,

Probing valley phenomena with gate-defined valley splitters

Juan Daniel Torres Luna1, 2, ${\dagger}$, Kostas Vilkelis1, 2, ${\ddagger}$,
Antonio L. R. Manesco2, *.

1 Qutech, Delft University of Technology, Delft 2600 GA, The Netherlands

2 Kavli Institute of Nanoscience, Delft University of Technology, Delft 2600
GA, The Netherlands

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
May 1, 2024

## Abstract

Despite many reports of valley-related phenomena in graphene and its
multilayers, current transport experiments cannot probe valley phenomena
without the application of external fields. Here we propose a gate-defined
valley splitter as a direct transport probe for valley phenomenon in graphene
multilayers. First, we show how the device works, its magnetotransport
response, and its robustness against fabrication errors. Secondly, we present
two applications for valley splitters: (i) resonant tunnelling of quantum dots
probed by a valley splitter shows the valley polarization of dot levels; (ii)
a combination of two valley splitters resolves the nature of order parameters
in mesoscopic samples.

##  1 Introduction

#### Recent progress in fabrication made valley phenomena experimentally
accessible.

The recent advances in the fabrication of graphene multilayer devices,
particularly boron nitride encapsulation [1, 2] and the use of graphite back
gates [3, 4, 5], improved the mobility and suppressed intervalley scattering
of bulk samples. Due to valley conservation in this new generation of clean
bulk graphene samples, many experimental works observed valley order arising
from spontaneous $SU(4)$ spin-valley symmetry breaking [6, 7, 8, 9, 10, 11].
In addition, the out-of-plane displacement field opens a gap in graphene
multilayers [12, 13, 14, 15], and therefore allows to fabricate gate-defined
nanodevices [16, 17, 18, 4]. The smooth potential of electrostatically-defined
devices avoids intervalley scattering caused by reflection at atomically-sharp
edges. As a consequence, valley-dependent phenomena were also observed in
nanostructures, such as the injection of valley-polarized states in quantum
point contacts [17, 19, 20, 21, 22], and valley-polarized in states quantum
dots [4, 5] with long lifetimes [23, 24].

#### Despite the recent observations, their measurements are often indirect
and require external magnetic fields.

The recent experiments on correlated graphene multilayer devices observe
various phases that suggest the presence of valley structure, such as nematic
superconductivity [9], isospin ferromagnetism [10], and correlated insulators
[11]. However, current transport experiments cannot resolve the valley
structure of the order parameter. For example, quantum oscillation experiments
cannot distinguish between valley polarized and intervalley coherent (IVC)
orders. In nanodevices, experiments probing valley-related phenomena rely on
breaking valley symmetry and therefore these probes are invasive and indirect.
For example, valley polarization of states injected from quantum point
contacts was probed with electron collimation [22] and via valley-Zeeman
splitting in quantum dots [25, 26]. Finally, transport signatures of valley
polarization using Andreev reflection [27, 28, 29, 30] are still limited by
device quality [31, 32, 33]. Thus, the only direct probes of valley phenomena
are via microscopy techniques [34, 35, 20, 11].

#### We explore the use of gate-defined valley splitters to probe valley
phenomenon in BLG.

In this manuscript, we propose the use of electrically defined valley
splitters in bilayer graphene as a magnetic field-free transport probe for
valley phenomena. Although the ingredients of this device were previously
explored both theoretically [36, 37, 38, 39] and experimentally [40, 41, 42],
previous works focused on the generation of valley-polarized currents rather
than its use as an experimental probe of valley polarization. We demonstrate
the use of valley splitters in two case uses: (i) a tool for measuring valley
polarization of confined states via resonant tunnelling, and (ii) a probe of
valley-dependent order parameters. We also evaluate the robustness of the
splitter’s efficiency against fabrication imperfections, show the requirements
for the alignment of splitting gates, and suggest changes in the device layout
to further relax the conditions.

##  2 Device

Figure 1:  Tight-binding simulations of a valley splitter. (a) Band structure
of the injecting one-dimensional channel. The presence of inversion and time-
reversal symmetry results in a valley-symmetric band structure. (b) Valley
expectation value of the scattering wave functions. The injecting states are
superpositions of states at both valleys. As they arrive at the helical edge
channels, each flavour leaves through one of the collector arms. The
difference between transmissions from upper and lower leads is proportional to
the valley polarization of the injected modes. Band structure of the (c) upper
and (d) lower collecting channels. The highlighted bands are valley-helical,
resulting in the splitting effect as injected modes encounter the collecting
arms.

#### We propose a valley splitter device which consists of a one-dimensional
injection channel and two valley-helical collector arms

We propose a bilayer graphene gate-defined device, depicted in Fig. 1, that
splits states according to their valley polarization. The devices consist of
three one-dimensional channels confined by gapped regions due to a non-zero
displacement field $D(\mathbf{r})$ as shown in Fig. 1(b). The injection
channel is confined by a region with $D>0$ and injects valley-symmetric
states. The collecting arms are valley-helical channels surrounded by regions
with opposite displacement field [36, 37, 40]. Because of the opposite sign of
the displacement field, one arm collects states at valley $K$ and the other
arm collects states at the opposite valley $K^{\prime}$. We propose the device
layout from Fig. 6(c) to achieve the required displacement field profile. The
control of the four gates provides independent control of the layer imbalance
$U\propto D$ and the chemical potential $\mu$.

#### We study the valley splitter using tight-binding simulations.

To demonstrate the transport properties of the valley splitter, we perform
numerical simulations using Kwant [43]. We implement the tight-binding model
for bilayer graphene described in Ref. [44]:

$\displaystyle\mathcal{H}=\sum_{n}\psi_{n}^{\dagger}\left[U(\mathbf{r}_{n})\mathrm{sign}(\mathbf{r}_{n}\cdot\hat{z})-\mu(\mathbf{r}_{n})+g\mu_{B}B\sigma_{z}\right]\psi_{n}+\sum_{\left\\{n,m\right\\}\in
S_{i}}\gamma_{i}e^{i\phi_{nm}}\psi_{n}^{\dagger}\psi_{m},$ (1)

where $\psi_{n}=(c_{n\uparrow},c_{n\downarrow})^{T}$, $c_{n,\sigma}$ is an
annihilation operator of an electron with spin $\sigma$ at the atomic site
$n$. The matrices $\sigma_{i}$ are Pauli matrices that act on spin degrees of
freedom. Here $\mu(\mathbf{r})$ and $U(\mathbf{r})$ are the position-dependent
chemical potential and layer imbalance, $g\approx 2$ is the $g$-factor in
bilayer graphene, $\mu_{B}$ is the Bohr magneton, and $B$ is the out-of-plane
magnetic field. Due to the magnetic field $B$, the hoppings between sites $n$
and $m$ are corrected by a Peierls phase $\phi_{nm}$. The hopping energies are
set according to Ref. [44]: each hopping energy $\gamma_{i}$ corresponds to a
set $S_{i}$ of hopping vectors between the site pairs $(n,m)$. The
$\gamma_{i}$ hoppings are schematically shown in Fig. 6(a). The electrostatic
potential set by $\mu(\mathbf{r})$ and $U(\mathbf{r})$ has a screening length
of $60\text{\,}\mathrm{nm}$ [45, 46]111To avoid the complexity of
electrostatic simulations, we pass a gaussian filter with
$\sigma=$30\text{\,}\mathrm{nm}$$ to step-functions that define the
electrostatic potential according to the layout shown in Fig. 6.. To reduce
the computational cost of the calculations, we rescale the in-plane lattice
constant to $\tilde{a}=sa$ by a factor $s=16$ and fix the hoppings accordingly
to preserve the low-energy dispersion.

Figure 2:  Magnetotransport simulations with a valley splitter. (a, c) Band
structure of the injecting arms at finite orbital field $B$ which lifts the
valley degeneracy. As a consequence, the number of propagating modes per
valley changes. (c) Transmission difference between the two helical collecting
arms. A finite difference indicates the valley polarization of the injected
modes. The transmission changes in integer steps, proportionally to the net
number of propagating modes per valley.

#### We confirm the splitter behaviour with tight-binding simulations.

We show the tight-binding simulation for the valley splitter device in Fig. 1.
In all simulations in this manuscript, we set the distance of
$100\text{\,}\mathrm{nm}$ between split gates in the injecting channel. The
layer imbalance away from the one-dimensional channels is fixed at $U=\pm 100$
meV. We confirm that the scattering wavefunctions $\psi$ are spatially split
according to their valley polarization by showing their valley expectation
value $\langle\psi|\mathcal{V}|\psi\rangle$ in Fig. 1(b). The valley operator
$\mathcal{V}$ is defined as a sublattice-dependent Haldane coupling [47, 48,
32]. The smooth electrostatic potential ensures adiabatic propagation of
wavefunctions and prevents both backscattering and valley mixing. As a result,
each injected mode is perfectly transferred to the lead with the matching
valley helicity according to the dispersions shown in Fig. 1(a, c, d).

Figure 3:  Misalignment effects on the valley helical channel. (a) Example
electrostatic potential used in the layers. The solid lines show the potential
in each layer. The dashed line shows the average potential, which corresponds
to a local modulation of the chemical potential. (b) The band structure of the
valley helical channel at $\delta x=$10\text{\,}\mathrm{nm}$$ and
$U=$50\text{\,}\mathrm{meV}$$. Because of the misalignment $\delta x$ and the
corresponding modulation of the chemical potential, non-helical bands approach
the Fermi energy. (c) The bandwidth of the valley helical states as a function
of misalignment $\delta{x}$ and layer imbalance $U$. (d) The bandwidth of the
valley helical states as a function of $\delta x$ and screening length $\chi$
with $U=$50\text{\,}\mathrm{meV}$$. Plots (a, c, d) use
$\chi=$60\text{\,}\mathrm{nm}$$.

#### We show that the transmission difference in the valley splitter measures
the valley polarization of injected modes.

We now prove that the valley splitter directly probes the valley polarization
of the injected current. We define the net valley polarization of a set of
modes $\Phi_{i}$ as

$V=\frac{\sum_{i}\langle\Phi_{i}|\mathcal{V}|\Phi_{i}\rangle}{\sum_{i}\langle\Phi_{i}|\Phi_{i}\rangle}~{}.$
(2)

Moreover, defining the transmission to arms that collect states in the valley
$K$ and $K^{\prime}$ as $T_{K}$ and $T_{K^{\prime}}$ and assuming a splitter
with perfect efficiency and adiabatic,

$T_{\alpha}=\sum_{i}\langle\Phi_{i}|P_{\mathcal{\alpha}}|\Phi_{i}\rangle~{},$
(3)

where $P_{\mathcal{\alpha}}$ is the projector for the valley $\alpha$. Thus,

$T_{V}=\frac{T_{K}-T_{K^{\prime}}}{T_{K}+T_{K^{\prime}}}\equiv V$ (4)

as long as the collecting arms are not saturated, i.e. $T_{\alpha}$ is smaller
than the number of valley-helical channels (2 for bilayer graphene). Our
numerical simulations demonstrate this result by lifting the valley degeneracy
in the injecting channel with an orbital magnetic field $B$. Because valley is
approximately a good quantum number in the Hamiltonian of Eq. 1, the imbalance
in the number of modes per valley $n_{\alpha}$ in the injector results in
$T_{V}=(n_{K}-n_{K^{\prime}})/(n_{K}+n_{K^{\prime}})$, as shown in Fig. 2. It
is important to notice that the equivalence between $T_{V}$ and $V$ does not
require valley conservation, and we explore the generality of our result in
Sec. 3.

#### The efficiency of the splitter depends on the geometry of the device.

Because one of the main fabrication challenges is the alignment between gates,
we simulate the effects of misalignments on the valley helical collecting
channels in Fig. 3. In Fig. 3(b), we observe that finite gate misalignment
$\delta{x}$ shifts bulk bands closer in energy to the valley helical states
due to a modulation of the chemical potential shown in Fig. 3(a). This shift
reduces the bandwidth of the valley helical states and reduces the energy
window for which the valley splitter is operational. To determine the regime
in which the valley splitter is operational, we define the bandwidth
$E_{\text{gap}}$ as the energy difference between the chemical potential and
the first bulk band below the valley helical state. Whenever
$E_{\text{gap}}>0$, the valley helical states are outside the chemical
potential of the system, and the valley splitter breaks down. Based on Fig.
3(c), we expect the valley splitter to tolerate misalignments of
$20-30$\mathrm{nm}$$, with a higher tolerance for smaller displacement fields.
In addition, the tolerance to misalignments is sensitive to screening length
$\chi$, and it increases with increasing screening length as shown in Fig.
3(d). To increase the tolerance against gate misalignment, we suggest
additional gates to control the charge density in the helical channels to
compensate for the local modulation of the chemical potential. We also confirm
through numerical simulations that the valley splitter is insensitive to the
angle between its arms [49].

##  3 Applications

###  3.1 Resonant tunneling

Figure 4:  Resonant tunnelling through a valley-polarized quantum dot. (a)
Device schematics showing the electrostatic potential of a quantum dot
connected to the valley splitter. (b) Low-energy dispersion of the injecting
channel at one of the valleys. (c) Total transmission as a function of the
tunnel gates $t_{\mathrm{dot}}$ and chemical potential $\mu_{\mathrm{dot}}$.
Resonant tunneling appears at $>$20\text{\,}\mathrm{meV}$$ tunnel gate values.
The vertical white line indicates the value chosen to generate panel (e). (d)
The energy levels of an isolated quantum dot as a function of an out-of-plane
magnetic field. The colour indicates the valley-polarisation of the states.
(e) $T_{V}$ as a function of an out-of-plane magnetic field and chemical
potential $\mu_{\mathrm{dot}}$.

#### We probe the valley polarization of a quantum dot directly with a valley
splitter.

An immediate application of a valley splitter is to probe the valley
polarization of quantum dot levels. We illustrate the operation of the device
with numerical simulations of resonant tunnelling through a quantum dot
connected to a valley splitter as shown in Fig. 4 (a). To define the quantum
dot, we add two additional layer imbalance $t_{\mathrm{dot}}$ regions that
provide confinement and also control the tunnelling rate. A valley degenerate
lead with dispersion shown Fig. 4 (b) injects electrons into the quantum dot
which in turn couples to the valley splitter. As $t_{\mathrm{dot}}$ increases,
we observe transmission peaks due to resonant tunnelling as a function of the
dot chemical potential $\mu_{\mathrm{dot}}$ (Fig. 4 (c)). Finally, we compare
the spectrum of an isolated quantum dot shown in Fig. 4 (d) with $T_{V}$ in
Fig. 4 (e). We observe that we not only measure the energy of the dot levels
but also their valley polarization. The small mismatch between the transport
experiment and the energy levels of a dot is due to the weak coupling between
the dot levels and the leads.

###  3.2 Probing valley-dependent order parameter

Figure 5:  A cavity with IVC probed by a double valley splitter setup. (a)
Band structure of the hole bands in bulk bilayer graphene with IVC. (b) Valley
texture of the hole pocket for a Fermi energy indicated by the dashed line in
panel (a). (c) Device layout showing two valley splitters connected to a
cavity with IVC. The colours indicate the electrostatic potential in the top
bilayer graphene layer. (d) Valley splitter’s transmission signal $T_{V}$ as a
function of the cavity’s chemical potential and magnitude of the IVC order
parameter. (e) Average of $T_{V}$ over the range of $\mu_{\mathrm{cavity}}$
shown in panel (d).

#### We propose a device to probe valley-dependent order parameters and
demonstrate with IVC.

We propose the setup shown in Fig. 5 (c) to directly probe the valley-
dependent order. The setup consists of a cavity with valley order and two
valley splitters: one injects valley-polarized current, and the other probes
the valley polarization. As polarized electrons travel through the cavity,
their valley isospin rotates according to the order parameter. The current
collected by the collector valley splitter directly relates to the rotation
electrons undergo in the cavity. As an example, we demonstrate its use by
probing a mesoscopic cavity with IVC order parameter $\Delta_{\mathrm{IVC}}$.

#### We simulate the cavity with a Kekulé onsite modulation to induce IVC

Intervalley coherence order couples the two valley degrees of freedom. We
induce IVC with a Kekulé onsite modulation to the cavity [11, 50]:

$\mathcal{H}_{\mathrm{IVC}}=\Delta_{\mathrm{IVC}}\sum_{m}\sum_{n\in\\{A2,B1\\}}\cos\left(\bm{\eta}_{m}\cdot\mathbf{r}_{n}\right)$
(5)

where $\bm{\eta}_{m}=\Delta K(\cos(2m\pi/3),\sin(2m\pi/3))$, and $\Delta K$ is
the momentum separation between valleys. The modulation only acts on the
sublattices $A2$ and $B1$, where the numbers label the first and second layers
in Fig. 6, because these sites set the low-energy manifold [44, 50]. We show
in Fig. 5(a) the valance band structure of a bulk region with a Kekulé
modulation. The modulation mixes valleys and results in avoided crossings in
the band structure. In Fig. 5 we show the $k$-space valley texture of a hole
pocket in Fig. 5(a), consistent with previous works [51, 10, 50].

#### We show that valley mixing increases with $\Delta_{\mathrm{IVC}}$.

We show the effects of the IVC cavity on the valley splitter’s transmission
signal $T_{V}$ in the upper panel of Fig. 5(d). In the absence of intervalley
coherence, $\Delta_{\mathrm{IVC}}=0$, the quasiparticles collected by the
splitter transmit only to the arm with the same valley polarization as the
injected state, thus $T_{V}=V_{\mathrm{injection}}$. As
$\Delta_{\mathrm{IVC}}$ increases, we observe that $T_{V}$ fluctuates due to
the valley-dependent dynamical phase accumulated by the quasiparticles in the
cavity. Furthermore averaging over the cavity’s chemical potential
$\mu_{\mathrm{cavity}}$ (Fig. 5(e)) we observe a monotonic increase of
transmission to the channel with opposite valley helicity. Therefore, we
conclude that $T_{V}\neq V_{\mathrm{injection}}$ indicates valley mixing in
the cavity.

#### Changing temperature and electrostatics allows us to distinguish between
internal coherence and intervalley scattering in an experiment.

At zero temperature, our scheme cannot distinguish between intervalley
coherence (caused by a spontaneous symmetry breaking) and intervalley
scattering (caused by short-range disorder). Therefore, we rely on temperature
dependence to distinguish between the two. Above the critical temperature of
the IVC phase, the cavity loses its valley polarization whereas short-range
scattering remains. Thus the absence of valley mixing above the critical
temperature would rule out intervalley scattering. Alternatively, it is
possible to control phase transitions electrostatically [6]. Thus, controlling
the correlated phases in the cavity also allows to verify the amount of valley
mixing in different phases and check for the presence of intervalley
scattering. These two checks allow us to unambiguously detect IVC orders.

##  4 Conclusion

We proposed a gate-defined valley splitter as a transport probe of valley
phenomena in graphene multilayers. Our layout tolerates gate misalignments
$\lesssim$30\text{\,}\mathrm{nm}$$, and we suggest directions for engineering
the device layout to relax the misalignment threshold even further. Moreover,
we proposed two applications of valley splitters relevant to recent
experimental progress in the field. Due to the correspondence between valley
polarization and normalized net transmission through the valley splitter arms,
we show that resonant tunnelling in gate-defined quantum dot probes directly
the polarization of the dot levels. Moreover, we show that a combination of
two valley splitters – one to inject valley-polarized states and the other to
probe valley polarization – allows us to distinguish between valley-polarized
and intervalley coherent order parameters in mesoscopic cavities.

## Author contributions

J.T. proposed to investigate gate-defined valley splitters. A.M. and J.T.
proposed possible applications. All authors conceived the layout of the
devices, performed numerical simulations and prepared the manuscript. A.M.
supervised the project.

## Acknowledgements

The authors acknowledge the inputs of Anton Akhmerov, Johanna Zijderveld, Timo
Hyart and Jose Lado, Josep Ingla-Aynés, Luca Banszerus, Valla Fatemi and
Andrea Young for useful discussions on theoretical and experimental
considerations. We thank Anton Akhmerov, Johanna Zijderveld, and Isidora Araya
Day for feedback on the manuscript.

### Funding information

The project received funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation program grant
agreement No. 828948 (AndQC).

## Data availablity

All code used in the manuscript is available on Zenodo [49].

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##  Appendix A Proposed device layout

We propose the device layout shown in Fig. 6. The bilayer graphene sample is
placed between four graphite gates, shown inf Fig. 6 (c). The four layers of
gates allow indepent control of the layer imbalance in the three regions and
the chemical potential $\mu$ in the injecting channel.

Figure 6:  (a) Lattice structure of bilayer graphene. We indicate the bottom
and top layers by indices $1$ and $2$ respectively and the sublattice $A$
($B$) by filled (empty) circles. We perform numerical simulations including
all the hoppings $\gamma_{i}$ indicated in the panel. Panels (b) and (c) show
a scheme of the valley splitter. (b) Target distribution of the displacement
field $D$. The three one-dimensional channels constitute the valley splitter.
The channel confined by two regions with $D>0$ is the injection channel. The
other two arms are valley-helical, and thus each arm only collects a single
valley species. (c) Proposal of gate layout. The setup consists of four gate
layers: two at the top, and two at the bottom of the bilayer graphene sample.
This set of gates allows for independent control of the layer imbalance $U$
and the chemical potential $\mu$.


# Characterisations for uniform amenability

Jingming Zhu and Jiawen Zhang  College of Data Science, Jiaxing University,
Jiaxing , 314001, P.R.China<EMAIL_ADDRESS>School of Mathematical
Sciences, Fudan University, 220 Handan Road, Shanghai, 200433, China.
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we provide several characterisations for uniform amenability
concerning a family of finitely generated groups. More precisely, we show that
the Hulanicki-Reiter condition for uniform amenability can be weakened in
several directions, including cardinalities of supports and certain operator
norms.

The first author is supported by National Natural Science Foundation of China
under Grant No.12071183. The second author is supported by National Natural
Science Foundation of China under Grant No.11871342.

Keywords: (Uniform) amenability, (Uniform) Property A, Hulanicki-Reiter
condition.

##  1\. Introduction

The notion of amenability for groups was introduced by von Neumann [7] as the
obstruction to the Banach–Tarski paradox [1]. Amenability admits a large
number of characterisations, and is now fundamental in many areas of
mathematics. There are a lot of excellent references on amenability and we
guide interested readers to, _e.g._ , [9, 11, 12].

On the other hand, Yu introduced the notion of Property A for discrete metric
spaces [14] to attack the coarse Baum-Connes conjecture, and hence the Novikov
conjecture in the case of groups. Property A can be regarded as a non-
equivariant version of amenability, and has gained a lot of attractions due to
its close relations with operator algebras and other areas of mathematics
(see, _e.g._ , [13]).

In this paper, we mainly focus on the case of a family of groups and study a
uniform version of amenability and Property A for the family. Roughly
speaking, a family of groups is uniformly amenable if the quantifiers in the
definition of amenability can be made uniform. More precisely, we consider the
following:

######  Definition 1.1.

Let $\\{G_{i}\\}_{i\in I}$ be a family of finitely generated groups with
finite generating set $S_{i}$ for $i\in I$. We say that
$\\{(G_{i},S_{i})\\}_{i\in I}$ is _uniformly amenable_ if for any
$\varepsilon>0$ there exists $D>0$ and a family of functions
$\\{f_{i}:G_{i}\to[0,1]\\}_{i\in I}$ such that $\sum_{\gamma\in
G_{i}}|f_{i}(\gamma)|^{2}=1$, $\sum_{\gamma\in
G_{i}}|f_{i}(\gamma)-f_{i}(\gamma s)|^{2}\leq\varepsilon^{2}$ for any $s\in
S_{i}$ and $\mathrm{supp}(f_{i})\subseteq B(1_{G_{i}},D)$.

The main contribution of this paper is to provide several characterisations
for uniform amenability. To be more precise, we first consider the general
case and obtain two conditions equivalent to uniform amenability (see Theorem
3.1), which indicate that the uniform boundedness on the diameter of
$\mathrm{supp}(f_{i})$ in Definition 1.1 can be relaxed to the uniform
boundedness on their cardinalities, or even to that on the $\ell^{1}$-norm of
$f_{i}$. Furthermore, we realise that when the group $G_{i}$ is amenable, the
square of the $\ell^{1}$-norm of $f_{i}$ coincides with the operator norm of
certain linear bounded operator on $\ell^{2}(G_{i})$ associated to the
function $f_{i}$ (see Proposition 3.6). Therefore, in this special case we
obtain two extra characterisations for uniform amenability, one of which also
relates to uniform Property A (see Theorem 3.8). Applying our main results to
a sequence of finite groups, we obtain characterisations for Property A on
their coarse disjoint union (see Corollary 3.9).

The paper is organised as follows: After recalling the background notions and
notation in Section 2, we state and prove our main results in Section 3.
Finally, we give some couterexamples and open questions in Section 4.

### Acknowledgements

The first-named author would like to thank Prof. Bolin Ma for helpful
discussions. The second-named author would like to thank Prof. Yijun Yao for
helpful discussions. We would also like to thank the anonymous referee for
pointing out a gap in the previous version.

##  2\. Preliminaries

Here we collect some necessary notions for this paper. Readers are guided to
[8, 13] for general background and more details.

###  2.1. Notions from coarse geometry

Let us start with some basic notions.

######  Definition 2.1.

Let $(X,d)$ be a discrete metric space.

  1. (1)

For $x\in X$ and $R>0$, denote the closed ball $B(x,R):=\\{y\in X:d(x,y)\leq
R\\}$.

  2. (2)

We say that $(X,d)$ has _bounded geometry_ if $\sup_{x\in X}\sharp B(x,R)$ is
finite for any $R>0$. Here we use the notation $\sharp A$ to denote the
cardinality of the subset $A\subseteq X$.

  3. (3)

For a subset $A\subseteq X$ and $R>0$, denote $\mathcal{N}_{R}(A):=\\{x\in
X:d(x,A)\leq R\\}$ the $R$-neighborhood of $A$ in $X$. Also denote
$\partial_{R}A:=\\{x\in X\setminus A:d(x,A)\leq R\\}$ the $R$-boundary of $A$
in $X$.

  4. (4)

For $R>0$, we say that a subset $A\subseteq X$ is _$R$ -connected_ if for any
$x,y\in A$ there exists a sequence $x_{0}=x,x_{1},x_{2},\cdots,x_{n}=y$ in $A$
such that $d(x_{i},x_{i+1})\leq R$ for $i=0,1,\cdots,n-1$.

  5. (5)

For a subset $A\subseteq X$, we denote $\chi_{A}$ the characteristic function
of $A$. For $x\in X$, also denote $\delta_{x}:=\chi_{\\{x\\}}$.

######  Definition 2.2.

Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces and $f:X\to Y$ be a map. We
say that $f$ is a _coarse embedding_ if there exist functions
$\rho_{\pm}:[0,\infty)\to[0,\infty)$ with
$\lim_{t\to+\infty}\rho_{\pm}(t)=+\infty$ such that for any $x,y\in X$ we have

$\rho_{-}(d_{X}(x,y))\leq d_{Y}(f(x),f(y))\leq\rho_{+}(d_{X}(x,y)).$

If additionally there exists $C>0$ such that $Y=\mathcal{N}_{C}(f(X))$, then
we say that $f$ is a _coarse equivalence_ and $(X,d_{X}),(Y,d_{Y})$ are
_coarsely equivalent_.

Discrete groups provide a large class of interesting examples of metric spaces
with bounded geometry (see, _e.g._ , [8, Chapter 1]). For a finitely generated
group $G$ with a finite generating set $S$, there is the _word length function
$\ell_{S}$_ defined by

$\ell_{S}(g)=\min\\{k:g=s_{i_{1}}^{\pm}s_{i_{2}}^{\pm}\cdots
s_{i_{k}}^{\pm}\mbox{~{}where~{}}s_{i_{j}}\in S\mbox{~{}for~{}}1\leq j\leq
k\\}.$

Note that $\ell_{S}$ is a proper function on $G$ in the sense that
$\sharp\\{g:\ell_{S}(g)\leq R\\}$ is finite for all $R\geq 0$. This leads to a
discrete left-invariant metric $d_{S}$ on $G$ by
$d_{S}(g,h):=\ell_{S}(g^{-1}h)$, which is clearly of bounded geometry. We call
$d_{S}$ the _word length metric_ on $G$ (associated to $S$). Gromov [4]
observed that different finite generating sets $S$ give coarsely equivalent
word length metrics on $G$ (see also [8, Theorem 1.3.12]).

Given a sequence of finite metric spaces, we can build a single metric space
encoding the information of coarse geometry of the sequence in a uniform way:

######  Definition 2.3.

Let $\\{(X_{n},d_{n})\\}_{n\in\mathbb{N}}$ be a sequence of finite metric
spaces. _A coarse disjoint union_ of $\\{X_{n}\\}_{n}$ is a metric space
$(X,d)$ such that $X=\bigsqcup_{n}X_{n}$ as a set and the metric $d$ coincides
with $d_{n}$ on each $X_{n}$ and satisfies:

$d(X_{n},X_{m})\geq
n+m+\mathrm{diam}(X_{n})+\mathrm{diam}(X_{m})\quad\mbox{for~{}each~{}}n\neq
m\mbox{~{}in~{}}\mathbb{N}.$

Note that the metric $d$ in Definition 2.3 is _not_ uniquely determined,
however, it is easy to see that they are mutually coarsely equivalent to each
other.

Also recall the following notion:

######  Definition 2.4.

Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces and $f:X\to Y$ be a map.
Given $\varepsilon,R>0$, we say that the map $f$ has _$(\varepsilon,R)$
-variation_ if for any $x,y\in X$ with $d_{X}(x,y)\leq R$, then
$d_{Y}(f(x),f(y))\leq\varepsilon$.

###  2.2. Amenability

The notion of amenability was originally introduced by von Neumann [7] as the
obstruction to the Banach-Tarski paradox. Here we use the following equivalent
definition due to Følner [2, 3]:

######  Definition 2.5.

Let $G$ be a finitely generated group equipped with a word length metric. We
say that $G$ is _amenable_ if for any $R>0$ and $\varepsilon>0$, there exists
a finite subset $F\subseteq G$ such that

$\frac{\sharp\partial_{R}F}{\sharp F}\leq\varepsilon.$

We also need an analytic characterisation for amenability. For a set $X$ and
$p\in[1,\infty)$, we denote

$\ell^{p}(X):=\big{\\{}f:X\to\mathbb{C}~{}\big{|}~{}\sum_{x\in
X}|f(x)|^{p}<\infty\big{\\}}.$

For $f\in\ell^{p}(X)$, we endow the $\ell^{p}$-norm by $\|f\|_{p}:=(\sum_{x\in
X}|f(x)|^{p})^{1/p}$. Also denote

$\ell^{p}(X)_{1,+}=\big{\\{}f\in l^{p}(X):\|f\|_{p}=1\text{ and }f(x)\geq
0\text{ for any }x\in X\big{\\}}.$

Given a group $G$, a function $f:G\to\mathbb{C}$ and $\gamma\in G$, denote the
_right $\gamma$-translation_ of $f$ by $\gamma\cdot f$. In other words,
$(\gamma\cdot f)(h)=f(h\gamma)$ for any $h\in G$. We also need to consider the
_left $\gamma$-translation_ of $f$ defined by $(\gamma\star
f)(h):=f(\gamma^{-1}h)$ for $h\in G$.

The following result was essentially due to Hulanicki and Reiter111They
originally consider the $\ell^{1}$-case, while this is easily seen to be
equivalent to the $\ell^{2}$-case shown in Proposition 2.6 using the Mazur
maps (see, _e.g._ , [13, Definition (Theorem) 6.1.1]). [6, 10]:

######  Proposition 2.6 (Hulanicki-Reiter condition).

Let $G$ be a finitely generated group with finite generating set $S$ and the
associated word length metric. Then the following are equivalent:

  1. (1)

$G$ is amenable;

  2. (2)

For any $\varepsilon>0$, there exists a function $f\in\ell^{2}(G)_{1,+}$ such
that $\|f-s\cdot f\|_{2}\leq\varepsilon$ for any $s\in S$ and
$\mathrm{supp}(f)$ is finite.

We remark that since word length metrics on groups are indeed graph metrics on
their Cayley graphs, condition (2) above is also equivalent to the following:
for any $\varepsilon>0$ and $R>0$, there exists a function
$f\in\ell^{2}(G)_{1,+}$ such that $\|f-\gamma\cdot f\|_{2}\leq\varepsilon$ for
any $\gamma\in G$ with $\ell_{S}(\gamma)\leq R$ and $\mathrm{supp}(f)$ is
finite. This form resembles the original Hulanicki-Reiter condition for the
$\ell^{1}$-case.

As mentioned in Section 1, our main focus will be the notion of uniform
amenability for a family of groups (see Definition 1.1). Roughly speaking, it
requires that condition (2) in Proposition 2.6 holds uniformly for the given
family. In fact, this is also equivalent to a uniform version of the Følner
condition (Definition 2.5):

######  Proposition 2.7.

Let $\\{G_{i}\\}_{i\in I}$ be a family of finitely generated groups with
finite generating set $S_{i}$ such that $\sup_{i}\sharp S_{i}<\infty$ and the
associated word length metric. Then the following are equivalent:

  1. (1)

$\\{(G_{i},S_{i})\\}_{i\in I}$ is uniformly amenable, _i.e._ , for any
$\varepsilon>0$ there exists $D>0$ and a family of functions
$\\{f_{i}\\}_{i\in I}$ such that each $f_{i}\in\ell^{2}(G_{i})_{1,+}$
satisfies $\|f_{i}-s\cdot f_{i}\|_{2}\leq\varepsilon$ for $s\in S_{i}$ and
$\mathrm{supp}(f_{i})\subseteq B(1_{G_{i}},D)$.

  2. (2)

For any $R>0$ and $\varepsilon>0$, there exists $D^{\prime}>0$ and a family of
finite subsets $\\{F_{i}\subseteq G_{i}\\}_{i\in I}$ such that
$\sharp\partial_{R}F_{i}/\sharp F_{i}\leq\varepsilon$ and $F_{i}\subseteq
B(1_{G_{i}},D^{\prime})$.

The proof follows directly from that of Proposition 2.6 (see, _e.g._ , [8,
Theorem 3.2.2]) while noticing that the quantifiers can be uniformly
controlled thanks to the assumption of $\sup_{i}{\sharp S_{i}}<\infty$, hence
we omit the details. It is worth noticing that the condition $\sup_{i}{\sharp
S_{i}}<\infty$ is also equivalent to that the coarse disjoint union $\bigsqcup
G_{i}$ has bounded geometry.

We also remark that changing the generating sets $S_{i}$ might change the
coarse geometry of the family $\\{G_{i}\\}_{i\in I}$, since the control
function $\rho_{\pm}$ in Definition 2.2 might alter according to $i\in I$.
Hence when talking about uniform amenability, we have to fix a generating set
for each group in the family.

###  2.3. Property A

The notion of Property A was introduced by Yu in his celebrated work [14] to
attack the coarse Baum-Connes conjecture. Intuitively speaking, Property A is
a non-equivariant version of amenability, hence these two notions enjoy
analogous descriptions.

######  Definition 2.8.

A discrete metric space $(X,d)$ is said to have property A if for any
$\varepsilon,R>0$ there exists a collection of finite subsets
$\\{A_{x}\\}_{x\in X}$ in $X\times\mathbb{N}$ and a constant $S>0$ such that

  1. (1)

$\frac{\sharp(A_{x}\Delta A_{y})}{\sharp(A_{x}\cap A_{y})}\leq\varepsilon$ for
$x,y\in X$ with $d(x,y)\leq R$;

  2. (2)

$A_{x}\subset B(x,S)\times\mathbb{N}$ for any $x\in X$.

Analogous to the case of amenability, we also need the following analytic
characterisation for Property A due to Higson and Roe222Again note that the
$\ell^{2}$-case shown in Proposition 2.9 is equivalent to the $\ell^{1}$-case
using the Mazur maps (see, _e.g._ , [13, Theorem 1.2.4]). [5].

######  Proposition 2.9 (Higson-Roe condition).

Let $(X,d)$ be a discrete metric space of bounded geometry. Then the following
are equivalent:

  1. (1)

$(X,d)$ has Property A;

  2. (2)

For any $\varepsilon>0$ and $R>0$, there exists a map
$\xi:X\to\ell^{2}(X)_{1,+}$ such that $\xi$ has $(\varepsilon,R)$-variation
and there exists a constant $S>0$ such that $\mathrm{supp}(\xi(x))\subset
B(x,S)$ for any $x\in X$.

######  Remark 2.10.

Note that when $X$ comes from a finitely generated group equipped with a word
length metric, then condition (2) above is also equivalent to the following:
for any $\varepsilon>0$, there exists a map $\xi:X\to\ell^{2}(X)_{1,+}$ such
that $\xi$ has $(\varepsilon,1)$-variation and there exists a constant $S>0$
such that $\mathrm{supp}(\xi(x))\subset B(x,S)$ for any $x\in X$.

It is well-known that for a finitely generated group $G$ with a word length
metric, amenability implies Property A. This can be easily seen either from
the definitions or the analytic characterisations.

Similarly, we would also like to consider the following notion of Property A
for the family case.

######  Definition 2.11.

Let $\\{X_{i}\\}_{i\in I}$ be a family of discrete metric spaces. We say that
$\\{X_{i}\\}_{i\in I}$ has _uniform Property A_ if for any $\varepsilon>0$ and
$R>0$, there exists $D>0$ and a family of functions
$\\{\xi_{i}:X_{i}\to\ell^{2}(X_{i})_{1,+}\\}_{i\in I}$ such that each
$\xi_{i}$ has $(\varepsilon,R)$-variation and
$\mathrm{supp}(\xi_{i}(x))\subset B(x,D)$ for any $x\in X_{i}$ and $i\in I$.

##  3\. Main results

In this section, we present several versions of the main results under
different hypotheses and provide detailed proofs. Let us start with the
general case.

######  Theorem 3.1.

Let $\\{G_{i}\\}_{i\in I}$ be a family of finitely generated groups with
finite generating set $S_{i}$ for each $i$. Consider the following conditions:

  1. (1)

For any $\varepsilon>0$, there exists $M>0$ and a family of functions
$\\{f_{i}\\}_{i\in I}$ such that each
$f_{i}\in\ell^{2}(G_{i})_{1,+}\cap\ell^{1}(G_{i})$ satisfies $\|f_{i}-s\cdot
f_{i}\|_{2}\leq\varepsilon$ for $s\in S_{i}$ and $\|f_{i}\|_{1}\leq M$.

  2. (2)

For any $\varepsilon>0$, there exists $N>0$ and a family of functions
$\\{f_{i}\\}_{i\in I}$ such that each $f_{i}\in\ell^{2}(G_{i})_{1,+}$
satisfies $\|f_{i}-s\cdot f_{i}\|_{2}\leq\varepsilon$ for $s\in S_{i}$ and
$\sharp\mathrm{supp}(f_{i})\leq N$.

  3. (3)

The family $\\{(G_{i},S_{i})\\}_{i\in I}$ is uniformly amenable.

Then (1) $\Rightarrow$ (2) $\Rightarrow$ (3). Additionally, if there exists
$L>0$ such that $\sharp S_{i}\leq L$ for each $i$, then the three conditions
above are equivalent.

Note that when the index set $I$ is finite, Theorem 3.1 holds trivially by an
elementary approximation argument. Hence the main contribution of this result
is on the uniformality of parameters.

To prove Theorem 3.1, we need the following auxiliary lemma:

######  Lemma 3.2.

Let $X$ be a set and $f:X\to[0,1]$ be a function in $\ell^{2}(X)_{1,+}$ with
finite support. Then for any $\varepsilon\in[0,1]$, there exists $c\geq 0$
such that $\|f_{c}\|_{2}=\varepsilon$ where $f_{c}(x):=\min\\{f(x),c\\}$ for
$x\in X$.

###### Proof.

It is clear that the function $F(c):=\|f_{c}\|_{2}$ for $c\in[0,1]$ is
continuous since $\mathrm{supp}(f)$ is finite. Note that $F(0)=0$ and
$F(1)=1$, hence the result follows from the Intermediate Value Theorem for
continuous functions. ∎

###### Proof of Theorem 3.1.

“(1) $\Rightarrow$ (2)”: Given $\varepsilon>0$, take $M>0$ and a family of
functions $\\{f_{i}\\}_{i\in I}$ satisfying condition (1) for $\varepsilon$
and $M$. Approximating each $f_{i}$ by finitely support functions, we can
assume that each $f_{i}$ has finite support. Set
$\hat{\varepsilon}:=\frac{\varepsilon}{2+2\varepsilon}$. By Lemma 3.2, there
exists $c_{i}>0$ for each $i\in I$ such that
$\|g_{i}\|_{2}=\hat{\varepsilon}$, where
$g_{i}(\gamma):=\min\\{f_{i}(\gamma),c_{i}\\}$ for $\gamma\in G_{i}$. Set
$\mathrm{supp}_{\hat{\varepsilon}}(f_{i}):=\\{\gamma\in
G_{i}:f_{i}(\gamma)\geq c_{i}\\}$.

We claim: $\sharp\mathrm{supp}_{\hat{\varepsilon}}(f_{i})$ is uniformly
bounded with respect to $i\in I$. In fact, we have:

$\hat{\varepsilon}^{2}=\sum_{\gamma\in G_{i}}g_{i}(\gamma)^{2}\leq
c_{i}\cdot\sum_{\gamma\in G_{i}}g_{i}(\gamma)$

since $g_{i}(\gamma)\leq c_{i}$ for $\gamma\in G_{i}$. Then we have

$M\geq\|f_{i}\|_{1}=\sum_{\gamma\in G_{i}}f_{i}(\gamma)\geq\sum_{\gamma\in
G_{i}}g_{i}(\gamma)\geq\frac{\hat{\varepsilon}^{2}}{c_{i}},$

which implies that $c_{i}\geq\hat{\varepsilon}^{2}/M$ for all $i\in I$. On the
other hand, we have

$\hat{\varepsilon}^{2}=\sum_{\gamma\in G_{i}}g_{i}(\gamma)^{2}\geq
c_{i}^{2}\cdot\sharp\mathrm{supp}_{\hat{\varepsilon}}(f_{i})\geq\frac{\hat{\varepsilon}^{4}}{M^{2}}\cdot\sharp\mathrm{supp}_{\hat{\varepsilon}}(f_{i}).$

Hence we obtain that $\sharp\mathrm{supp}_{\hat{\varepsilon}}(f_{i})\leq
M^{2}/\hat{\varepsilon}^{2}$, which conclude the claim.

Back to the proof, let us define
$h_{i}:=\frac{f_{i}-g_{i}}{\|f_{i}-g_{i}\|_{2}}$ for $i\in I$. Then
$\mathrm{supp}(h_{i})\subseteq\mathrm{supp}_{\hat{\varepsilon}}(f_{i})$, which
has uniformly bounded cardinality by the claim above. Also for any $s\in
S_{i}$, we have

$\|h_{i}-s\cdot h_{i}\|_{2}\leq\frac{\|f_{i}-s\cdot f_{i}\|_{2}+\|g_{i}-s\cdot
g_{i}\|_{2}}{\|f_{i}-g_{i}\|_{2}}\leq\frac{\varepsilon+2\hat{\varepsilon}}{1-\hat{\varepsilon}}=2\varepsilon.$

Hence we conclude condition (2).

“(2) $\Rightarrow$ (3)”: Given $\varepsilon>0$, take $N>0$ and a family of
functions $\\{f_{i}\\}_{i\in I}$ satisfying condition (2) for $\varepsilon$
and $N$. For each $i\in I$, let $\\{U_{i,k}\\}_{k=1}^{N_{i}}$ be the family of
$2$-connected components of $\mathrm{supp}(f_{i})$, and denote
$f_{i}^{(k)}:=f_{i}\cdot\chi_{U_{k}}$ for each $k$. Note that the family
$\\{U_{i,k}\\}_{k=1}^{N_{i}}$ is mutually at least $3$-apart from each other,
hence for any $s\in S_{i}$ we have:

$\varepsilon^{2}\geq\|f_{i}-s\cdot
f_{i}\|_{2}^{2}=\sum_{k=1}^{N_{i}}\|f_{i}^{(k)}-s\cdot f_{i}^{(k)}\|_{2}^{2}.$

Also note that $N_{i}\leq N$ and for each $i\in I$ and $k=1,\cdots,N_{i}$,
there exists $\alpha_{i,k}\in G_{i}$ such that $U_{i,k}\subseteq
B(\alpha_{i,k},2N)$.

If $\mathrm{diam}(G_{i})\leq 4(N+1)N$, then we simply take
$g_{i}:=\frac{1}{(\sharp G_{i})^{1/2}}\chi_{G_{i}}$. Otherwise, choose a point
$\beta_{i}\in G_{i}$ such that $d_{i}(\beta_{i},1_{G_{i}})=4(N+1)N$ and a
finite sequence of elements
$1_{G_{i}}=\alpha^{\prime}_{i,1},\alpha^{\prime}_{i,2},\cdots,\alpha^{\prime}_{i,N_{i}}$
on a geodesic connecting $1_{G_{i}}$ and $\beta_{i}$ such that
$d_{i}(\alpha^{\prime}_{i,k},\alpha^{\prime}_{i,l})=4(N+1)|k-l|$ for
$k,l=1,\cdots,N_{i}$ (note that we do not require
$\alpha^{\prime}_{i,N_{i}}=\beta_{i}$). Set
$\gamma_{i,k}:=\alpha^{\prime}_{i,k}\alpha_{i,k}^{-1}$ for $k=1,\cdots,N_{i}$.
Then we have the following:

  1. (i)

$d(\gamma_{i,k}\cdot U_{i,k},\gamma_{i,l}\cdot U_{i,l})\geq 3$ for any $k\neq
l$ in $\\{1,2,\cdots,N_{i}\\}$;

  2. (ii)

the union $\bigcup_{k=1}^{N_{i}}\gamma_{i,k}\cdot U_{i,k}$ is contained in
$B(1_{G_{i}},4(N+2)N)$.

Set $g_{i}:=\sum_{k=1}^{N_{i}}\gamma_{i,k}\star f_{i}^{(k)}$, where
$\gamma_{i,k}\star f_{i}^{(k)}$ is the left $\gamma_{i,k}$-translation of
$f_{i}^{(k)}$ defined in Section 2.2. It follows from the construction (see
condition (i) above) that

$\|g_{i}\|^{2}_{2}=\sum_{k=1}^{N_{i}}\|\gamma_{i,k}\star
f_{i}^{(k)}\|_{2}^{2}=\sum_{k=1}^{N_{i}}\|f_{i}^{(k)}\|_{2}^{2}=\|f_{i}\|^{2}_{2}=1,$

and

$\displaystyle\|g_{i}-s\cdot g_{i}\|^{2}_{2}$
$\displaystyle=\sum_{k=1}^{N_{i}}\|\gamma_{i,k}\star
f_{i}^{(k)}-s\cdot(\gamma_{i,k}\star
f_{i}^{(k)})\|^{2}_{2}=\sum_{k=1}^{N_{i}}\|\gamma_{i,k}\star(f_{i}^{(k)}-s\cdot
f_{i}^{(k)})\|^{2}_{2}$ $\displaystyle=\sum_{k=1}^{N_{i}}\|f_{i}^{(k)}-s\cdot
f_{i}^{(k)}\|^{2}_{2}\leq\varepsilon^{2}$

for $s\in S_{i}$, where we use the commutativity of the left translation with
the right translation in the second equality. Finally, it follows from
condition (ii) above that

$\mathrm{supp}(g_{i})\subseteq\bigcup_{k=1}^{N_{i}}\gamma_{i,k}\cdot
U_{i,k}\subseteq B(1_{G_{i}},4(N+2)N).$

Hence we conclude the proof. ∎

######  Remark 3.3.

As mentioned above, the main contribution of Theorem 3.1 is to provide
quantitative relations among these conditions. Indeed it follows from the
proof that for $\varepsilon,M$ in condition (1), we may choose
$2\varepsilon,N=\frac{4M^{2}(1+\varepsilon)^{2}}{\varepsilon^{2}}$ for
condition (2). For $\varepsilon^{\prime},N^{\prime}$ in condition (2), we may
choose $\varepsilon^{\prime},4(N^{\prime}+2)N^{\prime}$ for condition (3).

Note that in Theorem 3.1, we do not require the assumption of amenability on
each $G_{i}$ at the very beginning, while this becomes an outcome of
conditions therein. In the following, we will explore the situation when we
already know that each $G_{i}$ is amenable, _e.g._ , $G_{i}$ is finite.

Let us first recall the following, which is the $\ell^{2}$-version of [8,
Theorem 4.4.4].

######  Lemma 3.4.

Let $G$ be a finitely generated amenable group with finite generating set $S$.
Then for any $\varepsilon>0$ and $D>0$, the following are equivalent:

  1. (1)

There exists a function $f\in\ell^{2}(G)_{1,+}$ such that $\|f-s\cdot
f\|_{2}\leq\varepsilon$ for $s\in S$ and $\mathrm{supp}(f)\subseteq
B(1_{G},D)$.

  2. (2)

There exists a map $\xi:G\to\ell^{2}(G)_{1,+}$ such that $\xi$ has
$(\varepsilon,1)$-variation and $\mathrm{supp}(\xi_{x})\subseteq B(x,D)$.

###### Proof.

Applying the Mazur maps

$M_{2,1}:\ell^{2}(G)_{1,+}\to\ell^{1}(G)_{1,+}\mbox{~{}defined~{}by~{}}(M_{2,1}f)(x):=f(x)^{2},x\in
G$

and

$M_{1,2}:\ell^{1}(G)_{1,+}\to\ell^{2}(G)_{1,+}\mbox{~{}defined~{}by~{}}(M_{1,2}f)(x):=\sqrt{f(x)},x\in
G,$

it is easy to see (_e.g._ , [13, Theorem 1.2.4, (2) $\Rightarrow$ (3)]) that
condition (1) and (2) are equivalent to their $\ell^{1}$-counterparts (_i.e._
, replacing $\ell^{2}(G)_{1,+}$ by $\ell^{1}(G)_{1,+}$ therein) without
changing the supports, respectively. Now we conclude the result thanks to [8,
Theorem 4.4.4]. ∎

Note that the emphasis of the above lemma is on the parameter $D$. Hence it
helps to bridge the condition of uniform amenability in Theorem 3.1 with
uniform Property A, which will be packaged into Theorem 3.8.

On the other hand, we would also like to explore an alternative description
for condition (1) in Theorem 3.1 in terms of operator norms under the
assumption of amenability, where the operator is defined as follows:

######  Definition 3.5.

For a finitely generated group $G$ and a function
$f\in\ell^{1}(G)\cap\ell^{2}(G)=\ell^{1}(G)$ with range in $[0,1]$, the
_associated kernel of $f$_ is the $G$-by-$G$ matrix $K_{f}$ defined by
$K_{f}(x,y):=\langle x\star f,y\star f\rangle_{\ell^{2}(G)}$ for $x,y\in G$.

For any $G$-by-$G$ matrix $K$ with complex values, we can associate an
operator $T_{K}$ on the space $C_{c}(G)$ of finitely supported complex-valued
functions on $G$ to itself by formal multiplication:

$\left(T_{K}(h)\right)(\gamma):=\sum_{\gamma^{\prime}\in
G}K(\gamma,\gamma^{\prime})h(\gamma^{\prime})$

for any $h\in C_{c}(G)$ and $\gamma\in G$. To simplify the notation, we denote
$T_{f}$ instead of $T_{K_{f}}$ for $f\in\ell^{1}(G)$.

The following result relates the operator norm of $T_{f}$ with the
$\ell^{1}$-norm of $f$.

######  Proposition 3.6.

For a finitely generated amenable group $G$ and a function $f\in\ell^{1}(G)$
with range in $[0,1]$, the operator $T_{f}$ can be continuously extended to a
bounded linear operator on $\ell^{2}(G)$ with operator norm equals
$\|f\|_{1}^{2}$.

###### Proof.

Given $\alpha=\sum_{\gamma\in G}\alpha(\gamma)\delta_{\gamma}\in C_{c}(G)$, we
have:

$\displaystyle\langle$ $\displaystyle
T_{f}\alpha,\alpha\rangle_{\ell^{2}(G)}=\sum_{x,y\in
G}\alpha(x)\overline{\alpha(y)}K_{f}(y,x)=\sum_{x,y\in
G}\alpha(x)\overline{\alpha(y)}\cdot\langle y\star f,x\star
f\rangle_{\ell^{2}(G)}$ $\displaystyle=\sum_{x,y,z\in
G}\alpha(x)\overline{\alpha(y)}f(x^{-1}z)f(y^{-1}z)=\sum_{z\in
G}\left(\sum_{x\in G}\alpha(x)f(x^{-1}z)\right)\cdot\overline{\left(\sum_{y\in
G}\alpha(y)f(y^{-1}z)\right)}$ $\displaystyle=\sum_{z\in G}\left|\sum_{x\in
G}\alpha(x)f(x^{-1}z)\right|^{2}=\|\alpha\ast f\|_{2}^{2},$

where $\alpha\ast f$ is the standard convolution of functions on groups. Note
that $\alpha\in C_{c}(G)$ and $f\in\ell^{1}(G)$, hence it follows from Young’s
inequality that

$\|\alpha\ast f\|_{2}\leq\|\alpha\|_{2}\cdot\|f\|_{1}.$

Therefore using the polarization identity, $T_{f}$ can be extended to a
bounded linear operator on $\ell^{2}(G)$, still denoted by $T_{f}$. It is
obvious that $T_{f}$ is symmetric, hence we obtain:

$\|T_{f}\|=\sup\big{\\{}|\langle
T_{f}\alpha,\alpha\rangle|:\alpha\in\ell^{2}(G)\mbox{~{}with~{}}\|\alpha\|_{2}=1\big{\\}}\leq\|f\|_{1}^{2}.$

Since $G$ is amenable, we can choose a sequence of Følner sets
$\\{F_{n}\\}_{n\in\mathbb{N}}$, _i.e._ , each $F_{n}$ is a finite subset in
$G$ such that $F_{n}\subset F_{n+1}$, $\bigcup_{n\in\mathbb{N}}F_{n}=G$ and

$\frac{\sharp\big{(}(F_{n}\cdot g)\Delta F_{n}\big{)}}{\sharp F_{n}}\to
0\mbox{~{}for~{}each~{}}g\in G.$

It follows that for any $\varepsilon>0$, there exists $m\in\mathbb{N}$ such
that $\sum_{x\in F_{m}}f(x)\geq\|f\|_{1}-\varepsilon$. For the finite subset
$F_{m}$, the Følner condition above provides an $n\in\mathbb{N}$ such that

$\frac{\sharp(F_{n}\setminus F_{n}g)}{\sharp
F_{n}}\leq\frac{\varepsilon}{\sharp F_{m}}$

for each $g\in F_{m}$. Set $F_{n,m}:=\\{x\in G:F_{m}\subseteq
F_{n}^{-1}x\\}=\bigcap_{g\in F_{m}}F_{n}g$. Then we have

$\frac{\sharp(F_{n}\setminus F_{n,m})}{\sharp
F_{n}}=\frac{\sharp\left(\bigcup_{g\in F_{m}}(F_{n}\setminus
F_{n}g)\right)}{\sharp F_{n}}\leq\frac{\sum_{g\in F_{m}}\sharp(F_{n}\setminus
F_{n}g)}{\sharp F_{n}}\leq\varepsilon,$

which implies that

$\frac{\sharp F_{n,m}}{\sharp F_{n}}\geq\frac{\sharp(F_{n,m}\cap
F_{n})}{\sharp F_{n}}=1-\frac{\sharp(F_{n}\setminus F_{n,m})}{\sharp
F_{n}}\geq 1-\varepsilon.$

Now we have:

$\displaystyle\|T_{f}\|$ $\displaystyle\geq\left\|\frac{1}{(\sharp
F_{n})^{1/2}}\chi_{F_{n}}\ast f\right\|_{2}^{2}=\frac{1}{\sharp
F_{n}}\sum_{x\in G}\left|\sum_{y\in
G}\chi_{F_{n}}(xy^{-1})f(y)\right|^{2}=\frac{1}{\sharp F_{n}}\sum_{x\in
G}\left|\sum_{y\in F_{n}^{-1}x}f(y)\right|^{2}$
$\displaystyle\geq\frac{1}{\sharp F_{n}}\sum_{x\in F_{n,m}}\left|\sum_{y\in
F_{n}^{-1}x}f(y)\right|^{2}\geq\frac{\sharp F_{n,m}}{\sharp
F_{n}}\cdot(\|f\|_{1}-\varepsilon)^{2}\geq(1-\varepsilon)\cdot(\|f\|_{1}-\varepsilon)^{2}.$

Letting $\varepsilon\to 0$, we conclude that the operator norm of $T_{f}$
coincides with $\|f\|_{1}^{2}$. ∎

Thanks to Proposition 3.6, we introduce the following notion:

######  Definition 3.7.

For a finitely generated group $G$ and a function $f\in\ell^{1}(G)$ with range
in $[0,1]$, the _associated operator of $f$_ is defined to be
$T_{f}:=T_{K_{f}}$, which is a bounded linear operator on $\ell^{2}(G)$ with
operator norm $\|f\|_{1}^{2}$.

Consequently, combining Theorem 3.1 with Lemma 3.4 and Proposition 3.6, we
obtain the following:

######  Theorem 3.8.

Let $\\{G_{i}\\}_{i\in I}$ be a family of finitely generated amenable groups
with finite generating set $S_{i}$ and the induced word length metric for each
$i$. Consider the following conditions:

  1. (1)

For any $\varepsilon>0$, there exists $M>0$ and a family of functions
$\\{f_{i}\\}_{i\in I}$ such that each
$f_{i}\in\ell^{2}(G_{i})_{1,+}\cap\ell^{1}(G_{i})$ satisfies $\|f_{i}-s\cdot
f_{i}\|_{2}\leq\varepsilon$ for $s\in S_{i}$ and $\|f_{i}\|_{1}\leq M$.

  2. (2)

For any $\varepsilon>0$, there exists $M^{\prime}>0$ and a family of functions
$\\{f_{i}\\}_{i\in I}$ such that each
$f_{i}\in\ell^{2}(G_{i})_{1,+}\cap\ell^{1}(G_{i})$ satisfies $\|f_{i}-s\cdot
f_{i}\|_{2}\leq\varepsilon$ for $s\in S_{i}$ and the operator norm of the
associated operator $T_{f_{i}}$ in Definition 3.7 is bounded by $M^{\prime}$.

  3. (3)

For any $\varepsilon>0$, there exists $N>0$ and a family of functions
$\\{f_{i}\\}_{i\in I}$ such that each $f_{i}\in\ell^{2}(G_{i})_{1,+}$
satisfies $\|f_{i}-s\cdot f_{i}\|_{2}\leq\varepsilon$ for $s\in S_{i}$ and
$\sharp\mathrm{supp}(f_{i})\leq N$.

  4. (4)

The family $\\{(G_{i},S_{i})\\}_{i\in I}$ is uniformly amenable.

  5. (5)

The family $\\{(G_{i},S_{i})\\}_{i\in I}$ has uniform Property A.

Then (1) $\Leftrightarrow$ (2) $\Rightarrow$ (3) $\Rightarrow$ (4)
$\Leftrightarrow$ (5). Additionally, if there exists $L>0$ such that $\sharp
S_{i}\leq L$ for each $i\in I$, then all the conditions above are equivalent.

In the case of a sequence of finite groups, Theorem 3.8 can be simplified as
follows:

######  Corollary 3.9.

Let $\\{G_{n}\\}_{n\in\mathbb{N}}$ be a sequence of finite groups with finite
generating set $S_{n}$ and the induced word length metric for each
$n\in\mathbb{N}$, and $X$ be their coarse disjoint union. If there exists
$L>0$ such that $\sharp S_{n}\leq L$ for each $n\in\mathbb{N}$, then the
following are equivalent:

  1. (1)

For any $\varepsilon>0$, there exists $M>0$ and a family of functions
$\\{f_{n}\\}_{n\in\mathbb{N}}$ such that each $f_{n}\in\ell^{2}(G_{n})_{1,+}$
satisfies $\|f_{n}-s\cdot f_{n}\|_{2}\leq\varepsilon$ for $s\in S_{n}$ and
$\|f_{n}\|_{1}\leq M$.

  2. (2)

For any $\varepsilon>0$, there exists $M^{\prime}>0$ and a family of functions
$\\{f_{n}\\}_{n\in\mathbb{N}}$ such that each $f_{n}\in\ell^{2}(G_{n})_{1,+}$
satisfies $\|f_{n}-s\cdot f_{n}\|_{2}\leq\varepsilon$ for $s\in S_{n}$ and the
operator norm of the associated operator $T_{f_{n}}$ is bounded by
$M^{\prime}$.

  3. (3)

For any $\varepsilon>0$, there exists $N>0$ and a family of functions
$\\{f_{n}\\}_{n\in\mathbb{N}}$ such that each $f_{n}\in\ell^{2}(G_{n})_{1,+}$
satisfies $\|f_{n}-s\cdot f_{n}\|_{2}\leq\varepsilon$ for $s\in S_{n}$ and
$\sharp\mathrm{supp}(f_{n})\leq N$.

  4. (4)

The sequence $\\{(G_{n},S_{n})\\}_{n\in\mathbb{N}}$ is uniformly amenable.

  5. (5)

The space $X$ has Property A.

##  4\. Questions

Recall that our main results (Theorem 3.1 and 3.8) indicate that the uniform
Hulanicki-Reiter condition is indeed equivalent to some formally weaker
versions. One might ask whether the same phenomenon happens for the Higson-Roe
condition. More precisely:

######  Question 4.1.

Let $\\{G_{i}\\}_{i\in I}$ be a family of finitely generated groups with
finite generating set $S_{i}$ for each $i$. Assume that for any
$\varepsilon>0$, there exist $M>0$ and a family of maps
$\\{\xi_{i}:G_{i}\to\ell^{2}(G_{i})_{1,+}\cap\ell^{1}(G_{i})\\}_{i\in I}$ such
that each $\xi_{i}$ has $(\varepsilon,1)$-variation and
$\|\xi_{i}(x)\|_{1}\leq M$ for $x\in G_{i}$. Does the family
$\\{(G_{i},S_{i})\\}_{i\in I}$ have uniformly Property A?

It is easy to construct a counterexample and provide a negative answer:

######  Example 4.2.

Let $\mathbb{F}_{2}$ be the free group generated by $S=\\{a,b\\}$, and
$\\{1\\}=N_{0}\triangleleft N_{1}\triangleleft N_{2}\triangleleft\cdots$ be a
sequence of normal subgroups in $\mathbb{F}_{2}$ with finite index and trivial
intersection. Take $G_{n}:=\mathbb{F}_{2}/N_{n}$ and
$S_{n}=\\{aN_{n},bN_{n}\\}$ for each $n\in\mathbb{N}$. Since $\mathbb{F}_{2}$
is non-amenable, the sequence $\\{(G_{n},S_{n})\\}_{n}$ does not have uniform
Property A. However, taking
$\xi_{n}(g):=\delta_{N_{n}}\in\ell^{2}(G_{n})\cap\ell^{1}(G_{n})$ for each
$n\in\mathbb{N}$ and $g\in G_{n}$, we obtain that each $\xi_{n}$ has
$(0,1)$-variation and $\|\xi_{n}(g)\|_{1}=1$ for any $g\in G_{n}$. Hence we
obtain a negative answer to Question 4.1.

In fact, Question 4.1 has a negative answer even for finite $I$. One can just
take a group without Property A and choose a function $\xi$ as in Example 4.2.

We realise that the obstruction to generalise the Higson-Roe condition is due
to the lack of equivariance for the Higson-Roe functions, which allows too
much flexibility for the choice. Hence in order to obtain Property A, we need
some extra requirement on their supports. Currently we do not have any
appropriate candidate, therefore we would like to pose the following question:

######  Question 4.3.

Let $G$ be a finitely generated group. Is it possible to weaken the
requirement on the support of the Higson-Roe functions as in Theorem 3.1 or
3.8 while still guaranteeing that $G$ has Property A?

## References

  * [1] S. Banach and A. Tarski.  Sur la décomposition des ensembles de points en parties respectivement congruentes.  Fund. math., 6(1), 1924.
  * [2] E. Følner.  On groups with full Banach mean value.  Math. Scand., 3:243–254, 1955.
  * [3] E. Følner.  Note on groups with and without full Banach mean value.  Math. Scand., 5:5–11, 1957.
  * [4] M. Gromov.  Asymptotic invariants of infinite groups.  In Geometric group theory, Vol. 2 (Sussex, 1991), volume 182 of London Math. Soc. Lecture Note Ser., pages 1–295. Cambridge Univ. Press, Cambridge, 1993.
  * [5] N. Higson and J. Roe.  Amenable group actions and the Novikov conjecture.  J. Reine Angew. Math., 519:143–153, 2000.
  * [6] A. Hulanicki.  Means and Følner condition on locally compact groups.  Studia Math., 27:87–104, 1966.
  * [7] J. von Neumann.  Zur allgemeinen theorie des masses.  Fund. math., 13(1):73–116, 1929.
  * [8] P. W. Nowak and G. Yu.  Large scale geometry.  EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich, 2012.
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  * [11] V. Runde.  Lectures on amenability, volume 1774 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, 2002.
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# Bayesian causal inference via
probabilistic program synthesis

Sam Witty
University of Massachusetts
Amherst, MA 01002
<EMAIL_ADDRESS>
&Alexander Lew11footnotemark: 1
Massachusetts Institute of Technology
Cambridge, MA 02139
<EMAIL_ADDRESS>
David Jensen
University of Massachusetts
Amherst, MA 01002
<EMAIL_ADDRESS>
&Vikash Mansinghka
Massachusetts Institute of Technology
Cambridge, MA 02139
<EMAIL_ADDRESS>
Contributed equally.

###### Abstract

Causal inference can be formalized as Bayesian inference that combines a prior
distribution over causal models and likelihoods that account for both
observations and interventions. We show that it is possible to implement this
approach using a sufficiently expressive probabilistic programming language.
Priors are represented using probabilistic programs that generate source code
in a domain specific language. Interventions are represented using
probabilistic programs that edit this source code to modify the original
generative process. This approach makes it straightforward to incorporate data
from atomic interventions, as well as shift interventions, variance-scaling
interventions, and other interventions that modify causal structure. This
approach also enables the use of general-purpose inference machinery for
probabilistic programs to infer probable causal structures and parameters from
data. This abstract describes a prototype of this approach in the Gen
probabilistic programming language.

##  1 Introduction

Bayesian formulations of causal inference enable practitioners to explicitly
reason about uncertainty when answering structural questions (e.g., “What is
the probability that $X$ causes $Y$?”) as well as questions about the effects
of a specific intervention (e.g., “How much will intervening to make
$X=x^{\prime}$ increase the probability that $Y=y^{\prime}$?”). Bayesian
formulations have been developed for both the potential outcomes framework
[17] and the causal graphical models framework [7, 9, 11]. In principle,
Bayesian approaches make it possible to incorporate prior knowledge and make
efficient use of limited data [16, 18].

In this paper, we explore a new approach to implementing Bayesian causal
inference based on probabilistic programming, inspired by Bayesian synthesis
[22]. Probabilistic programming languages enable users to compactly specify
probabilistic models in code. Some languages, like Stan [2], have syntax that
closely resembles the statistical notation often used in the literature to
define probabilistic models: a list of equations of the form $x\sim\dots$.
Others, like Gen [3], allow users to include arbitrary program control flow in
their models; a model is represented by a program that simulates
stochastically from a distribution. In this paper, we represent hypothesized
causal models explaining some phenomenon as programs in MiniStan, a simple
probabilistic programming language designed to resemble Stan (Figure 1). Then,
we use a more expressive probabilistic programming language, Gen, to encode a
prior and likelihood over MiniStan programs, and to do inference. The Gen
model (i) stochastically generates MiniStan programs to encode a prior
distribution over causal model structures and parameters, (ii)
programmatically edits the generated MiniStan programs to reflect
interventions and experimental conditions, then (iii) interprets the MiniStan
programs to generate observational and experimental data. We can then use
Gen’s inference programming and conditioning features to condition the entire
process on actual observational and experimental data, and to obtain posterior
samples of the MiniStan code defining the original observational model—that
is, to perform both structure learning and parameter estimation.

$\displaystyle P$ $\displaystyle\rightarrow S\mid S;\,P$ Programs
$\displaystyle S$ $\displaystyle\rightarrow x=E\mid x\sim D$ Statements
$\displaystyle D$
$\displaystyle\rightarrow\texttt{normal}(E,E)\mid\texttt{uniform}(E,E)\mid\texttt{bernoulli}(E)$
Distributions $\displaystyle E$ $\displaystyle\in\text{deterministic Julia
expressions}$ $\displaystyle x$ $\displaystyle\in\text{Julia variable
identifiers}$ Figure 1: Grammar of MiniStan

Causal models are typically structured as a set of autonomous components [1,
10, 20], such that interventions in the system can be accurately represented
in the model as an alteration of a small number of model components, and all
other model components (and the causal relationships among them) remain
unchanged. In the formalism of causal graphical models, interventions are
typically expressed using the do-operator [20], which fixes the value of one
random variable and removes the influence of its parents. However, many
realistic interventions are not accurately represented by this particular
variety of model alteration [6, 13, 23]. For example, realistic interventions
might best be represented by altering the functional form of a particular
dependence, enabling or disabling specific causes, or enacting complex
combinations of these interventions. This paper demonstrates interventions
represented as modifications of probabilistic program source code and shows
how this representation enables the Bayesian synthesis approach to handle a
broad class of experimental data.

###  1.1 A conceptual example

(a) “Belief and skill matter” CGM.

(b) “Only skill matters” CGM.

⬇

quote

s $\sim$ normal(mu_s, sigma_s)

b $\sim$ normal(s, sigma_b)

logit_o = s * lambda_so + b * lambda_bo

o $\sim$ bernoulli(1/(1+exp(-logit_o)))

end

(c) “Belief and skill matter” model as source code.

⬇

quote

s $\sim$ normal(mu_s, sigma_s)

b $\sim$ normal(s, sigma_b)

logit_o = s * lambda_so

o $\sim$ bernoulli(1/(1+exp(-logit_o)))

end

(d) “Only skill matters” model as source code.

Figure 2: A conceptual example combining structure learning and parameter
estimation.

Consider the task of inferring whether a student’s belief in her ability is
causal for success at a research project. Observational data on student belief
and student success alone are insufficient to answer this question, due to the
confounding effect of skill (see Figures 2a & 2b).

We can imagine multiple types of experiments that would enable effective
causal inference despite the confounding effect of skill. For example, an
advisor could encourage a student, shifting her belief in her ability (but not
increasing her skill). An advisor could also administer an assessment on the
key skills needed for the project, before the student attempts it, and look at
the results. Unfortunately, although this might reveal the true skill level to
the advisor, this might also change the student’s belief in her own ability to
succeed. Hypothetically, one can imagine a miracle pill that modifies one’s
confidence to a fixed value, without changing anything else. Each of these
experiments corresponds to a different modification to the source code from
Figures 2c and 2d. Examples of these modifications are shown in Figures 3a-f.

This paper shows how to formalize this example, using probabilistic programs
that generate, edit, and interpret the source code of causal models. It also
presents results from an implementation in the Gen probabilistic programming
language, demonstrating the utility of incorporating diverse sources of
experimental data.

⬇

quote

s $\sim$ normal(mu_s, sigma_s)

b = 5

logit_o = s * lambda_so + b * lambda_bo

o $\sim$ bernoulli(1/(1+exp(-logit_o)))

end

(a) “Belief and skill matter” with belief pill.

⬇

quote

s $\sim$ normal(mu_s, sigma_s)

b = 5

logit_o = s * lambda_so

o $\sim$ bernoulli(1/(1+exp(-logit_o)))

end

(b) “Only skill matters” with belief pill.

⬇

quote

s $\sim$ normal(mu_s, sigma_s)

b $\sim$ normal(s \+ 3, sigma_b)

logit_o = s * lambda_so + b * lambda_bo

o $\sim$ bernoulli(1/(1+exp(-logit_o)))

end

(c) “Belief and skill matter” with encouragement design.

⬇

quote

s $\sim$ normal(mu_s, sigma_s)

b $\sim$ normal(s \+ 3, sigma_b)

logit_o = s * lambda_so

o $\sim$ bernoulli(1/(1+exp(-logit_o)))

end

(d) “Only skill matters” with encouragement design.

⬇

quote

s $\sim$ normal(mu_s \+ 2, sigma_s)

b $\sim$ normal(s, sigma_b / 100)

logit_o = s * lambda_so + b * lambda_bo

o $\sim$ bernoulli(1/(1+exp(-logit_o)))

end

(e) “Belief and skill matter” with assessment.

⬇

quote

s $\sim$ normal(mu_s \+ 2, sigma_s)

b $\sim$ normal(s, sigma_b / 100)

logit_o = s * lambda_so

o $\sim$ bernoulli(1/(1+exp(-logit_o)))

end

(f) “Only skill matters” with assessment.

Figure 3: Various interventions expressed as modifications of MiniStan source
code.

##  2 Priors on Causal Models

Figure 4: Graphical meta-model for the Bayesian synthesis approach to causal
structure and parameter learning. A set of global parameters $\theta$
determine the source code of the observational causal program $P_{obs}$, which
is modified via code-editing intervention functions to induce experimental
causal programs for the belief-pill ($P_{bp})$ encouragement design ($P_{e}$),
and the assessment ($P_{a}$) interventions. The code for each program is run
through an interpreter, which generates (observational or experimental) data.
The likelihoods of the various kinds of data under the different interpreted
programs can be used to infer the posterior distribution over $\theta$, and
therefore over the observational causal program $P_{obs}$.

To compute the posterior distribution over the two candidate causal models, we
first specify a prior distribution over a set of global latent variables. One
of these variables, $edge$, determines whether $Belief$ influences $Outcome$.

$\displaystyle\mu_{s}$ $\displaystyle\sim Normal(0,1)$
$\displaystyle\sigma_{s}$ $\displaystyle\sim Uniform(0,1)$
$\displaystyle\sigma_{b}$ $\displaystyle\sim Uniform(0,1)$
$\displaystyle\lambda_{so}$ $\displaystyle\sim Uniform(0,1)$
$\displaystyle\lambda_{bo}$ $\displaystyle\sim Uniform(0,1)$ $\displaystyle
edge$ $\displaystyle\sim Bernoulli(0.5)$

⬇

1@gen function generate_causal_model()

2 mu_s = @trace(normal(0, 1), :mu_s)

3 sigma_s = @trace(uniform(0, 1), :sigma_s)

4 sigma_b = @trace(uniform(0, 1), :sigma_b)

5 lambda_so = @trace(uniform(0, 1), :so_weight)

6 lambda_bo = @trace(uniform(0, 1), :bo_weight)

7 edge = @trace(bernoulli(0.5), :edge)

8

9 if edge

10 logit_o_expr = quote s * $so_weight + b * $bo_weight end

11 else

12 logit_o_expr = quote s * $so_weight end

13 end

14

15 causal_model = quote

16 s $\sim$ normal($mu_s, $sigma_s)

17 b $\sim$ normal(s, $sigma_b)

18 logit_o = $logit_o_expr

19 o $\sim$ bernoulli(1/(1+exp(-logit_o)))

20 end

21 return causal_model

22end

(a)

⬇

quote

s $\sim$ normal(0.237, 0.449)

b $\sim$ normal(s, 0.913)

logit_o = s * 0.137 + b * 0.852

o $\sim$ bernoulli(1/(1 + exp(-logit_o)))

end

⬇

quote

s $\sim$ normal(-0.592, 0.302)

b $\sim$ normal(s, 0.724)

logit_o = s * 0.503 + b * 0.491

o $\sim$ bernoulli(1/(1 + exp(-logit_o)))

end

⬇

quote

s $\sim$ normal(1.892, 0.108)

b $\sim$ normal(s, 0.301)

logit_o = s * 0.542

o $\sim$ bernoulli(1/(1 + exp(-logit_o)))

end

(b)

⬇

1@gen function generate_data(NObs, NBeliefPill, NEncouragement, NAssessment)

2 observational_model = @trace(generate_causal_model())

3 belief_pill_model = applyDoIntervention(observational_model, :b, 5)

4 encouragement_model = applyShiftIntervention(observational_model, :b, 3)

5 assessment_model =
applyVarianceScalingIntervention(applyShiftIntervention(observational_model,
:s, 2),

6 :b, 1/100)

7

8 observational_data = @trace(interpretMiniStan(observational_model,
n_runs=NObs), :obs)

9 belief_pill_data = @trace(interpretMiniStan(belief_pill_model,
n_runs=NBeliefPill), :belief_pill)

10 encouragement_data = @trace(interpretMiniStan(encouragement_model,
n_runs=NEncouragement), :encouragement)

11 assessment_data = @trace(interpretMiniStan(assessment_model,
n_runs=NAssessment, :assessment)

12end

(c)

Figure 5: Gen implementation of causal inference via Bayesian synthesis. The
generate_causal_model Gen program (a) encodes a prior distribution over
MiniStan models; (b) shows three samples from this prior. The generate_data
Gen program (c) encodes the likelihood: it samples a possible causal model
from the prior (line 2), modifies it to obtain MiniStan code representing
experimental conditions (lines 3-6), then simulates observational and
experimental data by running the MiniStan programs (lines 8-11). The
interpreter is itself a Gen probabilistic program.

In the Bayesian synthesis framework, a prior distribution over causal models
is a stochastic procedure generating programs in a domain specific language
(Figure 5). The grammar for our simple domain specific language, MiniStan, is
presented in Figure 1.

##  3 Likelihoods for Experiments

To incorporate experimental evidence of various forms, the Bayesian synthesis
approach requires an intervention library which consists of a set of code-
editing functions that modify causal model programs in the domain specific
language. For the conceptual example, our intervention library contains three
interventions: (i) an atomic intervention, which applies the do-operator; (ii)
a shift intervention, which changes the mean of a distribution by a fixed
increment; and (iii) a variance-scaling intervention, which modifies the
variance of a random variable assumed to be drawn from a normal distribution.
In principle, an intervention library could contain arbitrary rules for
modifying causal model source code, including changing the underlying
distribution for a random variable or adding variables (latent or observed)
that didn’t exist in the observational model.

These interventions can be freely composed to represent a diverse set of
experimental scenarios. We demonstrate this compositionality in the
“assessment” experiment, which is composed of a shift intervention (a
student’s skill may improve if she has to take a test) and a variance-scaling
intervention (a student’s belief in her ability has less noise after taking a
test).

When interpreted, a causal program in MiniStan represents a likelihood
function over observational data. To compute the likelihood of experimental
data, we simply modify the causal program using the intervention library
before subsequently interpretting the modified program.

⬇

1function applyDoIntervention(program, var, newValue)

2 walk(program) do expr

3 @match expr begin

4 :($x = $val) && if x == var end => :($var = $newValue)

5 :($x $\sim$ $dist) && if x == var end => :($var = $newValue)

6 _ => expr

7 end

8 end

9end

⬇

1function applyShiftIntervention(program, var, shiftValue)

2 walk(program) do expr

3 @match expr begin

4 :($x $\sim$ normal($mean, $std)) && if x == var end => :($x $\sim$
normal($mean + $shiftValue, $std))

5 :($x $\sim$ uniform($a, $b)) && if x == var end => :($x $\sim$ uniform($a +
$shiftValue, $b + $shiftValue))

6 :($x = $value) && if x == var end => :($x = $value + $shiftValue)

7 _ => expr

8 end

9 end

10end

Figure 6: Julia implementation of the atomic (“do”) intervention and the shift
intervention. Rather than perform graph operations such as removing edges, an
atomic intervention on a program walks the program’s code and replaces any
expression that assigns var with a new expression, implementing the
intervention (var = newValue). The shift intervention walks the program’s code
and adds shiftvalue to the mean argument for the normal distribution, the
lower and upper bound arguments for the uniform distribution, and the value of
any deterministic assignment.

##  4 Inference

We demonstrate the utility of this approach by performing approximate
posterior inference over synthesized causal model programs from our conceptual
example. In this example we: (i) generate a MiniStan program from the prior,
(ii) generate a set of observational and experimental data from the
interpreted MiniStan program, and (iii) perform approximate posterior
inference over synthesized causal models using sequential Monte Carlo [4] with
Metropolis Hastings rejuvination. We generated ten individuals’ skill, belief,
and outcome for each of the four observational and experimental settings from
a single causal model where
$\mu_{s}=-0.013,\sigma_{s}=0.776,\sigma_{b}=0.646,\lambda_{so}=0.734,\lambda_{bo}=0.717,$
and $edge=True$.

Figure 7: Posterior probability of the existence and strength of causal
dependence between a student’s belief and her subsequent outcome. The vertical
gray line is the actual value for lambda_bo

.

Using only observational data, the posterior probability of the edge variable
is low. This may be because the data can be explained only by appealing to
skill, and this simpler model could lead to a higher marginal probability than
one which introduces a new parameter (lambda_bo). (This phenomenon is
sometimes called “Bayesian Ockham’s Razor”.) However, as we incorporate
additional experimental evidence the posterior probability of the edge
increases. Similarly, the posterior distribution over $\lambda_{bo}$, the
effect of belief on outcome, concentrates around the true value as we leverage
experimental evidence.

##  5 Discussion

The Bayesian synthesis approach we have outlined in this paper provides
several advantages over alternative approaches to structure discovery and
parameter estimation in causal modeling: (i) an explicit characterization of
uncertainty over model structures; (ii) a principled way to model diverse
interventions; and (iii) a formalization that can be re-used in diverse
problems, with varying degrees of prior knowledge, without requiring
practitioners to design custom inferences for each use case.

Although this example uses parametric causal models, it is conceptually
straightforward to use Gaussian processes and/or Dirichlet process mixture
models for the functional forms of causal relationships [22]. It may thus be
fruitful to develop Bayesian variants of existing non-parametric techniques
for causal inference [12, 15].

The results reported here were obtained using vanilla sequential Monte Carlo
over the joint space of model structure, parameters, and the latent variables
in each observation or experiment. In order for this approach to scale to
complex models, hierarchical priors over models, and large datasets, we expect
more powerful techniques will be necessary. However, the Gen platform provides
programmable inference constructs [3], including hybrids of Hamiltonian Monte
Carlo [5] and Metropolis-Adjusted Langevin [21] approaches with sequential
Monte Carlo [4], that could potentially address some of these scaling
challenges.

##  6 Related Work

Probabilistic programs are often used to represent causal processes [8]. Some
languages, such as Omega [24], make this causal interpretation explicit,
including a semantics for interventional and counterfactual reasoning. It
would be interesting to consider whether the framework we present here, which
considers interventions to be arbitrary code-editing procedures, could also be
usefully applied to counterfactual reasoning problems.

Incorporating experimental evidence for structure learning and parameter
estimation can be thought of as the inner loop of an optimal experimental
design procedure. Probabilistic programs have been used to automate this
search over experiments [19], seeking to maximize the expected information
gain over some query given new evidence. In that work, experiments are modeled
as arguments to a probabilistic program. Our approach instead describes an
experiment as a modification of MiniStan programs, enabling a clean
abstraction between the specification of causal models (or distributions over
causal models) and interventions that modify those models.

Improving methodology for combining observational and experimental evidence
has far-reaching implications for a wide variety of scientific disciplines,
and has received significant attention in the graph-based causal inference
literature. For example, extensions of the do-calculus have been developed to
incorporate experiments expressed as atomic interventions given a known causal
graphical model structure [14]. Recent extensions of existing graph-based
structure discovery algorithms have been made to incorporate atomic
interventions [25] and imperfect interventions [26]. Our work proposes
characterizing imperfect interventions as code-editors acting on probabilistic
programs; this representation enables us to perform posterior inference (with
uncertainty estimates) over both structure and model parameters.

## Acknowledgements

We thank Javier Burroni, Dan Garant, Zenna Tavares, and Reilly Grant for
thoughtful discussion.

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