# Motivation:

Among Artificial Intelligence researchers there is a subject of debate that I’d like to address. This concerns the origin of the notion of Causal Path Entropy [4] and the associated maximum entropy principle which may be used to guide intelligent behaviour. Moreover, the authors make the following falsifiable claim in the fourth page of [4]:

The remarkable spontaneous emergence of these sophisticated behaviors from such a simple physical process suggests that causal entropic forces might be used as the basis for a general—and potentially universal—thermodynamic model for adaptive behavior. Namely, adaptive behavior might emerge more generally in open thermodynamic systems as a result of physical agents acting with some or all of the systems’ degrees of freedom so as to maximize the overall diversity of accessible future paths of their worlds (causal entropic forcing)

by the end of this article I believe the reader will have both a better appreciation of the origins of the Causal Path Entropy and be capable of evaluating the authors’ conjecture.

# Introduction:

In [4], the authors introduce the Causal Path Entropy as follows:

For any open thermodynamic system such as a biological organism we may treat phase-space paths taken by the system over a time interval $[0,\tau]$ as microstates and partition them into macrostates $\{X_i\}_{i\in I}$ using the equivalence relation:

$$x(t) \sim x’(t) \iff x(0)=x’(0)$$

As a result, we can identify each macrostate $X_i$ with a present system state $x(0)$.

We may then define the Causal Path Entropy $S_c$ of a macrostate $X_i$ associated with the present system state $x(0)$ as the path integral:

$$S_c(X_i,\tau)=-k_B \int_{x(t)} P(x(t)|x(0))\ln P(x(t)|x(0)) Dx(t)$$

where $k_B$ is the Boltzmann constant.

Now, the authors of [4] claim that this is a new kind of entropy although this is simply a conditional Boltzmann entropy. In fact, on the second page of [3] C. Villani introduces the Boltzmann entropy using the time-dependent density $f_t$ on particles in phase-space $(x,v) \in \Omega \times \mathbb{R}_v^3$:

$$S(f_t) = - \int_{\Omega \times \mathbb{R}_v^3} f_t(x,v) \ln f_t(x,v) dxdv$$

and we may analyse the dependence of the evolution of $S$ on particular initial conditions $p(0)=(x_0,v_0)$ by defining:

$$S(f_t|p(0)) = - \int_{\Omega \times \mathbb{R}_v^3} f(p(t)|p(0)) \ln f(p(t)|p(0)) dp$$

where $p \in \Omega \times \mathbb{R}_v^3$

# On the origin of the ‘Causal Path Entropy’:

In Boltzmann’s 1872 paper [2], he introduces the H-theorem which relies upon a time-dependent probability density $f_t$ on particles in phase-space as pointed out in [3] in order to demonstrate that in an isolated system entropy $S(f_t)$ tends to increase as a function of time.

Furthermore, Boltzmann’s equation (3) actually depends on a choice of initial conditions and in [2] it is shown that Boltzmann gave careful consideration to this question so the conditional Boltzmann entropy was certainly known to Boltzmann even if it was expressed in slightly different notation. We can say something more precise. Boltzmann showed that whatever the initial state of a closed gas system, it would on average tend to evolve to an equilibrium state. (Note: We wouldn’t attribute Newton’s Fundamental Theorem of Calculus to the first person who used different notation.)

Now, given equation (4) we may consider two possible extremal principles guiding the behaviour of organisms. One being entropy maximisation and the opposite principle being entropy minimisation. The authors of [4] introduce a maximum entropy principle guiding the behaviour of intelligent organisms and after demonstrating its application to toy mechanical problems with non-living components claim:

The remarkable spontaneous emergence of these sophisticated behaviors from such a simple physical process suggests that causal entropic forces might be used as the basis for a general—and potentially universal—thermodynamic model for adaptive behavior.

Such an extrapolation from non-living to living systems is not only unscientific but a source of a fundamental error as I explain in the following section.

Schrödinger and the search for extremal entropic principles for adaptive systems:

It’s of fundamental importance to realise that this maximum entropy principle due to [4] is demonstrably false as pointed out by Schrödinger in [6] and may be demonstrated as follows. The state-space $X$ which may be identified with an organism (coupled with its environment) may be partitioned so that $X = X_d \cup X_a$ where $X_a$ are states where the organism is alive and $X_d$ are states where the organism is not alive so $X_d \cap X_a = \emptyset$. Given that $|X_d| \gg |X_a|$ (i.e. there are more ways for an organism to be dead than alive), a maximum entropy principle where the agent is coupled with its environment(as is the case in this theory of intelligent systems) leads to a dead organism almost surely. Under ergodic assumptions, this conclusion may be reached by a different line of reasoning as pointed out by Karl Friston in [7].

In fact, let’s consider a concrete example starting with an explanation of what is going on:

1. The Causal Path Entropy maximisation formalism introduced in [4] isn’t a framework for learning so the agent has a static state-transition probability distribution $P$. In addition, it pre-supposes that the agent is coupled with its environment and for our example we may assume that state-transitions are fully determined by the actions of our agent.
2. If we don’t restrict ourselves to toy settings then an objective state-transition probability distribution is implausible. In complex settings, the agent might have a state-transition probability distribution where the probabilities must be subjective.
3. For concreteness let’s suppose the agent’s action-space is coarse-grained to two actions($L,R$) which represents turning left or turning right on a rectangular grid of infinite size. Furthermore, let’s suppose that the environment is first-order Markov.
4. If action $L$ always leads to two states with equal probability and action $R$ always leads to three states with equal probability, due to the maxent principle(which doesn’t identify states) if it can plan $\tau$ steps ahead then its plan is simply to execute action $R$, $\tau$ times.
5. To realise that the agent dies quickly it’s sufficient to assume that its state-space is countable and therefore it may be identified with the natural numbers $\mathbb{N}$. Now, assume that the agent is only alive when the state $x$ is an odd number. Then things end badly very soon. In fact, the probability that our agent is alive after executing a plan over $\tau$ time steps is $2^{-\tau}$.

However, long before Karl Friston in 2009, Schrödinger in 1944 in [6] postulated a minimum entropy principle for the following reasons:

How would we express in terms of the statistical theory the marvellous faculty of a living organism, by which it delays the decay into thermodynamical equilibrium (death)? We said before: ‘It feeds upon negative entropy’, attracting, as it were, a stream of negative entropy upon itself, to compensate the entropy increase it produces by living and thus to maintain itself on a stationary and fairly low entropy level.

It must be clarified here that Schrödinger isn’t claiming that an organism’s internal entropy $H_o$ should be minimised so that it’s non-existent but that there’s an extremal entropy minimisation principle which stabilises the organism’s internal entropy so it doesn’t deviate significantly from $H_o$ whose principal tendency is to increase due to the 2nd law of thermodynamics.

Conclusion:

Not only is the basis for the so-called ‘Causal Path Entropy’ not new; their supposedly original theory is also demonstrably false. This demonstration was made by none other than Schrödinger more than 50 years ago who found no reason to do experiments on toy mechanical systems because he realised that the thermodynamics of living systems is fundamentally different from that of non-living systems. Therefore, the same extremal principle can’t hold for both types of systems.

References:

1. G. C. Rota & D. Sharp. 1985. Mathematics, Philosophy, Artificial and Intelligence. Science.
2. Boltzmann, L. (1872). Weitere Studien über das Wärmegleichgewicht unter Gasmolekün, Sitzungberichte der Akademie der Wissenschaften zu Wien, mathematischnaturwissenschaftliche Klasse, 66, 275-370. Reprinted in Boltzmann (1909) Vol.
3. Villani. (2007) H-Theorem and beyond: Boltzmann’s entropy in today’s mathematics.
4. Gross, A. Wissner. (2013) Causal Entropic Forces. Physical Review Letters.
5. P. Schuster(2007) Boltzmann and Evolution: Some Basic Questions of Biology seen with Atomistic Glasses
6. Schrödinger E. (1944) What is Life?
7. Friston K. (2009) The free-energy principle: a rough guide to the brain?