###### Abstract

Bayesian optimization (BO) is an efficient method to optimize expensive black-
box functions. It has been generalized to scenarios where objective function
evaluations return stochastic binary feedback, such as success/failure in a
given test, or preference between different parameter settings. In many real-
world situations, the objective function can be evaluated in controlled
‘contexts’ or ‘environments’ that directly influence the observations. For
example, one could directly alter the ‘difficulty’ of the test that is used to
evaluate a system’s performance. With binary feedback, the context determines
the information obtained from each observation. For example, if the test is
too easy/hard, the system will always succeed/fail, yielding uninformative
binary outputs. Here we combine ideas from Bayesian active learning and
optimization to efficiently choose the best context and optimization parameter
on each iteration. We demonstrate the performance of our algorithm and
illustrate how it can be used to tackle a concrete application in visual
psychophysics: efficiently improving patients’ vision via corrective lenses,
using psychophysics measurements.

##  1 Introduction

Bayesian optimization (BO) has been used in many applications to optimize
systems whose performance is expensive to evaluate, such as a robot’s walking
motion (Antonova2017), deep brain stimulation (Grado2018), or machine learning
model hyperparameters (Snoek2012). BO has also been generalized to situations
where one can only obtain the performance of the system in the form of a
binary output, corresponding, for example, to whether the system passes/fails
a test (Tesch2013). Examples of binary Bayesian optimization (BBO) include
testing whether a robot is able to successfully maneuver across a given
environment, or the optimization of a model’s hyperparameters, by early-
stopping the training of under-performing models while continuing others
(Zhang2019a). In these cases, the way the system is tested is critical in
optimizing its performance. For example, if we ask a robot to move across an
impossibly complex environment (e.g. filled with obstacles and pot-holes), it
will always fail, leaving us none the wiser as to how to improve its
performance.

A similar problem arises in preferential Bayesian optimization (PBO), where
the performance of a system is evaluated via a set of comparisons between
different parameter settings (Brochu2010b). For example, PBO was used to
improve the performance of hearing aid devices, by asking patients to evaluate
which of two parameter settings they preferred, when listening to excerpts of
music (Nielsen2015a). In this example, the context (i.e. the audio stimuli)
clearly plays a role in determining how much information is gained from
subjects’ responses: an audio stimulus that was overly simple (e.g. a single
monotone pitch) or random (e.g. white noise) would result in non-informative
responses from subjects, impeding progress in optimizing the hearing device
for most daily settings.

Here we show how to extend previous BO algorithms to deal with these two
related problems, in which the measured performance of a system takes the form
of a binary, context-dependent, output. We then show how this problem can be
efficiently solved by applying a simple heuristic, where BO is used to
maximize the system’s performance, while Bayesian active learning is used to
choose a new context, such that the observed binary output (e.g. whether the
system passed/failed a given test, or a comparison between two parameter
settings) is maximally informative about the system’s performance. We show
that our resulting algorithm performs competitively on a range of benchmark
problems. Further, we illustrate the relevance of our algorithm by applying it
to a real-world problem, where the goal is to find the correct optical
correction required to optimize a patient’s vision, based on their binary
responses in a psychophysics task.

##  2 Problem statement

Consider a system, whose performance can be quantified by a function,
$g(\boldsymbol{x})$. For example, $g\br{\b{x}}$ could represent the visual
acuity (VA) of a patient fitted with glasses with optical parameters, $\b{x}$.
We are interested in solving the global optimization problem:

$\b{x}^{*}=\underset{\b{x}\in\mathcal{X}}{\arg\max}\ g\br{\b{x}},$ (1)

where $\mathcal{X}$ is a bounded set denoted as the search space.

We consider the case where the performance of the system cannot be evaluated
directly, but instead, can be assessed by observing whether it passes/fails a
test in a given context, parameterized by $\b{s}$. For example, we could test
whether a subject (fitted with glasses with optical parameters, $\b{x}$) can
correctly identify a visual stimulus, parameterized by $\b{s}$, presented
onscreen. The outcome of such a test is denoted by a binary variable, $c$,
with success probability given by:

$p\br{c=1|\b{x},\b{s}}=\Phi\br{f\br{\b{s},\b{x}}},$ (2)

where $\Phi$ is the normal cumulative distribution function, and
$f\br{\b{s},\b{x}}$ is a function describing performance of the system with
parameters $\b{x}$ and given a context $\b{s}$.

If we are to optimize the system based on observed binary outcomes, $c$, the
probability that the system passes a test in a given context, $\b{s}$
(determined by $f\br{\b{s},\b{x}}$) must be closely related to its overall
performance, $g\br{\b{x}}$. For simplicity, we begin by assuming that the same
configuration, $\b{x}^{*}$, maximizes performance in all contexts, such that:

$\forall\boldsymbol{s}\in\mathcal{S},\hskip
8.5359pt\underset{\boldsymbol{x}\in\mathcal{X}}{\arg\max}\
g\left(\boldsymbol{x}\right)=\underset{\boldsymbol{x}\in\mathcal{X}}{\arg\max}\
f\left(\boldsymbol{s,x}\right),$ (3)

where $\mathcal{S}$ is a bounded set denoted as the context space. In our
previous example, this would imply that the optical correction that maximizes
VA also maximizes the subject’s performance in identifying a range of
different visual stimuli, parameterized by $\b{s}$. This condition will be
true for a broad range of situations in which $f\br{\b{s},\b{x}}$ takes the
form:

$f\br{\b{s},\b{x}}=q\br{h\br{\b{s}}g\br{\b{x}}+b\br{\b{s}}}$ (4)

where $h\br{\b{s}}$ is non-negative for all $\b{s}$, and $q\br{\cdot}$ is a
monotonically increasing one-dimensional function. Plugging this back into Eqn
2, we have:

$p\br{c=1|\b{s},\b{x}}=\Phi\sqbr{q\br{h\br{\b{s}}g\br{\b{x}}+b\br{\b{s}}}}.$
(5)

We can think of $b\br{\b{s}}$ and $h\br{\b{s}}$ as controlling the difficulty
of the test. Without loss of generality, we can assume that $g\br{\b{x}}\geq
0$ for all $\b{x}$. For large $h\br{\b{s}}$ the probability of success will
saturate close to 1 for all $\b{x}$: i.e., the task is easy, so the system
always passes the test. In contrast, when $h\br{\b{s}}$ is close to zero, the
system will perform close to baseline, with success probability
$\Phi(q(b(\boldsymbol{s})))$. In both these extreme cases, little information
is gained about the parameters, $\b{x}$, that optimize performance, since the
success probability depends only weakly on $g\br{\b{x}}$. Our goal, therefore,
is to find a sampling rule that selects $(\b{s}$, $\b{x})$ on each trial so as
to appropriately set the difficulty of the task, and efficiently optimize
$g\br{\b{x}}$ in a limited number of trials.

##  3 Contextual binary Bayesian optimization

###  3.1 Inference

As is standard in BO, we will build a surrogate model of $f\br{\b{s},\b{x}}$,
by assuming a zero-mean Gaussian process (GP) prior, such that $f\sim
GP\br{0,k\br{\cdot,\cdot}}$, where $k$ is a kernel function. When the observed
outcome is binary, the posterior probability distribution over $f$ cannot be
written analytically but must be approximated. While a number of different
approximations schemes exist, here we use the Laplace approximation, since it
is standard and simple to implement (Rasmussen2006). However, our approach
could be applied to other approximation schemes such as expectation
propagation (Minka2001; Seeger2002).

Ideally, the kernel should reflect prior knowledge about the objective, such
as the fact that the optimum does not depend on the context. This can be done,
for example, by assuming a multiplicative structure of the function of the
form $f(\boldsymbol{s},\boldsymbol{x})=h(\boldsymbol{s})g(\boldsymbol{x})$,
which in practice can be incorporated by using a kernel that decomposes as:
$k(\boldsymbol{s},\boldsymbol{x},\boldsymbol{s}^{\prime},\boldsymbol{x}^{\prime})=k_{1}(\boldsymbol{s},\boldsymbol{s}^{\prime})k_{2}(\boldsymbol{x},\boldsymbol{x}^{\prime})$.
Moreover, $h$ should be constrained to be positive. In practice, we did not
use this latter constraint in our experiments, as this would make the
inference more difficult and it did not seem to impede performance.

###  3.2 Knowledge gradient acquisition rule

Having inferred a Bayesian model of the objective function,
$p\br{f|\mathscr{D}}$, the next step of any BO algorithm is to select new
parameters, $\b{x}$ and $\b{s}$, to evaluate the objective. To do this, we
extended the ‘knowledge gradient’ (KG) acquisition rule, developed previously
for BO with continuous outputs (Frazier2009), to contextual BBO.

We first explain our binary KG algorithm for standard BBO (i.e with no
contextual variable, $\b{s}$). As in standard KG, the reported solution of the
optimization routine at iteration $t$ is assumed to be the maximum of the
posterior mean of the GP after observing data
$\mathscr{D}_{t}=\br{\\{\b{x}_{1},c_{1}\\},\\{\b{x}_{2},c_{2}\\},\dots\\{\b{x}_{t},c_{t}\\}}$,
that is:

$\mu^{*}_{\mathscr{D}_{t}}=\underset{\b{x}\in\mathcal{X}}{\arg\max}\
\mathbb{E}\sqbr{g(\boldsymbol{x})|\mathscr{D}_{t}}.$ (6)

The KG corresponds to the expected increase in $\mu^{*}$ if we are allowed one
additional observation, $\\{\boldsymbol{x}_{t+1},c_{t+1}\\}$:

$\displaystyle\mathrm{KG}(\boldsymbol{x}_{t+1})$ $\displaystyle=$
$\displaystyle\sum_{c_{t+1}=0,1}p(c_{t+1}|\boldsymbol{x}_{t+1},\mathscr{D}_{t})\mu^{*}_{\mathscr{D}_{t},\b{x}_{t+1},c_{t+1}}$
(7) $\displaystyle-\mu_{\mathscr{D}_{t}}^{*}$

where
$p(c_{t+1}=1|\boldsymbol{x}_{t+1},\mathscr{D}_{t})=\mathbb{E}_{g}[\Phi(g(\boldsymbol{x}_{t+1}))|\mathscr{D}_{t}]$,
and $\mu^{*}_{\mathscr{D}_{t},\b{x}_{t+1},c_{t+1}}$ is the maximum of the
posterior mean after observing
$(\mathscr{D}_{t},\\{\boldsymbol{x}_{t+1},c_{t+1}\\})$. In supplementary
LABEL:app:KGgradient we show that the gradient of KG, required to maximize it
efficiently, can be computed when using the Laplace approximation (see
supplementary LABEL:app:Laplace) to approximate the GP posterior.

To extend the above KG acquisition function to contextual BBO, we simply add
the context, $\b{s}$, to the system parameters, $\b{x}$, and replace
$g\br{\b{x}}$ with $f\br{\b{s},\b{x}}$. In this case, the maximization in Eqn
6 is performed only over the system parameters, $\b{x}$, given a fixed
context, $\b{s}_{0}$ (see supplementary LABEL:supp:max_KG).

Computing the KG acquisition function is impractically slow for problems with
moderately high dimensions, as its computation requires performing two nested
optimizations: the optimization required for the Laplace approximation of the
GP posterior (see supplementary LABEL:supp:max_KG), and the maximization in
Eqn 6. Thus, we looked for an alternative, that could scale to larger
problems.

###  3.3 Sequential algorithm for choosing system parameters, $\b{x}$ and
context, $\b{s}$

In our set-up, the parameters, $\b{x}$, and $\b{s}$, play two very different
roles. The system parameters $\b{x}$ determine the system’s underlying
performance, $g\br{\b{x}}$, which we want to maximize. The context, $\b{s}$,
determines how much information we gain about the system’s performance from
binary observations, $c$. We therefore decided to choose $\b{x}$ and $\b{s}$
sequentially, using different heuristics: first, $\b{x}_{t+1}$ was chosen
using binary BO, so as to optimize the system’s performance; next,
$\b{s}_{t+1}$ was chosen using Bayesian active learning, to maximize the
information we obtain from the binary observation, $c_{t+1}$.

####  3.3.1 Choosing $\b{x}$ using Bayesian optimization

We tested two different acquisition rules for selecting $\b{x}_{t+1}$: GP
Upper Credible Bound (GP-UCB) (Srinivas2010), and Thompson Sampling (TS)
(Thompson1933).

Thompson sampling (TS) involves choosing $\boldsymbol{x}_{t+1}$ by sampling
from the distribution $p(\boldsymbol{x}^{*}|\mathscr{D}_{t}),$ where
$\b{x}^{*}$ is defined, for a given objective function, as
$\b{x}^{*}=\arg\max_{\b{x}}f\br{\b{x},\b{s}}$ (note that, from Eqn 3,
$\b{x}^{*}$ is independent of $\b{s}$). In supplementary section
LABEL:sub:efficiently_sampling_GP we describe how to efficiently sample from
$p(\boldsymbol{x}^{*}|\mathscr{D}_{t})$ in the context of GP classification
and preference learning.

Second, we implemented a generalization of UCB to binary outputs, where
$\b{x}_{t+1}$ is chosen by maximising the following acquisition function:

$\alpha_{\mathrm{UCB}}(\boldsymbol{x})=\mathbb{E}(\Phi(f(\boldsymbol{x,\b{s}}))+\beta\sqrt{\mathbb{V}(\Phi(f(\boldsymbol{x,\b{s}}))},$
(8)

where a constant $\beta$ (set to $\beta=\Phi^{-1}(0.95)$) sets the balance
between exploitation (first term on right hand side) and exploration (second
term on right hand side). See Fauvel2022 for details on how to analytically
compute this acquisition function and its gradient.

####  3.3.2 Choosing $\b{s}$ using Bayesian active learning

After choosing the system parameters, $\boldsymbol{x}_{t+1}$, the next step is
to select a context $\boldsymbol{s}_{t+1}$ such that the observation,
$c_{t+1}$ will be maximally informative about the underlying function
$f(\cdot,\boldsymbol{x}_{t+1})$. To do this, we used the Bayesian Active
Learning by Disagreement (BALD) algorithm, developed by Houlsby2011.

For a given $\boldsymbol{x}_{t+1}$, the context $\boldsymbol{s}_{n+1}$ is
chosen so as to maximize the mutual information between the observation
$c_{t+1}$ and the objective function $f\br{\cdot,\b{x}_{t+1}}$,
$I(c_{t+1},f|\boldsymbol{s}_{t+1},\boldsymbol{x}_{t+1})$, which is given by:

$\displaystyle I(c_{t+1},f|\boldsymbol{s}_{t+1},\boldsymbol{x}_{t+1})$
$\displaystyle=$ $\displaystyle
H(c_{t+1}|\boldsymbol{x}_{t+1},\boldsymbol{s}_{n+1})$
$\displaystyle-\mathbb{E}_{f}[H(c_{t+1}|\boldsymbol{x}_{t+1},\boldsymbol{s}_{n+1},f)],$

where $H(c_{t+1}|\boldsymbol{x}_{t+1},\boldsymbol{s}_{n+1})$ is the total
entropy of $c_{t+1}$, and
$\mathbb{E}_{f}[H(c_{t+1}|\boldsymbol{x}_{t+1},\boldsymbol{s}_{n+1},f)]$ is
the conditional entropy of $c$, given the latent function
$f\br{\cdot,\b{x}_{t+1}}$. (Note that all three terms are also conditioned on
the observed data, $\mathscr{D}_{t}$, which was removed from the equation for
notational simplicity). This acquisition criterion can be efficiently
approximated (Houlsby2011).

To see this, let $u(\mathscr{D}_{t})$ be some utility function of observed
data $\mathscr{D}_{t}$. The marginal utility of experiment
$(\boldsymbol{s},\boldsymbol{x},c)$ is:

$u(c,\boldsymbol{s},\boldsymbol{x})=u((c,\boldsymbol{s},\boldsymbol{x})\cup\mathscr{D}_{t})-u(\mathscr{D}_{t})$
(10)

With a myopic acquisition rule, the goal is to maximize at each iteration the
expected marginal utility, that is, to solve:

$(\boldsymbol{\hat{s}},\boldsymbol{\hat{x}})=\underset{\boldsymbol{(\boldsymbol{s},\boldsymbol{x})}}{\arg\max}\
\mathbb{E}_{c}[u(c,\boldsymbol{s},\boldsymbol{x}|\mathscr{D}_{t})]$

So, if we split the problem of selecting $(\boldsymbol{s},\boldsymbol{x})$, by
first selecting $\boldsymbol{x}$ for an arbitrary
$\boldsymbol{s}=\boldsymbol{s}_{0}$, and then select $\boldsymbol{s}$, we
have:

$\aligned\mathbb{E}_{c}[u(c,\boldsymbol{\hat{s}},\boldsymbol{\hat{x}}|\mathscr{D}_{t}))&=\underset{\boldsymbol{(\boldsymbol{s},\boldsymbol{x})}}{\max}\mathbb{E}_{c}[u(c,\boldsymbol{s},\boldsymbol{x}|\mathscr{D}_{t})]\\\
\geq\underset{\boldsymbol{s}_{0}}{\max}\underset{\boldsymbol{x}}{\max}\mathbb{E}_{c}[u(c,\boldsymbol{s},\boldsymbol{x}|\mathscr{D}_{t})|\boldsymbol{s}_{0}]$
(11)

So, if we split the problem of selecting $(\boldsymbol{s},\boldsymbol{x})$, by
first selecting $\boldsymbol{x}$, and then $\boldsymbol{s}$, the maximum
expected utility of the experiment decreases.
$\mathbb{E}_{c}[u(c,\boldsymbol{\hat{s}},\boldsymbol{\hat{x}}|\mathscr{D}_{t})]-\underset{\boldsymbol{s}_{0}}{\max}\underset{\boldsymbol{x}}{\max}\mathbb{E}_{c}[u(c,\boldsymbol{s},\boldsymbol{x}|\mathscr{D}_{t})|\boldsymbol{s}_{0}]$
can be seen as a second-order adaptivity gap (the first-order gap being due to
myopia). In practice, we select $\boldsymbol{s}_{0}$ by uniformly sampling in
$\mathcal{S}$.

Let $u(\mathscr{D}_{t})$ be some utility function of observed data
$\mathscr{D}_{t}$. The marginal utility of experiment
$(\boldsymbol{s},\boldsymbol{x},c)$ is:

$u(c,\boldsymbol{s},\boldsymbol{x})=u((c,\boldsymbol{s},\boldsymbol{x})\cup\mathscr{D}_{t})-u(\mathscr{D}_{t})$
(12)

Let $Q_{k}((\boldsymbol{s},\boldsymbol{x}))|\mathscr{D}_{t})$ be the expected
utility of querying the system at $(\boldsymbol{s},\boldsymbol{x})$ after
observing $\mathscr{D}_{t}$ when there are $k$ steps remaining, assuming all
later decisions are optimal. By definition:

$Q_{k}(\boldsymbol{s},\boldsymbol{x}|D)=E_{c}[u(c,\boldsymbol{s},\boldsymbol{x}|\mathscr{D}_{t})]+E_{c}[\underset{\boldsymbol{(s^{\prime},x^{\prime})}}{\max}Q_{k-1}((\boldsymbol{s}^{\prime},\boldsymbol{x}^{\prime})|D,\boldsymbol{s},\boldsymbol{x},c)]$
(13)

Ideally, we would like to select experiment $(\boldsymbol{s},\boldsymbol{x})$
that maximizes $Q_{k}(\boldsymbol{s},\boldsymbol{x}|\mathscr{D}_{t})$. With a
myopic acquisition rule, we give ourselves a more tractable objective: the
goal is to maximize at each iteration the first term in the right-hand side,
that is to solve:

$(\boldsymbol{s}^{*},\boldsymbol{x}^{*})=\underset{\boldsymbol{(\boldsymbol{s},\boldsymbol{x})}}{\arg\max}\
E_{c}(u(c,\boldsymbol{s},\boldsymbol{x}|\mathscr{D}_{t}))$ $\aligned
E(u(c,s^{*},x^{*}|\mathscr{D}_{t}))=\underset{\boldsymbol{(\boldsymbol{s},\boldsymbol{x})}}{\max}E_{y}(u(c,\boldsymbol{s},\boldsymbol{x}|\mathscr{D}_{t}))\\\
\geq\underset{\boldsymbol{s}}{\max}E_{y}(u(c,\boldsymbol{s},\boldsymbol{x}|\mathscr{D}_{t}))$
(14)

So, if we split the problem of selecting $(\boldsymbol{s},\boldsymbol{x})$, by
first selecting $x$, and then $s$, the maximum expected utility of the
experiment decreases.
$E_{c}(u(c,s^{*},x^{*}|\mathscr{D}_{t}))-\underset{\boldsymbol{s}}{\max}E_{c}(u(c,\boldsymbol{s},\boldsymbol{x})|\mathscr{D}_{t}))$
can be seen as a second-order adaptivity gap (the first-order gap being due to
myopia).

###  3.4 Preferential Bayesian optimization

We next considered how to extend our method to preferential optimization
(PBO). In standard PBO Brochu2010b; Gonzalez2017a; Dewancker2018, i.e. with no
contextual variable, $\b{s}$, the result of an evaluation, $c$, depends on the
relative value of the objective function, $g$, with two different parameter
settings, $\b{x}$ and $\b{x}^{\prime}$, according to:

$P(c=1|g,\b{x},\b{x}^{\prime})=\Phi\br{g\br{\b{x}}-g\br{\b{x}^{\prime}}}.$
(15)

The parameters to be compared, $\b{x}$ and $\b{x}^{\prime}$, are called a
duel. To extend this to the contextual case, we assume that the comparison
depends on both the system parameters, $\b{x}$, and context, $\b{s}$,
according to:

$P(c=1|f,\boldsymbol{s},\b{x},\b{x}^{\prime})=\Phi\left(f\left(\boldsymbol{s,x}\right)-f\left(\boldsymbol{s,x^{\prime}}\right)\right),$
(16)

where $f\br{\b{s},\b{x}}$ is defined as before, such that, for all
$\b{s}\in\mathcal{S}$,
$\arg\max_{\b{x}}g\left(\boldsymbol{x}\right)=\arg\max_{\b{x}}f\left(\boldsymbol{s,x}\right)$
(Eqn 3).

The generalization of the KG to this scenario is straightforward. Likewise,
our sequential acquisition rule can be generalized to this scenario by
selecting the duel $(\boldsymbol{x},\boldsymbol{x^{\prime}})$ using PBO, and
then choosing the context of the duel $\boldsymbol{s}$ using Bayesian active
learning.

In our experiments, we selected system parameters using either: (i) the
KernelSelfSparring (KSS) algorithm (Sui2017), which is a simple extension of
TS, described above, or (ii) the Maximally Uncertain Challenge (MUC)
acquisition rule, described in Fauvel2022. Contextual parameters, $\b{s}$,
were then selected using Bayesian active learning by disagreement (BALD; see
previous section).

##  4 Experiments with synthetic test functions

To evaluate the performance of our method, we ran optimization experiments on
a set of 34 functions from a widely used virtual library for optimization
experiments (Surjanovic). The functions in this library exhibit a diversity of
behaviors that occur in real-life optimization problems.

For each objective function, we inferred the hyperparameters for three
different kernels (squared exponential, Matérn 3/2 and Matérn 5/2) using
maximum likelihood estimation with 1000 randomly chosen samples. We then
determined for each function the kernel that best described the function by
measuring the root-mean-squared error on 3000 points. The benchmark functions
are listed in supplementary LABEL:app:benchmarks. In all cases, we used the
Laplace approximation to approximate the posterior over the objective,
$p\br{f|\mathscr{D}}$.

We introduced a scalar context variable $s\in[0,1]$, so that for a test
function $g$ the response of a system query is 1 with probability
$P(s,\boldsymbol{x})=\Phi(sg(\boldsymbol{x}))$ and 0 otherwise.

To take the context variable into account when building the GP surrogate
model, we used the following kernel: for a base kernel $k$ determined using
the aforementioned procedure, we used:
$k_{s}((\boldsymbol{x},s),(\boldsymbol{x^{\prime}},\b{s}^{\prime}))=ss^{\prime}k(\boldsymbol{x},\boldsymbol{x}^{\prime})$.

To compare the different algorithms, we used the stratified analysis method
proposed by Dewancker2016. Briefly, for each benchmark function, we performed
pairwise comparisons between acquisition functions using the Mann-Whitney U
test at $\alpha=5\times 10^{-4}$ significance on the best value found at the
end of the optimization sequence. This determines a partial ranking based on
the number of wins.

Ties are then broken by running the same procedure, but based on the Area
Under Curve, which is related to the speed at which the algorithm reaches the
optimum. This generates a new partial ranking, based on which a Borda score
(Dwork2001) is attributed to each acquisition function (the Borda score of a
candidate is the number of candidates with a lower rank). Then, rankings from
different benchmarks are aggregated by summing the Borda scores to establish a
global ranking. This can be seen as a weighted vote from each benchmark
function.

###  4.1 Binary Bayesian optimization in adaptive contexts

We then ran optimizations on each function for 40 different random number
generator seeds, on 60 iterations. The initial number of random samples was
set to 5.

We termed the sequential acquisition rules in the binary feedback scenario TS
with active learning by disagreement (TS-ALD) and GP-UCB with active learning
by disagreement (UCB-ALD). We termed the generalization of KSS and MUC to the
contextual scenario KSS-ALD and MUC-ALD.

We compared the different acquisition rules: the contextual binary knowledge
gradient (cBKG), UCB-ALD and TS-ALD, with the following controls:

  * •

Fully random, in which $(s,\boldsymbol{x})$ was chosen at random at each
iteration,

  * •

$\boldsymbol{x}$ chosen using TS and $s$ chosen randomly,

  * •

$\boldsymbol{x}$ chosen using UCB and $s$ chosen randomly,

  * •

$(s,\boldsymbol{x})$ selected using BALD.

To avoid saturation effects when transforming the benchmark functions through
the non-linearity, we scaled the functions so that they have mean 0 and
variance 1. Examples of regret curves are presented in figure 1, and the
results of the stratified analysis are summarized in table 1. More detailed
results showing pairwise comparisons between acquisition functions are
presented in table LABEL:fig:cBBO_matrices.

UCB-ALD and TS-ALD show superior performance compared to cBKG (1), despite the
fact that, compared to cBKG, where $\boldsymbol{x}$ and $\boldsymbol{s}$ are
jointly selected, these sequential decision strategies induce an adaptivity
gap (Jiang2019b). Here, we used the Laplace approximation, given that
Expectation Propagation (EP) is known to improve performance compared to the
Laplace approximation and that the gradient of cBKG with EP seems intractable,
this suggests that TS-ALD and UCB-ALD are better acquisition functions in
general. The three heuristics outperformed the different controls.

Figure 1: Results across three different objective functions and different methods in binary BO with adaptive contexts. Solid lines correspond to the mean value of the inferred maximum across the 40 repetitions. Shaded areas correspond to standard error on the mean. Either cBKG, UCB-ALD, or TS-ALD outperform the other acquisition rules, however, there is a lot of variability from one experiment to another. The cBKG acquisition rule tends to be less robust than the sequential heuristics. Table 1: Comparison of acquisition functions on benchmarks. UCB-ALD largely outperforms the other rules. Importantly, the sequential acquisition rules (TS-ALD and UCB-ALD) outperform cBKG. Acquisition rule | Rank | Borda score
---|---|---
UCB-ALD | 1 | 144
TS-ALD | 2 | 83
cBKG | 3 | 59
UCB (random context) | 4 | 46
BALD | 5 | 35
TS (random context) | 6 | 19
Random | 7 | 10
|  |

###  4.2 Preferential Bayesian optimization in adaptive contexts

We then evaluated the performance of our sequential acquisition rules in their
generalization to the case of preferential judgments: KSS-ALD and MUC-ALD.
Given the limited performance of BKG in the previous experiments and the fact
that the preferential version of Knowledge Gradient is impractically slow, we
did not evaluate this acquisition function.

We then ran optimizations on each test function for 40 different random number
generator seeds, on 60 iterations. We compared KSS-ALD and MUC-ALD to the
following controls:

  * •

Fully random, in which $(s,\boldsymbol{x}_{1},\boldsymbol{x}_{2})$ was chosen
at random at each iteration,

  * •

$\boldsymbol{x}$ chosen using KSS and $s$ chosen randomly,

  * •

$\boldsymbol{x}$ chosen using MUC and $s$ chosen randomly,

  * •

$(s,\boldsymbol{x}_{1},\boldsymbol{x}_{2})$ selected using BALD.

Examples of regret curves are shown in figure 2, and the results of the
stratified analysis are summarized in figure 2. More detailed results showing
pairwise comparisons between acquisition functions are presented in table
LABEL:fig:cPBO_matrices. Again, adaptive selection of the context leads to
superior performance compared to controls.

Figure 2: Results across three different objective functions in preferential BO with adaptive contexts. Solid lines correspond to the mean value of the inferred maximum across the 40 repetitions. Shaded areas correspond to standard error on the mean. Sequential acquisition rules (KSS-ALD and MUC-ALD) consistently outperform the others. Table 2: Comparison of acquisition functions on benchmarks. The sequential acquisition rules outperform the controls. MUC-ALD is the most efficient acquisition function. MUC-ALD | 1 | 99
---|---|---
KSS-ALD | 2 | 37
Kernel Self-Sparring | 3 | 22
BALD | 4 | 9
Random | 5 | 5
|  |

##  5 Application: adaptive optimization in psychometric measurements

To illustrate the relevance of our algorithm, we considered the problem of
optimizing the parameters of a lens to improve a patient’s vision based on
their responses in a visual task.

Different types of visual defects typically result in different types of error
(myopia results in ‘spherical refractive error’, $S$; astigmatism results in
‘cylindrical refractive error’, $C$). These errors induce image blur, but can
be corrected by optimizing different optical parameters of a patient’s lenses.
However, patients’ visual performance are typically assessed indirectly, for
example by testing their performance in a visual task (e.g. asking patients to
identify a letter, presented onscreen). Clearly, the task used to test
patients’ visual performance will play a critical role in optimizing their
vision. If the task is too easy/difficult (e.g. the letters presented onscreen
are too small/large) then patients will always succeed/fail, and little
information will be gained as to how to optimize their optics. In the
following, we show our contextual BO algorithm can be used to address this
problem.

###  5.1 Model of patients’ responses

A simple formula relates the spherico-cylindrical error (measured in diopters
($\delta$)) to image blur $\beta$ (Raasch1995; Blendowske2015):

$\beta=\sqrt{\frac{1}{2}(S^{2}+(S+C)^{2})}$ (17)

We are thus trying to find the parameters of the optics $\boldsymbol{x}=(S,C)$
that minimize $\beta$, or equivalently, minimize visual acuity (VA), defined
as the log of the minimum angle of resolution (MAR, measured in minutes of
arc). VA can be related to blur using the formula (Blendowske2015):

$\text{VA}(\boldsymbol{x})-\text{VA}(\boldsymbol{x^{\star}})=1+\beta^{2},$

where $\boldsymbol{x^{\star}}$ corresponds to the best correction. A VA of
20/20 corresponds to logMAR = 0, so we considered that
$\text{VA}(\boldsymbol{x^{\star}})=0$.

We consider a simple model of subjects performing an $n$-alternatives forced-
choice visual task, such as identifying a letter on a screen. The size of the
letter, $s$ (measured in log of the visual angle, in minutes of arc),
determines the difficulty of this task. For a given optical correction
$\boldsymbol{x}$, the probability of correct response is well described by a
psychometric function (see e.g. Fulep2019 or Tokutake2011):

$P(c=1|s)=\gamma+(1-\gamma)\Phi(a(\boldsymbol{x})s+b(\boldsymbol{x})),$ (18)

where $\gamma$ is the chance probability of success, $a$ and $b$ are the slope
and intercept of the psychometric curve. $a$ is assumed to be strictly
positive. Here, for simplicity and consistently with experimental evidence, we
assumed that the slope of the psychometric function $a(\boldsymbol{x})$
remains essentially constant with varying blur (Horner1985; Carkeet2001).

VA is usually defined as the value of $s$ at which
$P(c=1|s)=\frac{1+\gamma}{2}$, or equivalently the inflexion point of the
psychometric curve: $VA(\boldsymbol{x})=-\frac{b(\boldsymbol{x})}{a}$.

To apply our algorithm to this problem, we define a function:

$f(s,\boldsymbol{x})=\Phi^{-1}[\gamma+(1-\gamma)\Phi(as+b(\boldsymbol{x}))]$
(19)

such that: $P(c=1|s,\boldsymbol{x})=\Phi(f(s,\boldsymbol{x}))$ (as in Eqn 2).
The problem of improving VA thus satisfies condition 3, as for any $s$, the
maximum of $f$ is the minimum of VA.

###  5.2 Gaussian process model

The use of GP classification for psychometric function estimation has been
introduced by Gardner2015a; Song2017; Song2018 in the context of audiometry.
We follow the same approach and build a GP model of the subject’s responses in
the task.

For $\gamma$ large enough, we have
$P(c=1|s)\approx\Phi(a(\boldsymbol{x})s+b(\boldsymbol{x}))$. Since in
practice, with a letter chart, $\gamma^{-1}=n=26$, we will make this
simplifying assumption in building our surrogate model.

We put a GP prior on $f$, with zero mean and kernel $k_{\psi}$ defined as:

$k_{\psi}((s,\boldsymbol{x}),(s^{\prime},\boldsymbol{x}^{\prime}))=\theta
ss^{\prime}+k(\boldsymbol{x},\boldsymbol{x}^{\prime})$ (20)

This kernel reflects the structure of the function $f$ (see equation 19) with
the assumption $\gamma\approx 0$ (see supplementary
LABEL:supp:kernel_psychophysics).

We used a squared exponential kernel as $k$. To minimize the effect of kernel
hyperparameters, we simulated the response to 1000 random pairs
$(s,\boldsymbol{x})$ and inferred the kernel hyperparameters using maximum
likelihood estimation. The hyperparameters were then kept constant during the
experiments.

###  5.3 Results

We repeated the simulated experiment 20 times for 260 iterations, for 8
different slopes values (evenly spaced between between 1.0 logMAR-1 and 8.0
logMAR-1). The search space was
$\mathcal{X}=[-4\delta,4\delta]\times[-4\delta,4\delta]$ and the context space
$\mathcal{S}=[-1,2]$ logMAR. Regret curves in the experiments where the slope
of the psychometric function was set at 5.0 logMAR-1 are shown on figure 3,
and the results of the stratified analysis are summarized in table 3.

Both UCB-ALD and TS-ALD led to rapid and consistent improvement of the VA
closed to its optimal value of logMAR = 0. UCB-ALD is the best performing
algorith (see table 3), with a mean VA at the end of the optimization sequence
of $2.48\times 10^{-2}$ logMAR in the case of a slope of 5.0 logMAR-1(s.e.m =
8.22), 3). At the end of the optimization sequence, the an average spherical
correction error of $2.74\times 10^{-2}\delta$ (s.e.m. = $5.36\times
10^{-2}$). Note that this is closed to the precision that can be achieved
nowadays ($0.01\delta$). The average cylindrical correction error is
$1.53\times 10^{-1}\delta$ (s.e.m. = $8.22\times 10^{-2}$).

However, when using random sampling, the optimization consistently failed
(figure 3). Overall, the results are consistent with the one obtained on the
synthetic benchmarks (table 3).

Here, we considered that the objective is a black box. However, for a specific
application in a clinical setting, parameterizing the objective using domain
knowledge would likely lead to faster and more robust improvement (see e.g.
Cox2017).

Figure 3: A. Psychometric curves for various refractive errors corresponding to different visual acuities, with a slope of the psychometric function set at 5.0 logMAR-1. The baseline corresponds to the chance level in the n-alternatives forced-choice task (here, n = 26). VA of 20/20 corresponds to logMAR = 0. B. Visual acuity with the inferred best parameters throughout the optimization. The lines correspond to the average over 60 repetitions, whereas the shaded areas correspond to standard error on the mean. UCB-ALD and TS-ALD both lead to faster convergence towards optimal correction (logMAR = 0) compared to controls where the context (TS, UCB, random) or the correction (BALD, random) are not adaptively selected. Table 3: Comparison of acquisition functions on the refractive error correction experiment with 8 different slopes (evenly spaced between between 1.0 logMAR-1 and 8.0 logMAR-1). The two heuristics combining BO and active learning ( UCB-ALD and TS-ALD) outperform the controls where the context (TS, UCB, random) or the correction (BALD, random) are not adaptively selected. Acquisition rule | Rank | Borda score
---|---|---
UCB-ALD | 1 | 35
TS-ALD | 2 | 26
UCB (random context) | 3 | 15
BALD | 4 | 8
TS (random context) | 5 | 3
Random | 6 | 0
|  |

So far, we considered the case where the optimum of the function $f$ is the
same across all contexts. In general, the maximum of $f(\boldsymbol{s},\cdot)$
may depend on $\boldsymbol{s}$. In that case, several criteria may be
considered. Either we may want to maximize the averaged performance of the
system across contexts, or we may want to maximize the worst-case performance
of the device. The former scenario corresponds to the ConBO framework
(Pearce2020), whereas the second corresponds to Max-Min BO (Weichert2020). Our
approach is a generalization of these frameworks as we consider cases in which
the context directly determines the amount of information about $f$ at the
sampled points.

The generalization of the ConBO framework to the binary case is given by:

$g(\boldsymbol{x})=\mathbb{E}_{\boldsymbol{s}}\left[\pi[f(\boldsymbol{x},\boldsymbol{s})]\right],$
(21)

whereas the generalization to max-min optimization is:

$g(\boldsymbol{x})=\underset{\boldsymbol{s}\in
S}{\min}\pi[f(\boldsymbol{s},\boldsymbol{x})].$ (22)

So far, we considered the case where the optimum of the function $f$ is the
same across all contexts. We now relax this assumption 3 and consider the case
where the goal is to maximize the average performance of the system, that is
$g$ and $f$ are related in the following way:

$g(\boldsymbol{x})=\mathbb{E}_{\boldsymbol{s}}\left[\pi[f(\boldsymbol{s},\boldsymbol{x})]\right]$
(23)

Generalization of UCB-ALD to this case is not trivial, as it would require to
integrate the uncertainty on $\Phi(f(\boldsymbol{s},\boldsymbol{x}))$ over all
contexts. However, we can generalize TS-ALD by sampling a function
$\widetilde{f}$ from the posterior $p(f|\mathscr{D}_{t})$, then compute the
corresponding sample:

$\widetilde{g}(\boldsymbol{x})=\int\widetilde{f}(\boldsymbol{s},\boldsymbol{x})d\boldsymbol{s}$

We can then select $\boldsymbol{x}_{t+1}$ by solving:
$\boldsymbol{x}_{t+1}=\underset{\boldsymbol{x}\in\mathcal{X}}{\arg\max}\
\widetilde{g}(\boldsymbol{x})$. The context $\boldsymbol{s}_{t+1}$ is then
selected according to BALD, to maximize mutual information between the
response $c_{t+1}$ and the function $f(\boldsymbol{x}_{t+1},\cdot)$.

Again, we assessed our method on synthetic benchmarks, taking the first
dimension of the benchmark as the context variable, and the remaining
dimension as the variable with respect to which the function $g$ is optimized.
We assumed a uniform distribution of the context and computed the expectations
by using a built-in MATLAB function for global adaptive quadrature.

##  6 Discussion

###  6.1 Summary of contributions

In this paper, we introduced a new framework for binary and preferential BO:
BO in adaptive contexts, in which the experimenter can choose the context
where each function evaluation is performed. We proposed an acquisition
function to jointly select the optimization variable and the context and
showed how to generalize existing acquisition functions to this framework by
combining them with Bayesian active learning. We show on synthetic benchmarks
that the proposed algorithms outperform controls in which the context is not
adaptively selected, in both BOs with binary or preference feedbacks. Finally,
we showcase our framework on a concrete problem: tuning the parameters of the
lens for patients with refractive errors.

###  6.2 Related work

In the bandit setting, contextual GP bandit (Krause2011) extends the GP bandit
framework to scenarios in which at each round corresponds a given random
context, and actions are taken knowing this context. The goal is to identify
the best arm averaged over all possible contexts. This framework has been
generalized to dueling bandits (Dudik2015). Our work differs as we consider
the problem of both selecting a context and action at each iteration.

The methods most related to our work are BO with expensive integrands
(Toscano-Palmerin2018), conditional BO (Pearce2020), and contextual policy
search (Pinsler2019), where the goal is to maximize the expected performance
conditionally on a random context variable. During the optimization process,
this context variable is controlled by the experimenter. In BO with expensive
integrands (Toscano-Palmerin2018), the goal is to maximize a function averaged
over a set of random conditions, when function evaluation is noise-free or
with i.i.d. Gaussian noise. The authors used a generalization of the knowledge
gradient. Pearce2020 further generalized by considering the case where
function evaluations are performed in parallel and batch of inputs are
selected at each iteration. Our work is a generalization to the binary and
preferential feedback scenarios, which implies new challenges. Indeed, even
when the objective function is the same across different contexts, the amount
of information we get from each evaluation varies with the context in a way
that is a priori unknown. This problem arises in general when the amount of
noise in function evaluation varies with the context, but we only considered
the case of probit likelihoods for simplicity. We also introduced sequential
acquisition rules that allow the generalization of acquisition rules used in
BO. We did not consider whether sequential acquisition also improves
performance in the case of Gaussian likelihood, and this should be the topic
of future work. In this paper, we considered the case where the optimum of the
function $f$ is the same across all contexts. In general, the maximum of
$f(\boldsymbol{s},\cdot)$ may depend on $\boldsymbol{s}$. In that case,
several criteria may be considered. Either we may want to maximize the
averaged performance of the system across contexts, or we may want to maximize
the worst-case performance of the device. We let the study of these
generalizations for future work.

###  6.3 Code availability

Matlab implementation available at https://disclose_upon_release.


11affiliationtext: Department of Statistics and Actuarial Science, University
of Stellenbosch22affiliationtext: School of Actuarial Science, University of
Cape Town33affiliationtext: Department of Statistical Sciences, University of
Cape Town44affiliationtext: Peresec, Cape Town**affiliationtext: Corresponding
author<EMAIL_ADDRESS>
# Epistemic Limits of Empirical Finance: Causal Reductionism and Self-
Reference

Daniel Polakow  Tim Gebbie  Emlyn Flint

###### Abstract

The clarion call for causal reduction in the study of capital markets is
intensifying. However, in self-referencing and open systems such as capital
markets, the idea of unidirectional causation (if applicable) may be limiting
at best, and unstable or fallacious at worst. In this work, we critically
assess the use of scientific deduction and causal inference within the study
of empirical finance and econometrics. We then demonstrate the idea of
competing causal chains using a toy model adapted from ecological
predator/prey relationships. From this, we develop the alternative view that
the study of empirical finance, and the risks contained therein, may be better
appreciated once we admit that our current arsenal of quantitative finance
tools may be limited to ex post causal inference under popular assumptions.
Where these assumptions are challenged, for example in a recognizable
reflexive context, the prescription of unidirectional causation proves deeply
problematic.

‘Everything that is born is necessarily born through the action of a cause’

Timaeus, Plato c. 360 BCE

‘I can calculate the motion of heavenly bodies, but not the madness of people’

Sir Isaac Newton, 1720 CE

Keywords— Causal Investing; Reflexivity; Empirical Finance; Epistemology; Risk
Management; Spuriosity

##  1 Introduction

There is an obvious resurgence of interest in causal inference. For good
reason; causality is arguably the holy grail of any scientific enquiry
(Little, 1998) and has been deliberated since the time of Hume (1739).
Existing coverage on causality in the philosophy literature is comprehensive
and, for brevity, we will not be reviewing this in detail here.

Rigor and a prescription of causal relevance has also pervaded finance and
economics disciplines; including econometrics (Angrist and Pischke, 2010; Chen
and Pearl, 2013; Ahmed, 2022) and investment management (Wilcox and Gebbie,
2014; López de Prado, 2023a). Concurrently, Quantitative Finance (QF), which
generically includes the investment and econometrics disciplines, is embarking
on a purge of ‘incorrect’ statistical methodologies. Such flawed statistical
approaches have resulted in a proliferation of false claims and charlatanism
(Ziliak and McCloskey, 2008; Harvey and Liu, 2014; Bailey et al., 2014, 2017;
López de Prado, 2015, 2023a, 2023b).

The uptake of defensible statistical methods is a progressive movement within
QF that, along with more cautious distillation of causal pathways, supports
scientific realism as well as positive economics. However, while explaining
economic phenomena is the intended result111Sensu the positive economics of
Friedman (1953) and the apparent rise of economics as a ‘cyborg science’
(Mirowski, 2001)., the movement also has the potential to incorrectly totalize
disparate phenomena. In particular, compared with prescriptive normative
economics, such an approach can become uncoupled from market ontology, leading
to a disconnect between the reality of capital markets and the assumptions –
and thus output knowledge – of its reductionist studies.

This contribution has foci on financial risk, specifically model risk in
relation to a single common assumption pervasive in QF studies; namely, that
markets, being social systems, adhere sufficiently to epistemic norms. These
epistemic norms carry with them a priori the necessity of unique, well-defined
causal chains that can be meaningfully extracted from data in a positive
economics framing and a reductionist scientific sense.

##  2 Causal chains

To understand epistemic norms in markets, first we note that there are some
important assumptions underpinning economic thinking. Of course, there exist
several different economic schools of thought. The dichotomy between
neoclassical, or orthodox, and heterodox economics (arguably, everything else)
is frequently made to smooth over and sometimes smother the pragmatic
importance of these assumptions as a binary framing that casts idealisms and
normative approaches in opposition to positivist perspectives.

While orthodox approaches idealize an economic system in which the rationality
of participants is key, heterodox economics emphasizes inefficiencies and
imbalances generated by the system. Nonetheless, both modern orthodox and
heterodox economic approaches aim to incorporate a range of imperfections
within their standard frameworks to render them more realistic. In this
setting, realism is defined as the ability to provide narrative explanations
that are cohesive within a given ontology, or to make successful predictions.

At the heart of the positivist agenda, which resides within the broader
scientific realist school of thought, is the goal of causal explanation; and
unidirectional causation is the de facto supposition. In this regard, of
contemporary relevance is the work spearheaded by Pearl and co-workers (Pearl
(1995, 2009a, 2009b); Pearl et al. (2016)) who have formalized empirical
causal modeling and hence laid the foundations of much of modern artificial
intelligence.

Unidirectional causality refers to $A$ causing $B$, for example the price
action in $B$ being caused by market interest $A$, ceteris paribus. In terms
of the mathematics of causality (Pearl, 1995), we express $B$ as some function
of $A$:

$P[A=a]\leavevmode\nobreak\ \mathrm{do}[B=b]>P[A=a].$ (1)

The ‘do’ operator is forcing $B=b$ by intervention. A diagrammatic
representation of Eqn. [1] is provided in Figure 1(a).

Eqn. [1] and its associated framework accommodates both neoclassical
equilibria and heterodox dynamics, depending on the size and term of the
arbitrageable phenomenon under consideration. Mathematician Robert Buck
defined ‘self-frustrating’ predictions as forecasts that are initially true,
but become false on public dissemination (Buck, 1963). Moreover, Popper (1957)
in his writings on social sciences refers to ‘self-limiting’ predictions
influencing the predicted event in a preventative sense. Markets are similarly
understood to adapt as the outputs of scrutiny are assimilated. This
understanding is accommodated within the idealized neoclassic theory of
economics.

Returning to Eqn. [1], unidirectional causality does not permit $B$ to cause
$A$ and it does not facilitate $A$ being limited by its subsequent impact on
$B$ via self-limiting feedback. While causal graphs are typically linear and
unidirectional, self-limiting processes with dampening feedback are commonly
tolerated without being fatal to ex post inference. Importantly, contemporary
QF is already suitably cognizant of this ‘adaptation’ of the market to new
information (López de Prado, 2015; McLean and Pontiff, 2016; Falke et al.,
2022).

More generally, adaptation is a feature common to ‘reflexive’ systems; namely,
systems that refer to themselves. While proponents of causal rigor recognize
that markets are adaptive, the fact that they are, more broadly, self-
referential is commonly overlooked. Specifically, markets may be possessed by
dynamics that extend further than simply adaptation.

A self-referential system is reflexive in nature. This is a surprisingly
pervasive idea: the ‘Oedipus effect’ of Popper (1957), the ‘self-fulfilling
prediction’ of Buck (1963), or the ‘back-coupling’ of Morgenstern (1972).
MacKenzie (of performativity literature renown) noted that the study of
economics does more than simply describe, but rather shapes and changes the
conditions of the economy, and society more broadly (MacKenzie, 2006;
MacKenzie et al., 2007). It thus possesses the ability to analyze, describe,
and modify its own structure, behavior, and properties. To say this
phenomenology of self-reference is well-known, recurring and deeply studied is
an understatement.222Self-referential study originates from Spencer-Brown’s
Laws of Form (Spencer-Brown, 1969) and extends to the calculus developed by
Varela (1975). The field finds parallels in philosophy (radical
constructivism; Von Glasersfeld (1984)), control theory (e.g. second-order
cybernetics; Von Foerster (2003)), and economics (the Eigenform of markets;
Kauffman (2009)). Daniel Hofstadter discusses self-reference systems
extensively in his popularist book (Hofstadter, 1999).

Lehmann-Waffenschmidt and Sandri (2007) characterize two dynamics of reflexive
systems: self-reinforcing and destabilizing (via positive feedback
loop).333Goodhart’s law (Goodhart, 1984): any observed statistical regularity
will tend to collapse once pressure is placed upon it for control purposes.
Campbell (1979)’s law prior and the ‘Lucas Critique’ (Lucas, 1976) are further
examples of reflexive responses. Markets are thus not limited to preventative
feedback loops; instead, such loops may also be destabilizing. For example,
predictions such as fundamental market value can persist or change solely due
to participants’ beliefs. Furthermore, runs on banks, asset bubbles, the
volatility implied in option prices, booms-and-busts and the allure of factor
exposures are all characteristic of a system possessed by self-referential
properties.444Such examples may also, however, characterize systems that are
not self-referencing. Discerning a parsimonious likelihood function is
therefore not trivial. Within such systems, sufficient conviction in an
association can, in fact, result in (i.e. cause) that same association
(Polakow, 2010). The notion of ‘spuriosity’ thus also becomes ambiguous.

Tony Lawson, the eminent heterodox economist, explains why markets ultimately
cannot adhere to epistemic norms. Lawson (2003) defines closed systems to
refer to a situation in which a correlation (an event regularity) occurs. In
the absence of conditions supporting such correlations, the system is then
deemed to be ‘open’. Markets, being a social system, are described as an ‘open
system’.

“… modern mainstream economics is awash with assumptions of perfect foresight
or rational expectations, of efficient markets, and of market equilibrium,
all, at least in the manner they are typically employed, essentially premised
on the successful predictability of future outcomes; all supposing that in an
open system actual outcomes can be effectively anticipated. Indeed all such
endeavour in effect treats open systems as if there are actually closed” –
Lawson (2003).

Markets are unequivocally an open system.555Here, a closed system means a
closed system of regular events, and implies predictability. An open system is
then one in which prediction is itself not regularly achievable. Such a system
often involves recursive or circular relationships, where the system’s
components interact or define themselves in terms of other components within
the same system. Of relevance here, self-referential systems are possessed of
unusual properties, highly complex and challenging to understand or analyze,
especially using traditional reductionist approaches and assumptions of
causation (Soros, 2013).

The dynamics of causal chains within self-referential systems are often
paradoxical. Umpleby (2010) notes that in self-referencing systems we
transition from a classical paradigm of inferring A causes B, to higher-order
dynamics with very different notions of stability, independence, structure and
temporality. The very idea of model building and usage becomes nuanced. The
presence of reflexivity will frequently result in circular reasoning – deemed
fallacious by contemporary academic standards.666Reflexivity further violates
the ad hominem informal fallacy and the fallacy of accent: markets operate at
both ‘observing’ and ‘participating’ levels (Umpleby, 2010). The problem of
causal emergence will frequently present itself in the form where multiple
competing versions co-exist and compete (Hoel, 2017).

Markets comprise a system of interacting actors, with multiple levels of
causation (Wilcox and Gebbie, 2014). The debate is not whether markets, being
an obvious social system, are self-referential or not. Rather, we believe the
controversy revolves around two points. Firstly, to what extent does such a
self-referential system differ ontologically from the closed system we
commonly assume? Secondly, can such a self-referential system produce
behaviors and inefficiencies that are persistent and non-arbitrageable?

Neoclassical economic thought, arguably the dominant paradigm, is premised on
the basis that imbalances are only temporary and so the existence of short-
term market imperfections are tolerated in the system.777In this vein was
Soros’ view of reflexivity rebuffed by the economic establishment as ‘merely
stating the obvious’ (Soros, 2009).

Heterodox thinking is more permissible to tolerating longer-term
disequilibrium; e.g. simple ecological models applied to market dynamics
result in endogenous inefficiencies and deviations from notional fundamental
values (Galla and Farmer, 2013; Scholl et al., 2021; Sanders et al., 2018).
Bouchaud (2013) reviews modelling efforts that include heterogeneities as well
as interactions, and proposes a potential unifying framework using Random
Field Ising models for both endogenous ruptures and crises, including ‘self-
referential’ feedback loops for non-equilibrium phenomenology missing in
classical economic models. Hirshleifer et al. (2023) examine the extent to
which heterogeneity of investment styles can co-exist in the long-term, in
contrast to the predictions wrought from traditional theory, and in response
to social contagion. The study builds on the Adaptive Markets Hypothesis (Lo,
2004, 2017), under which market participants behave (mostly) rationally using
a set of natural-selection heuristics. Such heuristics can become maladaptive
during exogenous shocks.888The rise in popularity of dynamic stochastic
general equilibrium (DSGE) models has been catalyzed by the need for policy
analysis and forecasting away from classical economic core frameworks to
permit for more richness and flexibility, but DSGE models have not been as
successful as wished (see Stiglitz (2018)). We further note that a return to
some notional equilibrium is an inherent feature in all DSGE models.

AB+-$1$$2$

(a) Conventional causal graph

AB+-$3$$4$

(b) Alternative (dual) causal graph

Figure 1: A conventional unidirectional causal graph of $A$ causing $B$ (path
$1$) is shown in Fig. 1(a); here with the addition of a negative (self-
limiting feedback loop, denoted “-”), and a positive (destabilizing feedback
loop, denoted “+”), both of the latter dependencies along path $2$. Fig. 1(a)
(path 1 only) graphically represents Eqn. [1]. This can be compared to the
alternative unidirectional causal graph shown in Fig. 1(b) (path 3 only),
which represents Eqn. [2], where $B$ is shown causing $A$ along path $3$, with
the addition of negative (self-limiting feedback loop, denoted “-”); and
positive (destabilizing feedback loop, denoted “+”), the latter both along
path 4. Feedback dynamics between these causal chains, 1(a) and 1(b), may be
indistinguishable unless intervention is involved.

Yet, it appears that both schools of thought – neoclassical and heterodox –
retain the view that unidirectional causality remains relevant as a formalism,
even if these linear cause-effect or circular dampening graphs exist only as
component processes enmeshed in an ensemble of interactions of a more complex
system (Hardy, 2001; Hommes, 2021).

If causality is in fact more malleable in a reflexive system (perhaps to the
point of being wholly subjective) there is a competing reality that always
requires admission as a hypothesis in any system prone to reflexivity, and
that is: B causes A (ceteris paribus):999e.g. the price action in B causes the
market interest A.

$P[B=b]\leavevmode\nobreak\ \mathrm{do}[A=a]>P[B=b]$ (2)

A diagrammatic representation of Eqn. [2] is provided in Figure 1(b). It is
critical to realize that the feedback dynamics embedded in Eqn. [1] will
effectively be indistinguishable from those of Eqn. [2] unless intervention is
invoked.101010$B$ impacting $A$ via path 2 may be indistinguishable from $B$
causing $A$ via path 3, unless by control (intervention), and then only under
certain conditions. This intervention is inevitably top-down in nature (Craver
and Bechtel, 2007; Auletta et al., 2008).

Bouchaud explains how disappointing quantitative success has been within the
economic sciences, and motivates for a break away from classical economic
tools (Bouchaud, 2008, 2013). Lehmann-Waffenschmidt and Sandri (2007) note
that within economics, it is not possible to draw conclusions on the logical
legitimacy of any reflexive mechanism. Shaikh (2013) noted that even heterodox
economics maintains the notion of an equilibrium and that different economic
foundations are required for adequately dealing with reflexivity. The
scientific fallibility of economics was identified and discussed
comprehensively in the early 1980s (McCloskey, 1983). Soros (2013) argued
plainly that markets did not benefit from the same epistemic norms of science.
In particular, if financial models operate within a self-referential framework
they can be non-causal, resulting in an inability to predict (De Scheemaekere,
2009).

In summary, the presence of reflexivity poses a clear challenge to the
epistemology of QF since it is not accommodated within a unidirectional
causality paradigm. The reason why reflexivity does so is simply stated:
chains of causality are ostensibly malleable by the actors within markets. For
guidance (and illustration) we turn to a well-understood set of complex
systems: ecology.111111Ecosystems are usually thermodynamically open, but
causally closed with some regularity and predictability.

##  3 Causal predation

Extending ecology metaphors within finance has precedent (Scholl et al.,
2021). Contemporary empirical views provide examples of predator-prey models
explaining, for example, S&P500 dispersion in terms of contrarian and trend-
following agents competing with and predating upon each other (Lux, 2021). As
a useful metaphor, we explore the co-existing dynamics of two competing causal
chains within a similar self-referential system, adapted from ecology (May,
1976; Sugihara et al., 2012). This forms an illustrative toy model of simple
reflexive causal chains that can generate complicated dynamics.121212The
concept of ‘regularity’ can be further restricted – e.g. Granger causality as
the measure of predictability (Sugihara et al., 2012) – this does not change
our argument.

Consider a single self-dependent and time-dependent feature $x(t)$ that can be
well modeled by a logistics function, here with rate of change parameter $r$,
and carrying capacity $K$

$\dot{x}(t)=rx(t)(K-x(t)).$ (3)

where the dot represents a time derivative. The feature $x$ could be taken to
represent the number of individuals in a population, or the log-price of an
asset, or the wealth associated with the accumulated profit-and-loss of some
trading strategy conditioned on various information sources. This model can be
randomly perturbed, although we keep the argument deterministic.

Once discretized, the model demonstrates the well-understood bifurcation
phenomena shown by May (1976). This is shown by replacing the feature
derivative with the derivative of a logarithm of the feature from Eqn. [3] and
converting it into a difference equation: $\ln x_{n+1}-\ln x_{n}=r(K-x_{n})$.
By taking the exponential of the log-difference equation we find the Logistics
Difference Equation (LDE)131313The Logistics Map (LM)
$x_{t+1}=rx_{t}(1-x_{t})$ is the first order approximation to the LDE using
$e^{x}\approx 1+x-\frac{1}{2}x^{2}+\ldots$. :

$x_{n+1}=x_{n}e^{r(K-x_{n})}.$ (4)

We now progress the metaphor, but for coupled features that include an
interaction term. For simplicity we restrict ourselves to two features $x(t)$
and $y(t)$, with constant causal couplings141414The Lotka-Volterra (LV) model
has the form, for two chains $S_{1}$ and $S_{2}$:
$\dot{S_{1}}=aS_{1}-bS_{1}S_{2}$, and $\dot{S_{2}}=cS_{2}-dS_{1}S_{2}$.:

$\displaystyle\dot{x}(t)$
$\displaystyle=rx(t)\left({K-x(t)-a_{yx}y(t)}\right),$
$\displaystyle\dot{y}(t)$
$\displaystyle=ry(t)\left({(K-y(t)-a_{xy}x(t)}\right).$ (5)

There are two coupling parameters $a_{yx}$ and $a_{xy}$. If the two features
were correlated then this parameter would be symmetric. The argument can be
generalized to many such features. To generate causal chains, we discretize
the coupled differential equations (Hastings, 1993; Sugihara et al., 2012) and
find the causal chain update equations:

$\displaystyle x_{n+1}$ $\displaystyle=x_{n}e^{r_{x}(K_{x}-x_{n}-a_{y\to
x}y_{n})},$ $\displaystyle y_{n+1}$
$\displaystyle=y_{n}e^{r_{y}(K_{y}-y_{n}-a_{x\to y}x_{n})}.$ (6)

This model has two different growth rates, two different carrying capacities,
and two dependency variables. With $a_{y\to x}\neq 0$, but $a_{x\to y}=0$ this
would be unidirectional, from $y$ to $x$. These are now situated as part of a
sequence of unidirectional causal relationships in a chain, where each iterate
$x_{n}$, can be thought of as a realization of one node, and similarly,
$y_{n}$ for the other, as a realization of the graph in Fig. 1.

We can simulate the chain of unidirectional causal graphs and visualize key
features of the resulting dynamics by considering iterations of the coupled
LDE. The resulting coupled bifurcation map is shown in Fig. 2. This renders
our key points concrete.

We see that even if we explicitly know, with certainty, the deterministic and
non-stochastic causal relationship between chains $x_{n}$ and $y_{n}$ (or even
if we know the causal relationships between the underlying causal features
$x(t)$ and $y(t)$) we cannot easily predict $x_{n}$ at all values of $r$ (nor
$K$) even if $y_{n}$ is known with certainty itself.

Figure 2: A model of the iterative stability of sampled sequences of
unidirectional causal graphs. This models the causal predation of the system
using the Logistics Difference Equations (LDE) from Eqn. [6]. The model
includes self-interactions and cross-interactions where a feature $x$ has a
unidirectional causal dependency on another feature $y$, but where $y$ is not
dependent on $x$. The discretized features represent realizations from causal
chains. The couplings introduce further instability and complexity. The y-axis
is the feature value (population size, strategy price, or accumulated profit-
and-loss) and is dependent on the growth factor $r$ given on x-axis, and the
carrying capacity $K$. In this example $r_{x}=r$, $r_{y}=0.95r$, $K_{x}=0.95$,
$K_{y}=1.00$ and $a_{y\to x}=-0.1$; any variety of combinations can be chosen.
The rounding significance is $\epsilon=10^{-4}$.

As a simple representation for causal chain competition and predation, Eqn.
[6] and Fig. 2 have the further benefit of making clear the role of
randomness.151515Difference equations can be thought of as filters, this
implies that erratic behavior due to sampling could be ameliorated using
filtering e.g. low pass filtering. For example, either $r$ (or $K$) can be
changed to be uniformly distributed, and then sampled at exponentially
distributed waiting times; perhaps representing exogenous shocks from top-down
causal sources from the broader environment, either changing $r$, or $K$. We
can also randomly change the interaction terms, and so on. These introductions
of randomness do not improve the situation and can undermine attempts to close
the system using additional filtering. In this framework, it is unreasonably
optimistic to assume that one can meaningfully infer stable causal graphs
uniquely from the data.

That is not to say causal graphs are not useful. In fact, for exploratory data
analysis, or visualizing the hypothesized system state, or rendering
assumptions transparent, causal graphs can be informative and should be
encouraged. However, even if causal chains are unique and can be reliably
extracted, this does not imply that the representation can faithfully render
the dynamic properties of the system itself.

In an open system, causality ultimately remains phenomenological. Markets are
reflexive, adaptive and saturated with agents responding strategically to each
other. It is thus understood that coherence of perception around a common
cause will often lead to forms of tight coupling with high cohesion, and thus
fragility, being written into the financial system itself.

##  4 Conclusion

Research may retrospectively support unidirectional chains of causality and
their corresponding inference. However, the conditions in which ex post
understanding can be extended to support ex ante prediction are limited. This
is caused by the market’s adaptation mechanism combined with the understanding
that chains of causality are malleable by market actors. This necessitates a
distinction of practical significance between “model estimation”, wherein the
model is presumed to be normatively true and the data representative of the
future, and “model calibration”, where one rather seeks to facilitate reliable
decision-making under uncertainty using a model that may be misspecified or
simply incorrect. This theme – a default necessity for calibration despite the
presence of ex post confidence – is well-known from derivative pricing
(Schoutens et al., 2004). It is unlikely that these chains can be used without
considerable risk for prediction or control in a forward-looking way.

If reflexive, the degree to which markets behave as a closed system is likely
to be a function of only teleological conviction. This means that there may
well be nothing tangible that supervenes various model representations. To the
extent that certain QF actors dominate markets, a causal chain supporting
their particular narrative will become established. The risk then exists of
other actors benefiting from distinct causal chains coming to bear. The system
is therefore prone to instability even in the absence of exogenous shocks. In
the age of hyperconnectedness, the shifting of perception and sentiment has
never been as fungible, or as inexpensive.161616Baudrillard (1981) called this
the ‘precession of simulacra’. It should be anticipated that without a more
complete theory of complexity, QF will remain lost in a spiral of tautological
self-reference.

As scientists, and as academics of QF, we naturally aspire to high standards
of critical thinking. Despite markets exhibiting unusual epistemic dynamics,
scientific reductionism in QF is deployed as an inviolate prescription.
However, the assumption of unidirectional causation will not always lead to
the appropriate causal inference. In contrast, within a highly reflexive
system, the pursuit of causality may, in fact, be doomed to failure.

Empirical finance may frequently find itself searching incorrectly for answers
under the only epistemological light within which it can navigate. To the
extent that markets closely proxy closed systems, the risk of violating
directional causal assumptions is arguably limited. Yet, if markets do have
the proclivity to ‘switch’ between different casual directions (as in Fig. 2),
QF is presented with an existential challenge. In such a scenario of overt
reflexivity, empirical finance is expected to awaken within a deeply
paradoxical associational relativity. Here, as in other contexts, realistic
attempts at hypothesizing more complex causal models will become multifaceted
and are less likely to give rise to useful inference (Rasmussen et al., 2019;
Saylors and Trafimow, 2020).

None of this is novel in the real-world application of QF i.e. in option
pricing, trading and asset management. One of the early approaches to dealing
with the causation problem in factor investing – path-dependencies and
unidirectional dependencies – was to introduce ad hoc models, such as those of
Haugen and Baker (1996) and Ferson and Harvey (1999), and relax the linearity
assumption of the pricing kernel (Wilcox and Gebbie, 2015). This early
modeling of dynamic temporal and multi-layered dependencies may appropriately
be couched as attempts at “machine learning” and are not essentially
inventive, nor do they progress the central argument here. Hence, we
deliberately avoid discussion of the “data-science problem” or the merits or
fallibility of any specific algorithm. The veracity of modeling tools in
reflexive systems takes on a life of their own, and invariably has less to do
with any computational tool or trick, and more to do with the nature of
adoption and dissemination of the same. The fundamental nature of the problem
remains, irrespective of what tools are fashionable.

We believe causal and scientific thinking may ultimately be limited within the
prescription of current QF tools for two reasons. The first presupposes that
markets are not sufficiently reflexive to matter and reductionist approaches
assuming unidirectional causality are valid. Within this paradigm, causal
inference is limited to ex-post, unless the phenomenon under study is non-
arbitrageable (because of the ‘tragedy of the commons’ (Hardin, 1968)). The
ability to forecast ex ante may ultimately be a rarity, and QF should be
cognizant of overreach. Arguably, this understanding would accomplish much of
the clean-up required within QF. The second limitation is that statistical and
causal inference will plainly be applied incorrectly where reflexivity is
present. For these reasons, we argue that empirical finance is not at risk of
becoming a pathological science (sensu López de Prado (2015)). Rather,
empirical finance is categorically a pathological science where it is able to
act as one at all (i.e. where not sufficiently reflexive to matter), and
patently not a science elsewhere.

We close with words from one of the foremost thinkers in causal economics,
Joshua Angrist who noted: ‘… Perhaps it’s worth restating an obvious point.
Empirical evidence on any given causal effect is always local, derived from a
particular time, place, and research design. Invocation of a superficially
general structural framework does not make the underlying variation or setting
more representative. Economic theory often suggests general principles, but
extrapolation of causal effects to new settings is always speculative.
Nevertheless, anyone who makes a living out of data analysis probably believes
that heterogeneity is limited enough that the well-understood past can be
informative about the future.’ (Angrist and Pischke, 2010). Heterogeneity is
certainly limited in some economic systems - for example where consumer
preferences are simple and demand forecastable; similarly in commodity driven
economies, single-currency zones and in traditional societies etc. Yet, the
challenge remains the surprising frequency with which the well-understood past
does not in fact inform the future elsewhere. For example, it has been noted
that the only consistent feature of economic and quantitative finance
disciplines is the distinct inability to predict and avert crises (Bouchaud,
2008).

The disparity between the two broad schools of economic thought is perhaps
dialectical, being two sides of a causally confused coin. As yet, we do not
have a viable alternative paradigm that accounts for the anomalies, and that
can provide both a general and operational approach to the heterodox
scientific project (Ghilarducci et al. (2023)). Such a viable alternative is a
key requirement for any Kuhnian paradigm shift. In deeply reflexive systems,
the heterodox project may in fact lack the necessary scientific foundation -
since it is without the predictive reach of causality. This acknowledgement is
belated to the discipline, and the perils of ignoring this phenomenological
reality are potentially grave. Causal predation (and the conditions of causal
chain co-existence) are not satisfactorily assimilated in current thinking. It
is clear that QF (and economics more broadly) would benefit from a better
understanding of what knowledge is likely to be impacted by such reflexivity,
and consequently the risks of attaching policy intervention, regulation or
investment to the same.

##  5 Acknowledgements

We thank Diane Wilcox and Michael Jennions for engaging conversations on the
topic. The views expressed here are solely those of the authors.

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