z_{{\rm
L},i}=x_{{\rm L},i}+iy_{{\rm L},i}$ denote the positions of the source, image,
and lens expressed in complex notions, respectively, and $\overline{z}$
denotes the complex conjugate of $z$. With the constructed map, the finite-
source magnification for a given position of a source with a normalized radius
$\rho_{\ast}$ is computed as the ratio of the number density of rays within
the source to that on the image plane. This method saves computation time by
enabling to produce many light curves resulting from various source
trajectories based on a single map. In addition, the method enables to speed
up computation by allotting computation into multiple CPUs. We further
accelerate computation by using semi-analytic hexadecapole approximation
(Pejcha & Heyrovský, 2009; Gould, 2008) for finite magnification computations.

In our finite-source computations, we consider the limb-darkening effect of
the source star by modeling the surface brightness profile as
$S_{\lambda}\propto 1-\Gamma_{\lambda}(1-3\cos\phi/2)$, where
$\Gamma_{\lambda}$ is the linear limb-darkening coefficient, $\lambda$ is the
passband, and $\phi$ denotes the angle between the line of sight toward the
source star and the normal to the source surface. The limb-darkening
coefficients are adopted from Claret (2000) considering the source type that
is determined based on the source locations in the color-magnitude diagrams.
We find that the source star of OGLE-2006-BLG-277 is a K-type giant star. For
OGLE-2012-BLG-0031, the lensed star is located in a very reddened region,
causing difficulties in precisely characterizing the star based on its color
and brightness. Nevertheless, it is found that the source is a giant. The
adopted coefficients are $\Gamma_{V}=0.74$, $\Gamma_{R}=0.64$ and
$\Gamma_{I}=0.53$ for both events. For data sets obtained without any filter,
we choose a mean value of the $R$ and $I$ band coefficients, i.e.,
$\Gamma_{N}=(\Gamma_{R}+\Gamma_{I})/2$, where the subscript “$N$” denotes that
no filter is used.

In Table 2, we list the best-fit solutions of the lensing parameters obtained
from standard modeling for the individual events. In Figures 1 and 2, we also
present the residuals from the fits. It is found that even though the fits
basically describe the main features of the observed light curves, there exist
important residuals that last throughout both events.

###  3.2. Higher-order Effects

Long-term residuals from the standard models suggest that one needs to
consider higher-order effects in order to better describe the lensing light
curves. Since it is known that such long-term residuals are caused by the
parallax and/or lens orbital effects, we conduct additional modeling
considering both higher-order effects.

To describe parallax effects, it is necessary to include two parameters
$\pi_{{\rm E},N}$ and $\pi_{{\rm E},E}$, which represent the two components of
the lens parallax vector $\mbox{\boldmath$\pi$}_{\rm E}$ projected onto the
sky along the north and east equatorial coordinates, respectively. The
magnitude of the parallax vector, ${\pi}_{\rm E}=(\pi_{{\rm
E},N}^{2}+\pi_{{\rm E},E}^{2})^{1/2}$, corresponds to the relative lens-source
parallax scaled to the Einstein radius of the lens, i.e. $\pi_{\rm E}=\pi_{\rm
rel}/\theta_{\rm E}$ (Gould, 2004). The direction of the parallax vector
corresponds to the relative lens-source motion in the frame of the Earth at a
reference time of the event. In our modeling, we use $t_{0}$ as the reference
time. Parallax effects cause the source motion relative to the lens to deviate
from rectilinear.

To first order approximation, the lens orbital motion is described by two
parameters $ds/dt$ and $d\alpha/dt$, which represent the change rates of the
normalized binary separation and the source trajectory angle, respectively
(Albrow et al., 2000; An et al., 2002). In addition to causing the relative
lens-source motion to deviate from rectilinear, the orbital effect causes
further deviation in lensing light curves by deforming the caustic over the
course of the event due to the change of the binary separation.

Figure 3.—  Geometry of the best-fit models for OGLE-2006-BLG-277 (upper
panel) and OGLE-2012-BLG-0031 (lower panel). Small dots and closed solid
curves represent the lens positions and caustics at two different times
$t_{1}$ and $t_{2}$. Black solid curves with arrows represent source
trajectories. The size of the small empty circle at the tip of the arrow of
each source trajectory represents the source size. The abscissa and ordinate
are parallel with and perpendicular to the binary axis, respectively. All
lengths are normalized by the Einstein ring radius.

In Table 2, we list the results of modeling considering the higher-order
effects. For each event, we conduct 3 sets of additional modeling in which the
parallax effect and orbital effect are considered separately (“parallax only”
and “orbital only”) and both effects are simultaneously considered
(“orbit+parallax”). In the lower panels of Figures 1 and 2, we present the
residuals of the individual models. In Figure 3, we present the geometry of
the lens systems of the best-fit solutions, where the source trajectory with
respect to the lens components and the resulting caustics are shown. We note
that the relative lens positions and caustics vary in time due to the orbital
motion of the lens and thus we mark the positions at two different moments.

For both events, we find that the dominant second-order effect is the lens
orbital motion. The dominance of the orbital effect is evidenced by the fact
that the models considering only the orbital effect result in fits as good as
those considering both the parallax and orbital effects. It is found that the
consideration of orbital effects improves the fits by $\Delta\chi^{2}=1124.4$
and $158.4$ compared to the standard models of OGLE-2006-BLG-277 and
OGLE-2012-BLG-0031, respectively. On the other hand, the improvements by
additionally considering the parallax effect are merely $\Delta\chi^{2}=16.3$
and $2.5$ for the individual events.

Figure 4.—  Distributions of $\chi^{2}$ in the space of the parallax
parameters $\pi_{{\rm E},E}$ and $\pi_{{\rm E},N}$ for OGLE-2006-BLG-277
(upper panels) and OGLE-2012-BLG-0031 (lower panels). For each event, the
distribution in the left panel is obtained from modeling considering only the
parallax effect, while the distribution in the right panel is constructed by
considering both the orbital and parallax effects. Different contours
correspond to $\Delta\chi^{2}<1$ (red), $4$ (yellow), $9$ (green), $16$ (sky
blue), $25$ (blue), and $36$ (purple), respectively.

To be noted is that parallax effects can mimic orbital effects to some extent
for both events. We find that the improvements of the fits by the parallax
effect are $\Delta\chi^{2}=841.6$ (cf. $\Delta\chi^{2}=1124.4$ improvement by
the orbital effect) and $150.1$ (cf. $\Delta\chi^{2}=158.4$ by the orbital
effect) for OGLE-2006-BLG-277 and OGLE-2012-BLG-0031, respectively. In
addition, the values of the lens parallax determined without considering
orbital effects substantially differ from those determined by considering
orbital effects. This can be seen in Figure 4 where we present $\chi^{2}$
distributions in the space of the parallax parameters. For OGLE-2006-BLG-277,
the measured lens parallax is $\pi_{\rm E}=1.13\pm 0.16$ when both parallax
and orbital effects are considered, while $\pi_{\rm E}=0.70\pm 0.05$ when only
the parallax effect is considered. For OGLE-2012-BLG-0031, modeling
considering only parallax effects results in a lens parallax $\pi_{\rm
E}=0.31\pm 0.08$, while the lens parallax is consistent with zero in 3$\sigma$
level in the model considering additional orbital effects. These facts imply
that orbital effects can masquerade as parallax effects and thus lens parallax
values measured based on modeling not considering orbital effects can result
in wrong values. This leads to wrong determinations of physical lens
parameters because masses and distances to lenses are determined from measured
values of the lens parallax.

It was pointed out by Batista et al. (2011) and Skowron et al. (2011) that the
parallax component perpendicular to the relative lens-source motion,
$\pi_{{\rm E},\bot}$, is strongly correlated with the orbital parameter
$d\alpha/dt$, causing a degeneracy between $\pi_{{\rm E},\bot}$ and
$d\alpha/dt$. They argued that this degeneracy occurs because the lens-source
motion in the direction perpendicular to the Sun-Earth axis induces deviations
in lensing light curves similar to those induced by the rotation of the
binary-lens axis. For both events OGLE-2006-BLG-277 and OGLE-2012-BLG-0031,
the direction of the relative lens-source motion is similar to east-west
direction, and thus $\pi_{{\rm E},\bot}\sim\pi_{{\rm E},N}$. According to this
degeneracy, the lens parallax vectors estimated by the “parallax only” and the
“orbit+parallax” models should result in similar values of $\pi_{{\rm E},E}$,
while values of $\pi_{{\rm E},N}$ can be widely different. However, both the
events analyzed in this work do not conform to the previous prediction. This
implies that the parallax-orbit degeneracy is much more complex than
previously thought, and thus it is essential to study the degeneracy in all
cases where higher-order effects are detected.

We determine the physical lens parameters based on the best-fit solutions
(orbit+parallax models). For this, we first determine the Einstein radius. The
Einstein radius is determined by $\theta_{\rm E}=\theta_{\ast}/\rho_{\ast}$,
where the normalized source radius $\rho_{\ast}$ is measured from the modeling
and the angular stellar radius is determined based on the source type. The
measured Einstein radius of the lens of OGLE-2006-BLG-277 is is $\theta_{\rm
E}=1.35\pm 0.11$ ${\rm mas}$. This corresponds to the relative lens-source
proper motion $\mu=\theta_{\rm E}/t_{\rm E}=13.0\pm 1.0$ ${\rm mas}$ ${\rm
yr^{-1}}$. With the measured mass ratio between the lens components, then the
masses of the individual lens components are $M_{1}=M_{\rm tot}/(1+q)=0.049\pm
0.014$ $M_{\odot}$ and $M_{2}=qM_{\rm tot}/(1+q)=0.097\pm 0.027$ $M_{\odot}$,
respectively. Therefore, the lens is composed of a low-mass star and a brown
dwarf. The distance to the lens is $D_{\rm L}=0.60\pm 0.14$ ${\rm kpc}$. The
close distance explains the relatively high proper motion ($13.0\pm 1.0$ ${\rm
mas}$ ${\rm yr^{-1}}$). With the physical parameters combined with orbital
parameters, we evaluate the ratio of tranverse kinetic to potential energy

$\left({\rm KE\over\rm PE}\right)_{\bot}={(r_{\bot}/{\rm AU})^{2}\over
8{\pi}^{2}(M_{\rm tot}/M_{\odot})}\left[\left({1\over s}{ds\over
dt}\right)^{2}+\left({d\alpha\over dt}\right)^{2}\right],$ (2)

where $r_{\bot}$ denotes the projected binary separation (Dong et al., 2009).
The ratio should obey $\rm(KE/PE)_{\bot}\leq KE/PE<1$ for kinetically stable
binary orbit. We find $\rm(KE/PE)_{\bot}=0.20\pm 0.04$. For
OGLE-2012-BLG-0031, it is difficult to determine the physical lens parameters
not only because the source type is uncertain but also because the lens
parallax is consistent with zero.

##  4\. Summary and Conclusion

We analyzed two binary-lens events OGLE-2006-BLG-277 and OGLE-2012-BLG-0031
for which the light curves exhibit significant residuals from standard binary-
lens models. From modeling considering higher-order effects, we found that the
residuals were greatly removed by considering the effect of the lens orbital
motion. We also found that parallax effects could mimic orbital effects to
some extent and the parallax values measured not considering the orbital
effect could result in dramatically different value from true values, and thus
wrong determinations of physical lens parameters. We also found that the lens
of OGLE-2006-BLG-277 was a binary composed of a low-mass star and a brown
dwarf companion.

Work by CH was supported by Creative Research Initiative Program
(2009-0081561) of National Research Foundation of Korea. AG was supported by
NSF grant AST 1103471. The OGLE project has received funding from the European
Research Council under the European Community’s Seventh Framework Programme
(FP7/2007-2013) / ERC grant agreement no. 246678 to AU. S.D. was supported
through a Ralgh E. and Doris M. Hansmann Membership at the IAS and NSF grant
AST-0807444. DMB, MD, MH, RAS and YT would like to thank the Qatar Foundation
for support from QNRF grant NPRP-09-476-1-078. Dr. David Warren gave financial
support to Mt. Canopus Observatory.

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