# On the Map-Territory Fallacy Fallacy

Maxwell J D Ramstead<EMAIL_ADDRESS>VERSES Research Lab, Los
Angeles, CA, 90016, USA  Wellcome Centre for Human Neuroimaging, University
College London, London, UK  Dalton A R Sakthivadivel
<EMAIL_ADDRESS>VERSES Research Lab, Los Angeles, CA,
90016, USA  Department of Mathematics, Stony Brook University, Stony Brook,
NY, USA  Department of Physics and Astronomy, Stony Brook University, Stony
Brook, NY, USA  Department of Biomedical Engineering, Stony Brook University,
Stony Brook, NY, USA  Karl J Friston<EMAIL_ADDRESS>VERSES Research Lab,
Los Angeles, CA, 90016, USA  Wellcome Centre for Human Neuroimaging,
University College London, London, UK

(28th August 2024)

###### Abstract

This paper presents a meta-theory of the usage of the free energy principle
(FEP) and examines its scope in the modelling of physical systems. We consider
the so-called ‘map-territory fallacy’ and the fallacious reification of model
properties. By showing that the FEP is a consistent, physics-inspired theory
of inferences of inferences, we disprove the assertion that the map-territory
fallacy contradicts the principled usage of the FEP. As such, we argue that
deploying the map-territory fallacy to criticise the use of the FEP and
Bayesian mechanics itself constitutes a fallacy: what we call the map-
territory fallacy fallacy. In so doing, we emphasise a few key points: the
uniqueness of the FEP as a model of particles or agents that model their
environments; the restoration of convention to the FEP via its relation to the
principle of constrained maximum entropy; the ‘Jaynes optimality’ of the FEP
under this relation; and finally, the way that this meta-theoretical approach
to the FEP clarifies its utility and scope as a formal modelling tool. Taken
together, these features make the FEP, uniquely, the ideal model of generic
systems in statistical physics.

##  I Introduction

Bayesian mechanics and the free energy principle (FEP) have been proposed as a
universal modelling methodology for physical things, and in particular, for
self-organising systems [1, 2, 3]. The FEP is a mathematical statement (or
indeed a physical principle) which provides a unifying perspective on formal
approaches to behaviour and dynamics [4]. Bayesian mechanics, a corollary of
the FEP, comprises a set of tools to write mechanical theories (i.e., laws of
motion) that relate the dynamics of states or paths of a physical system to
the dynamics of probabilistic beliefs (or conditional probability densities)
that are encoded by some (so-called internal) subsets of the states of the
system [5]. These internal states (or their dynamics) are coupled to other
(external) subsets of the system in such a way that they can be read as
parameters for probability densities over the external subset. Bayesian
mechanics unifies first-principles-based modelling efforts in physics by
integrating the FEP with one of the cornerstones of contemporary physics,
namely, the constrained maximum entropy principle (CMEP) [6]. As a
consequence, these are tools that allow us to model any particle, or
‘thing’—in the sense that it exists as a separable, identifiable thing [7,
8]—as if it were inferring the causes of its sensory perturbations and
(re)acting so as to remain in system-like configurations. This inference
proceeds by the minimisation of a free energy functional of such beliefs as
parameterised by internal states, which guarantees—or describes—the
minimisation of the surprisal of particular states. Using this approach, we
can model the dynamics (i.e., the observable behaviour) of things or particles
as realising a path of least surprisal. Bayesian mechanics formalises the
truism in physics that, on average, systems will do what it is that they do on
average. Behind this truism lies a formal apparatus that enables us to write
many of the core formulations of contemporary physics in a parsimonious
manner. The hope is that this will allow one to take some first steps towards
a mechanistic explanation of how self-organisation arises in physical systems.

In this paper, we begin the development of a bona fide meta-theory, and
philosophy for the usage, of the FEP. A good starting point for this project
would be the so-called map-territory fallacy, which some have alleged is
committed by the architects of the FEP [9, 10]. Indeed, some scholars have
argued that FEP-theoretic modelling conflates the metaphorical ‘map’ (i.e.,
the scientific model that scientists use to make sense of some target
phenomenon) and ‘territory’ (i.e., the actual natural system that is being
modelled). See [11] and [5] for a related critical discussion of such claims.
On this view, using the technology of FEP (and Bayesian mechanics more
broadly) to claim that things or particles engage in inference constitutes a
case of model reification. That is, some have alleged that, in describing the
dynamics of physical systems as inference, FEP theorists mistakenly conflate
their metaphorical ‘map’ of the territory (i.e., the scientific FEP-theoretic
or Bayesian mechanical model, which is a set of mathematical structures) with
the territory or target system itself, thereby reifying aspects of their model
into what are now assumed to be real features of the territory.

We will argue that deploying the map-territory fallacy to criticise the FEP
and Bayesian mechanics in this manner itself constitutes a fallacy: what we
will call the map-territory fallacy fallacy. We believe that the map-territory
fallacy is fallacious for two main reasons.

Firstly, we will argue that the FEP commits one to an unproblematic111The
story in this paper rests on a deflated and naturalised account of
representation and semantics, which is reviewed in [12, 13, 14]. As we detail
below, any system that can be modelled as an estimator i.e., as engaging in
the statistical estimation of some quantity, is a representation of the
estimated system. We are aware that proponents of anti-representationalist
philosophy of mind might want to resist this notion. Below, we explain why
this resistance is not justified, mathematically speaking. nested
representationalist account of physical dynamics: mathematically, there simply
is no conflation of map and territory, i.e., of modelled system and scientific
model (see also [15, 5, 16, 11]). Indeed, the technology of the FEP may be
used to write down a statistical (generative) model of the states of a system
(or paths of a system through its state space), which constitutes our
scientific models of the target system (as modellers). The model that we
construct contains a partition of the states or paths of the system (a
‘particular’ partition, i.e., into particles); this partition captures
relations of conditional independence relations amongst subsets of the system,
which track each others’ statistical structure dynamically. These subsets of
states engage in inference about (or represent the statistical structure of)
other subsets of states. Thus, we can think of the FEP, metaphorically, as a
map of that part of the territory that behaves as if it were a map. Crucially,
at this level of analysis, there is no conflation between our FEP-theoretic
model and the statistical model that is embodied by a particle in a self-
organising system: under the assumption that systems model their environments,
we can understand their dynamics in virtue of the model they hold. As such,
when we use the FEP to model the way a system models its environment, we
simultaneously model the system itself in a high-fidelity manner. This duality
is key to dismissing the map-territory fallacy and is an _essential_ feature
of the success of the FEP. That this exists in the FEP by consequence of the
fundamental posits of the theory, in particular, a symmetric boundary across
which bidirectional interactions occur, is no accident.

Secondly, we will argue that, when deployed in the context of FEP-theoretic
modelling, the map-territory fallacy rests on some rather deep
misunderstandings about the nature of modelling in physics, and about the
existence and nature of constraints in modelling physical systems in general.
Specifically, we leverage the mathematical fact that, via the principle of
maximum entropy, an FEP-theoretic model of belief updating is the optimal
model of systemic dynamics, to argue that the FEP provides ultimate
constraints on what it means to be an optimal (physical, i.e., embodied) model
of some target physical phenomenon.

This allows for the following: we can leverage this fact to claim that the
‘thing’ we are modelling can also be expected to use the principle of maximum
entropy to model its environment; since the FEP can be read as the application
of the CMEP to model maximum entropy agents modelling their environment, then,
referencing the first point, we can produce the high-fidelity model that we
have indicated. All this means is that, under the assumption that systems
model their environments optimally given some constraints on that model, they
will model their environment using the FEP or CMEP. In turn, this entails that
we can understand their dynamics in virtue of the model they hold by ourselves
using the FEP to reproduce that model. More particularly, we will always
reproduce a map of that part of the territory that makes maps.

Additionally, we argue that (in a metaphorical sense) the FEP and Bayesian
mechanics also provide a map of any possible map whatsoever of, or held by, a
physical system. This undermines any strict notion that maps and territories
must always be held distinct conceptually; and thus defeats the map-territory
fallacy (at least as applied to the FEP). However, this is not to suggest that
the map and the territory are never conflated; they obviously are in cases of
genuine model reification [11]. Instead, we suggest that this leads, in a
Kantian manner (in terms of logically necessary preconditions for sense-
making, and thus the parsing of communicated sensory streams on which
cognition is contingent [13]), to constraints on possible maps, which arise
from what it means to be a map or model of a physical process at all.

We first present a technical introduction to the FEP and Bayesian mechanics,
and examine the claim that the FEP is, metaphorically, a map of that part of
the territory that behaves as if it were a map. It turns out that most things
behave, at some level of description, like maps in this sense. We defuse the
map-territory fallacy at a first level of analysis, showing that there is no
simply ambiguity between model and modelled system in the core FEP formalism.
We then discuss the duality of the FEP and the CMEP, and examine the
implications of this duality for modelling physical systems. We further defuse
the map-territory fallacy regarding the FEP by noting that the FEP/CMEP
duality entails constraints on what counts as an optimal model of a physical
process. Here, the very distinction between map and territory is blurred; and
the metaphor itself is thereby shown to be less useful than might have been
hoped initially. Finally, we summarise our critical discussion and open onto
future research questions that ought to be taken up by philosophers of the
FEP.

Altogether, we present a thesis that modelling internal states modelling
external states constitutes the best possible model of the internal states of
a ‘thing,’ by proving that

  1. 1.

things with internal states model external states, and that this model-of-
models is captured by the FEP (Sections II.1 and II.2),

  2. 2.

the FEP is an optimal model at the particle-level (Section II.3),

  3. 3.

we can achieve a relevant model of those internal states by leveraging
particle-environment symmetry (Section III.1), and

  4. 4.

abiding by this symmetry, the FEP-theoretic model of the beliefs of a particle
is the optimal model at the modeller-level (Section III.2).

Together these justify the _utilisation_ of the FEP at a semi-axiomatic level,
and lay out a set of schema driving its applicability.

##  II The FEP and Bayesian mechanics

###  II.1 Introduction to the FEP

The FEP can be summarised as a principle of least surprisal [1, 5, 17] for
systems that possess a particular partition. A particular partition simply
takes a system and separates it into a set of internal states and external
states by an intervening set of blanket states (also called a _Markov
blanket_). This separation will be central to what follows in two senses.
Firstly, it is an existential imperative for anything. In other words, if the
states of something could not be individuated from everything else, the thing
would not exist. Secondly, when talking about maps—in the formal sense of a
function or morphism—we require the domain and codomain (i.e., the image) of
the map to exist. Therefore, the use of a map in this setting presupposes the
existence of a partition.

Formulated most generally, the FEP says that the paths taken by a system
through its state or phase space are characterised by a Lyapunov function, a
function that varies systematically with its dynamics. The utility of Lyapunov
functions is they allow us to understand those dynamics varying with such a
function as a gradient flow on the function—in other words, it is crucial to
understanding the ‘inference’ component of surprisal minimisation. Under the
FEP, this Lyapunov function is surprisal or variational free energy, depending
on the setting. Surprisal is simply negative log probability. Variational free
energy is a tractable (i.e., easily computed) upper bound on surprisal.
Minimising surprisal implies that the variational free energy is minimised,
and likewise, minimising the variational free energy places an upper bound on
the surprisal. Though not a minimum _per se_ , minimising the variational free
energy is a necessary first step, and a good proxy, for surprisal
minimisation.222In the limit of exact Bayesian inference, when the functional
form of posterior beliefs encoded by internal states coincides with the
posterior density over external states, the variational free energy is just
the surprisal. See e.g. [6, Lemma 4.2]. That the dynamics of a system encode
an inference whenever they minimise this quantity is the consequence of
reading surprisal as a Lyapunov function for those dynamics. This gradient
flow is what we identify as actually being the process (i.e., the performance)
of inference.

As stated, the FEP partitions a ‘system’ into three sets of states: the
particular states of a ‘thing’ or ‘particle’ (which is equipped with internal
states $\mu$), an environment (which possesses external states $\eta$,
relative to the thing in question), and an interface separating but coupling
the two (the so-called Markov blanket, $b$), which mediates the influence of
$\eta$ on $\mu$ and _vice-versa_. When we say ‘recognition model’ or ‘Bayesian
belief’ we mean a parameterised probability density over external states
conditioned on particular states $\mu$ and $b$. The _variational_ free energy
is based on a relative entropy, measuring the divergence between a particle’s
recognition model of its environment and the true posterior conditional
density over external states given blanket states. The recognition model is
parameterised by (the dynamics of) internal states in such a way that the
particle synchronises with the environment. More specifically, suppose there
exists a parametric model of the probability of external states,
$q(\eta;\hat{\eta}_{b},\Sigma_{\eta\mid b})$, where we have parameterised the
recognition model by the average333In fact this is a conditional average,
since the state of the environment is dependent on the state of the boundary
connected to it; hence we denote this as $\hat{\eta}_{b}$. More generally, the
parameter controlling the peak of the density—the conditional _maximum a
posteriori_ estimate—will do. state $\hat{\eta}_{b}$ and variance
$\Sigma_{\eta\mid b}$ of the environment.

If there exists a boundary of the form of a Markov blanket, then there exists
a function $\sigma$ relating (the dynamics of) internal states to the
environment across the boundary [6, Lemma 4.3]. In that case, we can convert
these parameters to $\sigma(\hat{\mu}_{b})$ and
$[\eta-\sigma(\hat{\mu}_{b})]^{2}$, meaning that the _particle_ models its
environment. Ultimately this simply means that, in virtue of being coupled
(vicariously, i.e., via blanket states) to its environment, the particle
reflects data about its environment—or, what it believes that its environment
is like. Under the assumption that a particle which exists in an environment
models that environment, free energy is implicitly a measurement of coherence.
If the particle stops modelling its environment in a particular way (i.e., for
a particular $\sigma$), it must have ceased to exist with the mean that
$\sigma$ was constructed for; conversely, if a particle ceases to exist, it
will eventually stop modelling its environment. The variational free energy
measures how far this recognition model of what lies beyond the blanket is
from the true probability density over external states, given blanket states.
If the recognition model is bad, and the free energy is high, it is likely
that the system will move into surprising (i.e., implausible or
uncharacteristic) states; this would render it non-existent, in the sense that
it would be occupying states that were not characteristic of the thing in
question (e.g., a fish out of water). These statements tie together the
reasoning that things which exist in an environment reflect the statistics of
their environment in a particular way based on the way they are coupled to
their environment, or, they do not exist that way.

Mathematically, we can express this by observing the following: since the
joint probability $p(\eta,b,\mu)$ factorises into $p(\eta\mid b,\mu)p(b,\mu)$
(see the probabilistic chain rule) and logarithms decompose additively, we
have

$\displaystyle\int$ $\displaystyle
q(\eta;\sigma(\hat{\mu}_{b}),\Sigma_{\eta\mid b})\log
q(\eta;\sigma(\hat{\mu}_{b}),\Sigma_{\eta\mid b})\differential{\eta}-\int
q(\eta;\sigma(\hat{\mu}_{b}),\Sigma_{\eta\mid b})\log
p(\eta,b,\mu)\differential{\eta}=$ $\displaystyle\int
q(\eta;\sigma(\hat{\mu}_{b}),\Sigma_{\eta\mid b})\log
q(\eta;\sigma(\hat{\mu}_{b}),\Sigma_{\eta\mid b})\differential{\eta}-\int
q(\eta;\sigma(\hat{\mu}_{b}),\Sigma_{\eta\mid b})\log p(\eta\mid
b,\mu)\differential{\eta}-\log p(\mu,b)$ (1)

where the left-hand side measures a divergence between
$q(\eta;\sigma(\hat{\mu}_{b}),\Sigma_{\eta\mid b})$ and the true joint
density; whilst on the right-hand side, the divergence between that
recognition model and the true _conditional_ density on external states bounds
the surprisal of internal states from above. (Note that this difference term,
which we will denote $F$, is always non-negative, since it is a KL
divergence—as such, it is _added_ to the surprisal in general. From that we
deduce $-\log p(\mu,b)+F\geq-\log p(\mu,b)$.)

The FEP can be used to reformulate the least action principles that appear in
both classical and quantum physics, due to intimate connections between
physical law and probability [2, 7]. In contemporary physics, classical laws
are actually the deterministic limit of underlying probabilistic
specifications of the behaviours of a system; indeed, we can quantify the
degree of divergence from the most probable path of a system by keeping track
of quantum effects. The classical path of least action is, definitionally, the
most probable path a system will take, i.e., in the Gaussian case, the average
path, and thus classical physics is quantum physics modulo what one might call
randomness.444Whilst different approaches to quantum mechanics emphasise
different aspects of the theory, this is especially evident in the path
integral formulation of quantum physics, which is reflected in the path
integral formulation of Bayesian mechanics, cf. [17].

We have known since Prigogine that we can always reformulate the physical laws
that pertain to systemic dynamics in terms of surprisal (and related metrics,
such as entropy) [18, 19]. That is, we can always take laws of physics
formulated in terms of probability densities, as just described, and
reformulate them in terms of surprisal and entropy. In this setting, rather
than speak of the most probable path or state, one speaks of the least
surprising one, where surprisal is parameterised by some ‘attractor state.’
This is not specific to FEP-theoretic modelling; it is a hallmark of all
contemporary approaches to physics.

Bayesian mechanics explains why every physical thing looks as if it were
performing inference over the causes of its perturbations, forming (Bayes)
optimal beliefs about the causes of those perturbations [6]. The interesting
thing about this is that surprisal is suited to act as a kind of ‘ontological
potential’ for the system, in the sense that the surprisal is equivalent to a
constraint on what states are accessible to the system; the trajectories of a
system are curves on this surprisal, which satisfy some equation of motion
derived from the surprisal itself. As has been asserted in recent literature
[17], one reading of the FEP would hold that the crux of the ubiquity of the
FEP is that every _thing_ which is coupled to another thing holds a model of
that thing in the following very trivial sense: in virtue of this coupling,
one system estimates the parameters of the probability density over states of
the other system (and vice-versa). Since the FEP regards these systems as
estimators, they are performing inference in the sense of parametric Bayesian
inference. Given that the term ‘belief’ is quite theoretically loaded
(especially in philosophy), we ought perhaps to say ‘estimator,’ instead. Once
more, our contention is that anything which estimates the parameters of a
density is engaged in a form of Bayesian inference, by mathematical
definition; i.e., it is mathematically indisputable that systems that estimate
statistics represent, or encode, or otherwise hold probabilistic
beliefs—whether the practitioner likes the word ‘beliefs’ or not, and
irrespective of what commitments one makes with encodings and representations.

There are two main ways to implement FEP-theoretic technology to model the
dynamics of a system; with the former being the asymptotic limit of the latter
(for detailed discussion, see [5, 17]). First, we can formulate the
probability density dynamics of a system [2, 6]. In this setting, surprisal
quantifies how unlikely it is that the system will visit some states, given
the kind of thing that it is. The least surprising states are those that are
characteristic of the particle. Second, we can formulate the FEP in terms of
paths [1, 17]. In this formulation of the FEP, the surprisal quantifies how
far a path deviates from the expected path. For instance, under constraints
that reproduce the correspondence principle of quantum physics, the least
surprising path is simply the classical path; this is like saying that
particles in quantum physics do inference over the forces acting on them at a
macroscopic level and act accordingly on average. That this higher-level
inference is generally conducive to self-organisation has been remarked on in
[20].

###  II.2 A map of that part of the territory that behaves as if it were a map

We will now argue that the FEP evinces a ‘map’ of sorts: a map of that part of
the territory which behaves as if it were a map. In other words, the FEP
provides us with tools to understand mathematically the representational
capacities of self-organising systems. Even more simply, it gives us tools to
model systems that look as if they are modelling the world, producing a map of
the mapmaker, as it were.

How does one get beliefs and inference from FEP-theoretic models? How does
FEP-theoretic technology allow us to model self-organising systems as
themselves engaging in inference and belief updating? The joint probability
$p(\eta,b,\mu)$ plays a central role here. This generative model can be read
as a probabilistic specification of the states characteristically occupied by
the joint particle-environment system. In this sense, it provides constraints
on the kind of states the joint system can be found in. In other words, it is
the ‘territory’ that characterises the system. On another reading, it can be
regarded as a generative model that gives rise to the free energy in (II.1),
and thereby the free energy gradients that underwrite particular dynamics. On
this view, it is a generative model that is entailed by the recognition model,
where we can read the recognition model as a map; namely, a map from internal
states to Bayesian beliefs about external states (i.e., the probabilistic
image of the mapping). If we now move to a meta-theoretical perspective and
consider one particle (e.g., a scientist or philosopher) observing another
particle, we have the interesting situation where the philosopher may impute a
particular generative model that best explains the mechanics of the observed
particle. Is this meta-model a map or a territory?555This meta-theoretical
move is commonplace in practical applications of the FEP and is sometimes
referred to as meta-Bayesian inference or, more simply, observing the observer
[21]. Practically, it involves optimising the parameters of a generative
model, such that under the FEP (the ideal Bayesian observer assumption) the
observable behaviour of the particle is rendered the most likely. This
application of the FEP is often described as computational phenotyping [22].

The meta-model is our scientific model of the system, as observers or
modellers. We generally assume that the system is its ‘own best model.’ In
other words, our scientific model is just the joint probability density that
describes the state of the system (i.e., the territory). This can be read as a
new take on the _nouvelle AI_ idea that physical systems are their own best
models; or as a new take on the good regulator theorem. For instance, in work
on robotics following the tradition of embodied cognition [23, 24],
practitioners have eschewed the construction of agents with internal
representations of some environment. In these systems, the physics and
geometry of the situation were sufficient to endow these agents with the
capacity to couple to an environment. The world is, on this view, its own best
representation. In our view, this approach dovetails nicely with FEP-theoretic
modelling, where the generative model in FEP-theoretic constructions is just a
joint probability density defined over all the states (or paths) of a system.
In other words, the generative model is just our representation of the world
as we believe it to be; and this model need not be encoded directly in the
particle or agent. Given a generative model or Lagrangian of the appropriate
sort, we can show that the system evinces a particular partition (i.e.,
contains particles); and we can show that subsets of the system track each
other, where tracking means inferring or becoming the sufficient statistics of
probabilistic beliefs about their external states. We interpret this tracking
as a form of inference, namely, variational or approximate Bayesian inference
under a generative model (i.e., the map).

It is important to say explicitly that the sense in which a system is its own
best model (_qua_ generative model) is not the same sense in which particles
of the system are models (i.e., maps) of their environment, or of themselves
acting in their environments (_qua_ recognition model); they are of course
connected, in the sense that the dynamics of internal states or paths (and
thereby, of particular recognition models) entail a generative model (via
(II.1), for instance) [15].

It is important to defuse a possible (but in our view naïve) objection. It is
a truism that physical systems need not explicitly calculate their
trajectories of motion, to be modelled as pursuing such trajectories. In
developing a Bayesian mechanical account of the dynamics of systems, we need
not assume that the particle itself is literally performing inference.666Ways
of defending stronger, increasingly literal versions of this positions are
available as well. For a defence of the claim that physical systems are quite
literally in the business of inference, see [25, 14]. Related to this, it is
possible to articulate a version of the notion of notion of implementation,
whereby a physical process implements a computation provided that a so-called
interpretation map exists, and commutes in the appropriate manner, such that
the physical process can be interpreted as implementing a computation; see
[26] for the general account, and [7, 27] for its application to the FEP. What
is at stake is an ‘as if’ description. We simply assume that there exists a
quantity that varies systematically with the dynamics of the system. It just
so happens that, under the FEP, this quantity turns out to be surprisal (or,
equivalently, variational free energy); and that the minimisation of this
quantity is mathematically equivalent to inference, in the sense of being an
estimator, as discussed above.

Thus, our map (as scientists and modellers) can be identified in an
unproblematic way as the generative model or Lagrangian of the system, which
FEP-theoretic technology enables us to write down. And it so happens that our
model is a model of the representational capacities of subsets of the target
system: that is, our map is precisely a map of systems that behave as if they
were maps, allowing us to construct scientific models of the maps that are
encoded by the internal subset of the system considered.

In summary, our scientific model is, metaphorically, a map that allows us to
say that some (internal) subset of the system looks as if it possesses a map;
which we interpret formally as tracking the statistics of another (external)
subset. This arguably defuses the first aspect of the map-territory fallacy:
namely, it shows that there is no conflation of the map and the territory, at
a first level of analysis.

###  II.3 Maps that are optimal employ the CMEP; maps of maps that are optimal
employ the FEP

Mathematically, entropy is the unique functional of probability densities
which yields consistent, unbiased inferences [28, 29] and is sufficiently
general to recover all of equilibrium thermodynamics [30, 29, 31] and much of
information and probability theory [32, 33]. Maximum entropy inference, in
which we find the probabilistic model that maximises entropy, given some
specific constraints (e.g., some data), is designed as the least biased
inference which accounts for known information, making it optimal in the sense
of being neither overfitted nor incomplete for any given scenario. As a
modelling principle, the constrained maximum entropy principle (CMEP) is
canonically the soundest [32, 31]. For a complete review of the principles
behind the CMEP, see [34]. For convenience, we refer to this host of results
as ‘Jaynes optimality.’ A model is Jaynes optimal if it is optimal for a set
of known unknowns, i.e., it produces the model which neither overfits those
constraints nor ignores them, by assigning probability in accord to the
preference induced by constraints [6]. Contrast this with Bayes optimality,
which is a model that provides a lower bound on classification error: Bayesian
updates can be derived from maximum entropy (see [35] and references therein,
or more recently, [36]), so we assume these are closely related. Under the
assumption that particles are Bayes optimal (perhaps justified by the complete
class theorem that Bayes optimal agents do exist), they are Jaynes optimal,
and _vice-versa_. In the simpler case of conditional independence of inputs,
for instance, it is known that naïve Bayes classifiers are CMEP models.

It stands to reason, then, that systems which model their environments use the
principle of maximum entropy. Recall the use case driving the FEP: we need to
use a principled modelling framework to form (scientific) models of things
using principled modelling frameworks to form (recognition) models. In this
sense, it would be ideal if the FEP is an elegant shorthand for applying
maximum entropy to model things using maximum entropy. Recent literature [6,
Theorem 4.1] has shown that this is precisely the case.

Let us assume that a particle models its environment, justified by the
deflationary approach to estimation-as-modelling discussed above. The FEP is
what allows us to model this model; we want to prove that it is equivalent to
the sort of model that such a particle actually could be expected to use. Take
(II.1) and negate it once, then maximise it instead of minimising it. Grouping
the last two terms in the expression, we then have

$-\int q(\eta;\sigma(\hat{\mu}_{b}),\Sigma_{\eta\mid b})\log
q(\eta;\sigma(\hat{\mu}_{b}),\Sigma_{\eta\mid
b})\differential{\eta}-\left(-\int
q(\eta;\sigma(\hat{\mu}_{b}),\Sigma_{\eta\mid b})\log p(\eta\mid
b,\mu)\differential{\eta}-\log p(\mu,b)\right).$

Contrast this with maximising the entropy of the recognition model $q$ subject
to the constraint that the surprisal of the environment given the particle is
(on average) no greater than the intrinsic surprisal of the particular states:
$-\operatorname{\mathbb{E}}_{q}[\log
q]-\left(\operatorname{\mathbb{E}}_{q}[J(\eta)]-C\right)$ where $J(\eta)=-\log
p(\eta\mid b,\mu)$ and $C=-\log p(\mu,b)$, and this proves the claim. That is
to say, since maximising a positive quantity is the same as minimising a
negative quantity, if things model their environments by using the CMEP, then
the FEP is an equivalent (scientific) model of that (scientific) model. As
such, the FEP is not merely a model of models, but is _the_ model of the
models that all such systems that model their environments use, due to its
relationship to the CMEP. The Jaynes optimality of the chosen recognition
density in the FEP rests precisely on using constrained maximum entropy to
choose that density.

The point of this section has ultimately been that, if particles use the CMEP
to model their environments (which is reasonable given they are optimal in the
sense discussed above) then understanding what our FEP-theoretic model does is
like inhabiting the system and understanding what the particle does. Recall
that by constructing a model of the particle modelling its environment, there
exists an induced model of the states of the particle $\mu$ doing that
modelling under $\sigma$. Next, we will prove that the induced model on
internal states is also optimal, for largely the same reason: ourselves, as
observers in an environment, also ought to employ the CMEP to model particles
in _our_ environment—a symmetry which is brought to life by the FEP.

##  III Of maps of maps

###  III.1 The duality of FEP and CMEP

There are two ways to model a particle modelling its environment: (i) to model
the environment as though we are the particle modelling its environment (from
its own first-person perspective), or, (ii) to model the particle holding a
model of the environment. These perspectives reduce to ‘forming a model of
external states as if we were the particle’—what we have called a map of the
piece of the territory which behaves like a map, which allows us to understand
the inference performed by other particles, which is in turn the core of the
FEP—and ‘forming a model of the internal states which model external states,’
the actual application of the FEP, which allows us to write down those
inferences on paper. Paradoxically, these are equivalent statements, in that
both ‘access’ the generative model enailed by the system. However, as methods
they go about that goal in radically different—even directly opposing—ways.
Opposites that are equal in some sense are known in mathematics as adjoint
pairs.

It has long been remarked that the FEP-theoretic problems are equivalent to
maximum entropy problems with a constraint, where the constraint is a log-
probability function supplied by a generative model. Recent work has
demonstrated that the FEP is, in a specific sense, mathematically equivalent
to the CMEP. They are two sides of the same metaphorical coin. More precisely,
they are dual to each other [6, 5]; see in particular [6, Theorem 4.2]. The
existence of a duality relationship between two structures means (informally)
that those two structures are dual to each other, i.e., that they form an
adjoint pair. Adjoint pairs are structures with identical ‘objects,’ but where
things that act on those objects act in an opposite manner to each other.
Here, the adjunction is semi-obvious: the objects (internal and external
states) are the same, but our model either (i) looks ‘outwards’, from the
particle’s first-person perspective, by forming a model of beliefs and
inferences about external states, or (ii) looks ‘inwards’ at the system from a
scientific (second person) perspective by forming a model of the dynamics of
internal states in relation to external states.

We can understand this as the following: suppose we have the recognition
density $q(\eta\mid\mu)$, a parameterised family of proposed recognition
models held by the particle, each of which is conditioned on the particular
value of the internal state holding that belief. Then one value of $\mu$
yields the optimal model, $\mu=\hat{\mu}_{b}=\sigma^{-1}(\hat{\eta}_{b})$. Our
model of a thing which models its environment under maximum entropy can thus
be formed by maximising the entropy of our own scientific model under the
constraint that our model should model a system modelling its environment.
This is the meta-Bayesian application of the constrained maximum entropy
principle, where the constraints are supplied by ideal Bayesian observer
assumptions (i.e., that the behaviour of the observed observer complies with
the FEP [21]). This simply states that our model of particular states should
consider that the particle as an observer or estimator, and reproduce our
intended aim of modelling the particle given our knowledge that the particle’s
behaviour is contingent on its model. Thus, we can maximise the following
entropy functional:

$-\operatorname{\mathbb{E}}_{p(\mu\mid b)}[\log p(\mu\mid
b)]-\lambda\left(\operatorname{\mathbb{E}}_{p(\mu\mid
b)}J(\mu)-\Sigma_{\eta\mid b}\right)$ (2)

where $J(\mu)=[\mu-k]^{2}$ and $k=\sigma^{-1}(\hat{\eta}_{b})$—in other words,
a model of a system that (on average, though how often precisely is controlled
by the Lagrange multiplier $\lambda$) holds an optimal model of its
environment. Note that this particle also matches the variance of the
environment, not just the mode, in order to model the full suite of
environmental states. Notice also that we have switched the locations of
$\eta$’s and $\mu$’s compared to (II.1), now maximising the entropy of the
model of internal states modelling external states, rather than the model of
external states. This reflects the duality referenced. Together, we implicitly
have a maximum entropy model of the system holding a maximum entropy or FEP-
theoretic belief.777For completeness, note that we often assume the
synchronisation map applies to the conditional _maximum a posteriori_
estimate, such that the system infers the parameter controlling the placement
of the peak of the probability density: the most likely state of the
environment. Our maximum entropy model of this model assumes the particle is
performing a Laplace approximation [3]. Importantly, note that the log-
probability constraint in (II.1) is precisely the variance term in (2) under a
Laplace approximation, meaning that (II.1) and (2) are identical up to
dualisation in that case. More general forms for (2) are possible which
generalise beyond the Laplace assumption.

This leads us to another, absolutely core, point—one which is central to the
FEP itself, and hence to any candidate philosophy of the FEP and Bayesian
mechanics. The duality we have leveraged arises from the fact that the
boundaries of a particle are symmetrical. In turn, this means the environment
of a particle is also an agent that models the particle. More precisely, the
particle and environment each model the actions of the other on their mutually
shared blanket states. Nature is constantly dissipating organised structures
in the world, and the inference that is performed by the natural world is how
best to dissipate the energy held in any such structure. The only way to avoid
this is to mirror nature herself; this is the point of us modelling ‘things’
as being unsurprising. This is the game that is characteristic of self-
organisation, and only by understanding both sides can we understand self-
organisation; such an attitude has been referred to as a relational approach
to biological physics [37].

In the end, map and territory are simply dual perspectives; the fact that the
FEP applies to us, and that we articulate models from our perspective as
external observers, may often blind us to this duality. However, it
underwrites the whole argument on offer: the FEP packages together an optimal
model of a thing and an optimal model of the thing’s model to achieve a new
view on self-organisation. This is also what we mean by the FEP being the
application of maximum entropy to things that maximise entropy, again making
it unique amongst modelling methods. By modelling a particle modelling its
environment, we achieve a dual model of the particle automatically, induced by
the symmetry between external and internal states. However, a further
uniqueness of the FEP is that, by emphasising the viewpoint of internal
states, it is uniquely suited modelling action and cybernetic systems. The
dual, CMEP-theoretic viewpoint is a good (i.e., Jaynes-optimal) account of the
dynamics of particles that stay together, in virtue of looking at an anomalous
‘thing’ that organises (i.e., the viewpoint of some scientist or heat bath),
but does not include an account of action or policy explicitly [6]. As
indicated, this agent-centred perspective is what the FEP brings to the table.

That the FEP and the CMEP turn out to be two complementary perspectives on the
physical existence of systems with particular partitions has profound
implications, which have not been remarked upon previously in great detail
(although see [5] for an exposition of the mathematical nature of this
duality). FEP-theoretic models of complex physical systems, as we have seen,
are mechanical theories, which provide us with an explanation (i.e., laws of
motion) of the observable behaviour (or dynamics) of a system, from the point
of view of a particle that self-organises (i.e., from the point of view of a
‘self’). Dually, the CMEP provides a description of self-organising systems,
this time from the point of view of an observer, located in the external
environment. Now, it is a mathematical fact that one can appeal to this
duality to convert any FEP-theoretic model into an equivalent CMEP-theoretic
one. More precisely, an FEP-theoretic model of belief updating (i.e., a model
of the dynamics of beliefs, encoded by the internal states of a particle) can
always be converted to a dual CMEP-theoretic model of systemic dynamics (i.e.,
a model of the dynamics of the states or paths of the system _per se_ , as
opposed to the beliefs that they encode). Moreover if we have one, we
implicitly have the other. This argument extends the dual aspect information
geometry of _internal states_ —describing their thermodynamics [19]—and the
information geometry that arises when internal states are read as the
sufficient statistics of Bayesian beliefs _about external states_ [12].

Besides being a well-appreciated and established theory in physics, as we will
go on to discuss, the relationship to the CMEP justifies the FEP as a
statistical-mechanically optimal model of some unknown data-generating
process.

###  III.2 FEP-theoretic models of encoded beliefs are CMEP-theoretically
optimal models of systemic dynamics

The effectiveness of the CMEP in statistical physics and beyond is not to be
taken lightly, and the implications of this duality are far reaching. Some
commentators have argued that the FEP is not special [38]: on this view, the
FEP is merely one formal approach to the dynamics of complex systems, without
any special epistemic status. The duality of FEP and CMEP negates this claim.
Indeed, as we have outlined, maximum entropy inference is a way of arriving at
the best model possible, mathematically, given our state of knowledge about
the system being modelled. So, the duality of FEP and CMEP has a dramatic and
deeply significant implication: given the duality of FEP and CMEP, we can
state in full generality that a FEP-theoretic model of a particle’s belief
updating is the optimal model of systemic dynamics, from the point of view of
an external observer. We know that an FEP-theoretic model of a particle’s
beliefs about a system simply is a maximum entropy model of the states of the
system, albeit seen from another perspective. We know that maximum entropy
inference enables us to arrive at the optimal model of that system, given our
state of knowledge. The implication is that, given some constraints, the FEP-
theoretic model is the optimal model of the physics of a system. This makes
intuitive sense: due to the way that the generative model or Lagrangian is
defined (i.e., over all the subsets of the systems), the FEP-theoretic model
contains every ingredient needed to fully account for the manner in which
subsets coupled to each other.

Of course, practically, it does not suffice to write down a path of least
surprisal. Minimally, without some additional specification, such a scientific
model cannot be complete; since the boundary conditions (e.g., initial
conditions and constraints) have to be added. More directly relevant to our
present concerns, one must specify the form of the generative model or
Lagrangian, which is often far from a trivial exercise. It is always a model
in context. Indeed, it is the model or energy function which does most of the
heavy lifting when it comes to modelling specific, empirical systems. We still
need to find the appropriate set of constraints; what the technology of
FEP/CMEP enables is doing the best inference possible, given our state of
knowledge. Indeed, given that they are principles (i.e., mathematical
structures with no empirical content), one cannot falsify the FEP/CMEP
directly [39, 5]. This would be like attempting to falsify calculus or group
theory: this constitutes a category error. One can only falsify specific
models that have been derived from FEP/CMEP-theoretic technologies. When we
arrive at incorrect results using these methods, the mistake usually lies in
the application: the inference was optimal, given the non-optimal data set
that we were using to constrain our inference procedure. What counts, in some
sense, then, is the function that we put the model to use. Assuming we have
adequate knowledge of the target system, and have written that knowledge down
adequately as a set of sensible constraints, then the FEP-theoretic model is
the optimal model; and if we have not done so, then we ought to be prepared
for our model to not be optimal. The same problem has been known in Bayesian
statistics for some time, and hence ought not to be particularly striking
[40].

These considerations serve to decisively defuse the second horn of the map-
territory fallacy, which is that map and territory are always separate.
Indeed, the FEP is not a mere model of physical processes: it is the optimal
model given our state of knowledge, and therefore, poses absolute constraints
on what it means to be a model of some physical process.

##  IV Towards a philosophy of the FEP and Bayesian mechanics

We now briefly summarise and elaborate on the points discussed above. We
propose these few points as the foundations of a philosophy of Bayesian
mechanics, which has yet to be constructed.

###  IV.1 A map of map-like territories, and a map of possible maps: Towards a
dynamical, nested representationalism

In summary, we have argued that the map-territory fallacy, as it has been
leveraged against the FEP (e.g., by [9, 10]) simply does not apply to FEP-
theoretic modelling: it constitutes a fallacy, which we have called the map-
territory fallacy fallacy. Mathematically, there is no ambiguity or conflation
of map and territory in FEP-theoretic modelling. Although the language used to
narrate the mathematics of the FEP are at times ambiguous (which may have led
to the confusion of some segments of the philosophical commentary and
reporting on the FEP), the mathematical structure itself is not. That the
target system itself is a generative model or Lagrangian ought to be read as
shorthand, to mean that the model that we deploy as scientists is meant to
capture all the dependencies that exist within the system. The particle itself
deploys a recognition model, which its internal states encode; their dynamics
are given in our generative model, which is meant to capture the real
dependence relations of the target system [15]. Applications of the FEP thus
commit to a kind of ‘nested representationalism,’ in the sense that it is a
scientific model (implemented as a generative model of states or Lagrangian of
paths), the content of which is precisely a mechanistic theory for the
representational capacities of certain particular kinds of systems. Here, and
in subsequent publications, we argue that the correct interpretation of the
FEP is a dynamical, nested representationalist one.

Moreover, as we have seen, the FEP is not merely one possible way to model
physical systems. Instead, it is the optimal model of systemic dynamics, given
our state of knowledge. Thereby, the FEP/CMEP duality imposes ultimate
constraints on what it means to be a physical model of some target system.
Thus, FEP/CMEP provides us with core mathematical constraints on the
optimality of modelling strategies. This determination of constraints on what
it means to physically realise a model from within the formal constraints of
the mathematics of physical modelling itself is reminiscent of Kant’s attempt
to arbitrate the limits of reason from within the system of reason itself
[41]; or again, Wittgenstein’s attempt to formulate the structure of any
proposition in general, within the language of propositional calculus [42]: it
is an attempt to provide constraints on any possible meaningful maps, which
arise from what it means to be a map or model of a physical process.

What is unique about the FEP/CMEP approach is that it is ultimately a model
that is apt to model the modeller. Indeed, the Markov blanket is a symmetric
boundary between particle and environment. We see the world through our own
Markov blanket; and any possible observer constitutes the environment for the
particle that we are modelling. From the point of view furnished by the
FEP/CMEP duality, our scientific models are attempts to infer what reveals
itself vicariously through our Markov blanket. What we, as modellers, call
systems or particles ultimately reflect what we, as modellers, can observe,
and identify as persistent entities that sketch themselves out on our Markov
blanket.

This position can be illustrated by revisiting the map–territory fallacy
through the lens of the FEP. The foregoing argument suggests that one should
not ask whether the FEP is subject to the map-territory fallacy; rather, one
should ask whether the map-territory fallacy subject to the FEP. With a
careful and formal definition of map and territory, the answer is yes, in the
following sense. One can associate the ‘territory’ with the generative model
describing the joint probability over causes and consequences, from the
perspective of some ‘thing.’ The ‘map’ may be best ascribed to the recognition
model in the sense that a map denotes a mapping; here, a map from internal
states to Bayesian beliefs about external states. Note that for a ‘map’ to
exist in this sense, there has to be a partition of states: in the absence of
a Markov blanket—which distinguishes internal states from external
states—there can be no map, and therefore, no fallacy. With these definitions
in place, the map–territory fallacy reduces to a failure to distinguish
between recognition and generative models. This distinction is precluded by
the FEP, whose core claim is that the map depends in a lawful way on the
territory—and that this mapping has existential implications, i.e., that there
exists a particular partition or something that holds a recognition model
about external states [15].

So, can there be a map-territory fallacy in applications of the FEP to form
scientific models? Again, the answer is yes, with some interesting examples.
In applications of the FEP, we are building recognition models (i.e., maps) of
generative models (i.e., territories) that best explain the behaviour of
something. To reify one of these maps is to falsely infer that our scientific
recognition model is the true generative model. Examples of this in statistics
could include the fallacy of classical inference [43]; in which the false
inferences rest upon using the wrong kind of (point null) hypotheses. Perhaps
a more interesting example of false inference is psychopathology; namely, the
type I errors when we falsely infer something is there when it is not (e.g.,
hallucinations and delusions) or type II errors when we falsely infer
something is not there when it is (e.g., dissociative symptoms and neglect
syndromes). In fact, there is a large literature in computational psychiatry
[44, 45, 46, 47, 48, 49] that could be read as one kind of map–territory
fallacy (i.e., conflating one’s false inferences with reality)—a very special
kind of fallacy that inherits from applications of the FEP. Holding an optimal
map for the wrong territory is the same sort of failure to be a proper
Bayesian that we discussed in Section III.2—not a failure of Bayesian
inference _per se_ , but a misapplication of such an inference.

###  IV.2 On the theoretical appropriation of the FEP in philosophy

The core publications on the FEP and Bayesian mechanics have mostly been in
mathematical biophysics [50, 6, 2] and computational or theoretical
neuroscience [4, 51]. Since the middle of the last decade, he FEP has also
generated much interest in philosophy [52, 53]. Despite these efforts, and
with a few exceptions (e.g, [39, 12]), there has been very little work on the
philosophical commitments of the FEP per se; and, given that it has only
emerged self-consciously over the last few years, there is only one
publication to date discussing the philosophy of Bayesian mechanics [5].
Indeed, arguably, most extant philosophical work on the FEP starts from a
given set of assumptions—for example, about the extended mind [54], the
enactive approach to cognition [15], or neo-Kantian, indirect accounts of
perception [55]—and then proceeds to present a reading of the FEP that fits as
well as possible with those assumptions. This seems especially common in
scholarship that is independently committed to philosophical theses that are,
in many crucial ways, contradictory to the core commitments of the FEP, such
as staunch anti-representationalism and the rejection of computational and
information theoretic tools to study self-organising systems [56, 57].

We do not think that this is conducive to understanding the theory and
commitments of the FEP and Bayesian mechanics themselves (although we have
been guilty of this ourselves, in some cases). Indeed, we believe that the
variegated theoretical appropriation of the FEP is one central reason why
there seem to be several different interpretations or versions of the FEP—with
wildly conflicting theoretical assumptions and broader philosophical
commitments. We urge researchers in the tradition to pursue their examinations
of the FEP on its own merits, setting aside questions about its compatibility
with specific, independent philosophical perspectives (or the lack thereof).

##  V Conclusion

This paper has taken some first steps towards the development of a semi-
axiomatic meta-theory and philosophy of the free energy principle (FEP) and
Bayesian mechanics, providing a justification of its use in the modelling of
physical systems, and an explanation of its scope in terms of first
principles. We argued that, in describing ‘things’ or ‘particles’ as
estimators or (deflated) representations of the systems to which they are
coupled, the FEP-theoretic apparatus does not commit the map territory
fallacy, i.e., it is false that the FEP reifies aspects of the metaphorical
map (i.e., our scientific model), mistakenly taking them to be part of the
territory (i.e., the target system that we wish to model). We have argued that
this allegation itself constitutes a map-territory fallacy fallacy. We have
argued that, in distinguishing the generative and recognition models, FEP-
theoretic modelling allows us to construct a map of that part of the territory
that behaves as if it were a map, without reification. Second, we have
leveraged the duality of the FEP and the CMEP to argue that, in a Kantian or
Wittgensteinian manner, the FEP also provides us with ultimate constraints on
what it means to be an optimal model of physical processes. The FEP us thus,
metaphorically, a map of any possible map whatsoever of, or held by, a
physical system.

We ought to celebrate the territory map mapping. The FEP is really, at its
core, a principled approach to the formalisation of this mapping. The FEP is
utterly unique as a ‘Jaynes-optimal’ model of physical systems that model
(i.e., are estimators of the statistics of) their environments. This restores
convention to the FEP, via its relation to the CMEP: the FEP is not a strange
theoretical beast residing in a faraway corner of theoretical biology, but
rather, turns out to be a core principle of physical science. Our meta-
theoretical approach to the FEP clarifies its role and scope as the uniquely
ideal or optimal modelling tool for generic systems in statistical physics.

### Acknowledgements

We are thankful to Mahault Albarracin, Mel Andrews, Lancelot Da Costa, Chris
Fields, James Glazebrook, Alex Kiefer, Gabriel René, and Adam Safron, as well
as the regular attendees of the Theoretical Neurobiology Group Meetings at
University College London’s Wellcome Centre for Human Neuroimaging, for
fruitful discussions that helped to shape the content of this paper.

## References

  * [1] Karl J Friston, Lancelot Da Costa, Noor Sajid, Conor Heins, Kai Ueltzhöffer, Grigorios A Pavliotis, and Thomas Parr.  The free energy principle made simpler but not too simple.  2022\.  Preprint arXiv:2201.06387.
  * [2] Karl J Friston.  A free energy principle for a particular physics.  2019\.  Preprint arXiv:1906.10184.
  * [3] Lancelot Da Costa, Karl J Friston, Conor Heins, and Grigorios A Pavliotis.  Bayesian mechanics for stationary processes.  Proceedings of the Royal Society A, 477(2256), 2021.
  * [4] Karl J Friston.  The free-energy principle: a unified brain theory?  Nature Reviews Neuroscience, 11(2):127–138, 2010.
  * [5] Maxwell J D Ramstead, Dalton A R Sakthivadivel, Conor Heins, Magnus Koudahl, Beren Millidge, Lancelot Da Costa, Brennan Klein, and Karl J Friston.  On Bayesian mechanics: a physics of and by beliefs.  2022\.  Preprint arXiv:2205.11543.
  * [6] Dalton A R Sakthivadivel.  Towards a geometry and analysis for Bayesian mechanics.  2022\.  Preprint arXiv:2204.11900.
  * [7] Chris Fields, Karl J Friston, James F Glazebrook, and Michael Levin.  A free energy principle for generic quantum systems.  Progress in Biophysics and Molecular Biology, 173:36–59, 2022.
  * [8] Dalton A R Sakthivadivel.  Weak Markov blankets in high-dimensional, sparsely-coupled random dynamical systems.  2022\.  Preprint arXiv:2207.07620.
  * [9] Jelle Bruineberg, Krzysztof Dolega, Joe Dewhurst, and Manuel Baltieri.  The emperor’s new Markov blankets.  Behavioral and Brain Sciences, pages 1–63, 2020.
  * [10] Thomas van Es.  Living models or life modelled? On the use of models in the free energy principle.  Adaptive Behavior, page 1059712320918678, 2020.
  * [11] Mel Andrews.  Making reification concrete: a response to Bruineberg et al.  2022\.
  * [12] Karl J Friston, Wanja Wiese, and J Allan Hobson.  Sentience and the origins of consciousness: from Cartesian duality to Markovian monism.  Entropy, 22(5):516, 2020.
  * [13] Maxwell J D Ramstead, Karl J Friston, and Inês Hipólito.  Is the free-energy principle a formal theory of semantics? From variational density dynamics to neural and phenotypic representations.  Entropy, 2020.
  * [14] Alex B Kiefer.  Psychophysical identity and free energy.  Journal of the Royal Society Interface, 17(169):20200370, 2020.
  * [15] Maxwell J D Ramstead, Michael D Kirchhoff, and Karl J Friston.  A tale of two densities: active inference is enactive inference.  Adaptive Behavior, 28(4):225–239, 2020.
  * [16] Maxwell J D Ramstead.  The empire strikes back: some responses to Bruineberg and colleagues, 2021.  Preprint arXiv:2112.15528.
  * [17] Dalton A R Sakthivadivel.  A worked example of the Bayesian mechanics of classical objects.  In The Third International Workshop on Active Inference, 2022.  Preprint arXiv:2206.12996. To appear.
  * [18] Ilya Prigogine.  Time, structure, and fluctuations.  Science, 201(4358):777–785, 1978.
  * [19] Gavin E Crooks.  Measuring thermodynamic length.  Physical Review Letters, 99(10):100602, 2007.
  * [20] Maxwell J D Ramstead, Casper Hesp, Alexander Tschantz, Ryan Smith, Axel Constant, and Karl J Friston.  Neural and phenotypic representation under the free-energy principle.  Neuroscience & Biobehavioral Reviews, 120:109–122, 2021.
  * [21] Jean Daunizeau, Hanneke E M Den Ouden, Matthias Pessiglione, Stefan J Kiebel, Klaas E Stephan, and Karl J Friston.  Observing the observer (I): meta-Bayesian models of learning and decision-making.  PLoS ONE, 5(12):e15554, 2010.
  * [22] Philipp Schwartenbeck and Karl J Friston.  Computational phenotyping in psychiatry: a worked example.  eNeuro, 3(4), 2016.
  * [23] Randall D Beer.  Computational and dynamical languages for autonomous agents.  In Mind as Motion: Explorations in the Dynamics of Cognition, pages 121–147. 1996.
  * [24] Rodney A Brooks.  Intelligence without representation.  Artificial Intelligence, 47(1-3):139–159, 1991.
  * [25] Alex B Kiefer.  Literal perceptual inference.  In Philosophy and Predictive Processing, 2017.
  * [26] Clare Horsman, Susan Stepney, Rob C Wagner, and Viv Kendon.  When does a physical system compute?  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 470(2169):20140182, 2014.
  * [27] Chris Fields, James F Glazebrook, and Michael Levin.  Neurons as hierarchies of quantum reference frames.  Biosystems, 219:104714, 2022.
  * [28] John Shore and Rodney Johnson.  Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy.  IEEE Transactions on Information Theory, 26(1):26–37, 1980.
  * [29] Steve Pressé, Kingshuk Ghosh, Julian Lee, and Ken A Dill.  Principles of maximum entropy and maximum caliber in statistical physics.  Reviews of Modern Physics, 85(3):1115, 2013.
  * [30] Edwin T Jaynes.  Information theory and statistical mechanics.  Physical Review, 106(4):620, 1957.
  * [31] Steve Pressé, Kingshuk Ghosh, Julian Lee, and Ken A Dill.  Nonadditive entropies yield probability distributions with biases not warranted by the data.  Physical Review Letters, 111(18):180604, 2013.
  * [32] Edwin T Jaynes.  Probability Theory: The Logic of Science.  Cambridge University Press, 2003.
  * [33] Petr Jizba and Jan Korbel.  When Shannon and Khinchin meet Shore and Johnson: equivalence of information theory and statistical inference axiomatics.  Physical Review E, 101(4):042126, 2020.
  * [34] Adom Giffin.  Maximum entropy: the universal method for inference.  2009\.  Preprint arXiv:0901.2987. Doctoral thesis.
  * [35] Edwin T Jaynes.  The relation of Bayesian and maximum entropy methods.  In Maximum-Entropy and Bayesian Methods in Science and Engineering, pages 25–29. Springer, 1988.
  * [36] Ariel Caticha and Adom Giffin.  Updating probabilities.  In AIP Conference Proceedings, volume 872, pages 31–42. American Institute of Physics, 2006.
  * [37] Robert Rosen.  A relational theory of biological systems.  The Bulletin of Mathematical Biophysics, 20(3):245–260, 1958.
  * [38] Vicente Raja, Dinesh Valluri, Edward Baggs, Anthony Chemero, and Michael L Anderson.  The Markov blanket trick: on the scope of the free energy principle and active inference.  Physics of Life Reviews, 39:49–72, 2021.
  * [39] Mel Andrews.  The math is not the territory: navigating the free energy principle.  Biology & Philosophy, 36(3):30, 2021.
  * [40] David H Wolpert.  Reconciling Bayesian and non-Bayesian analysis.  In Maximum Entropy and Bayesian Methods, pages 79–86. Springer, 1996.
  * [41] Immanuel Kant.  Critique of Pure Reason.  1781\.  1908 edition, Houghton Mifflin.
  * [42] Ludwig Wittgenstein.  Tractatus Logico-Philosophicus.  1922\.  2013 edition, Routledge.
  * [43] Dennis V Lindley.  A statistical paradox.  Biometrika, 44(1/2):187–192, 1957.
  * [44] Karl J Friston.  Precision psychiatry.  Biological Psychiatry: Cognitive Neuroscience and Neuroimaging, 2(8):640–643, 2017.
  * [45] David Benrimoh, Thomas Parr, Rick A Adams, and Karl J Friston.  Hallucinations both in and out of context: an active inference account.  PLoS ONE, 14(8):e0212379, 2019.
  * [46] Ryan Smith, Richard D Lane, Thomas Parr, and Karl J Friston.  Neurocomputational mechanisms underlying emotional awareness: insights afforded by deep active inference and their potential clinical relevance.  Neuroscience & Biobehavioral Reviews, 107:473–491, 2019.
  * [47] Adam Linson, Thomas Parr, and Karl J Friston.  Active inference, stressors, and psychological trauma: A neuroethological model of (mal) adaptive explore-exploit dynamics in ecological context.  Behavioural Brain Research, 380:112421, 2020.
  * [48] Philipp Sterzer, Rick A Adams, Paul Fletcher, Chris Frith, Stephen M Lawrie, Lars Muckli, Predrag Petrovic, Peter Uhlhaas, Martin Voss, and Philip R Corlett.  The predictive coding account of psychosis.  Biological Psychiatry, 84(9):634–643, 2018.
  * [49] Vivien Ainley, Matthew A J Apps, Aikaterini Fotopoulou, and Manos Tsakiris.  ‘Bodily precision’: a predictive coding account of individual differences in interoceptive accuracy.  Philosophical Transactions of the Royal Society B: Biological Sciences, 371(1708):20160003, 2016.
  * [50] Karl J Friston.  A free energy principle for biological systems.  Entropy, 14(11):2100–2121, 2012.
  * [51] Karl J Friston, Thomas Parr, and Bert de Vries.  The graphical brain: belief propagation and active inference.  Network Neuroscience, 1(4):381–414, 2017.
  * [52] Andy Clark.  Surfing Uncertainty: Prediction, Action, and the Embodied Mind.  Oxford University Press, 2015.
  * [53] Jakob Hohwy.  The Predictive Mind.  Oxford University Press, Oxford, 2014.
  * [54] Micheal D Kirchhoff and Julian Kiverstein.  Extended Consciousness and Predictive Processing: A Third-Wave View.  Routledge, New York, 2019.
  * [55] Jakob Hohwy.  How to entrain your evil demon.  In Philosophy and Predictive Processing, 2017.
  * [56] Thomas van Es and Inês Hipólito.  Free-energy principle, computationalism and realism: a tragedy.  2020\.
  * [57] Inês Hipólito and Thomas van Es.  Enactive-dynamic social cognition and active inference.  Frontiers in Psychology, 13, 2022.