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Below is a draft, and I'm publishing it now because I probably won't get around to finishing it in the near future. Ideally this would be illustrated (like this post explaining Embedded Agency), but I'm not going to do that myself. (Lmk:)

I wanted to create a distillation of Part 3a: Defining boundaries as directed Markov blankets, but I ended up creating some other kind of conceptual explanation of why it makes sense to represent «boundaries» via Markov blankets. It incorporates a lot of the ideas from this post.

There's also an idea in this draft about boundaries being what form agents (— as opposed to "agents having boundaries"). Similarly, that agents maintain homeostasis over their boundaries. I quite like this framing.

This is an agent: {graphic: person}

And she exists in an environment: {graphic: person in environment, with regions separated by colors}

But where is the agent? Precisely how is she separate from the environment? Where's the math?!

First consider: an agent most likely exists today because she survived yesterday. She's not a coincidence or a Boltzmann brain. She has agency (a process of making decisions), and she has successfully kept herself separate from the world so far

She hasn't allowed herself to starve, and she hasn't allowed other forces to overpower her. She's successfully fought entropy, and for her own sake she must continue to do so.

{graphics: not starving (ie exfiltration); not become a puppet to other agents eg zombie (ie infiltration)}

That is, she's maintaining some boundary that separates herself from everything else.

{graphic: agent and environment, her boundary/skin is highlighted in color}

So,

1. She must be preventing being controlled by outside influences (eg getting infected by something) and not letting bad, dangerous things in. She must somehow be keeping bad things out.
{graphic: shows agent with force arrows pointing outwards blocking pathogens or something}
2. She must somehow be keeping her guts/organs/information/heat/energy in.
{graphic: shows agent with force arrows pulling her organs etc inwards}
Bear in mind that diffusion would be the natural consequence of entropy, so something interesting must be happening here.
{graphic showing diffusion/entropy}
Somehow, she is keeping her good stuff and her sensitive stuff in.

This inwards force on good/sensitive things, and outwards force on bad/dangerous things is mediated by her boundary. That includes her skin, but also her food hole, waste hole, eyes and ears, and her ability to move.

{graphic}

If she didn't have these two inwards and outwards 'forces', she would die.

But instead, she persists through time.

{graphic: Markov chain: agent_t-1 --> agent_t --> agent_t+1}

We call this a Markov chain, where each node represents that state of the agent at one point in time. Each next node is fully determined by the nodes that point into it.

The same is of course true of the environment, which obeys ~deterministic laws of physics:

{graphic: Markov chain: E_t-1 --> E_t --> E_t+1}

Now, the agent and the environment also interact with each other.

And because the agent and the environment are part of the same universe, they interact:

{Both chains, proper time-respecting arrows in between: E_t --> agent_t+1; agent_t --> E_t+1}

However, our agent is also separating herself from the environment, so let's represent that too. We will split our representation of the agent into her guts (insides/viscera) (V) and her boundary (B).

{three chains: V, B, E: self-connected, but no interconnections between chains. Also there's a colored glow behind V and B that shows that V+B are the agent, and E is not the agent.}

And as we've been saying, the environment doesn't directly control the insides of the agent, so we're not going to draw those arrows

{graphic showing what arrows won't be drawn: E-->V)

Meanwhile, this would be true for something like a rock because rocks don't maintain homeostasis over their boundary. {graphic?} But she does. Again, she must maintain her boundary to separate herself from the environment if she is to live.

{something something antifragility?}

Instead, (for the most part) the environment influences the agent's boundary and then the boundary influences her insides. This is how she is able to keep unwanted things out (minimize infiltration).

{previous diagram, but with added interconnections ∀t: E_t-->B_t+1-->V_t+2}

Moreover, ideally her good/sensitive stuff doesn't leak out (minimize exfiltration). Matter comes out through the waste hole, not through a gash in one's stomach. So we will model causal influence outwards as entirely mediated by the boundary.

{second-to-last diagram, but with added interconnections ∀t: V_t-->B_t+1-->E_t+2"}

Now everything together:

{last two diagrams together, all added connections. I.e. this diagram, but B in place of A,P:  ~ like this}

Again, notice the connections that are not present:

{~ like this}

Now, the boundaries of real organisms can be subdivided into what handles inputs, and what handles outputs. That is: perceptions and actions. We will call this "passive", "active", and use this going forward

{something like this}

{update diagram accordingly by splitting B --> A ∪ P: ~ like this}

now, we can actually state the states represented by the nodes and measure the effect. {introduce and explain conditional mutual information}

define infiltration: "How does the environment directly affect the agent? ("Directly" = "subtracting (conditioning on) the effect of everything else")"

{equation from Critch's 3a}

define exfiltration: "How does the agent directly affect the environment? ("Directly" = "subtracting (conditioning on) the effect of everything else")"

{equation from Critch's 3a}

(referring to this—)

1. "Infiltration" of information from the environment into the active boundary & viscera:
2. "Exfiltration" of information from the viscera into the passive boundary & environment:

In our model, infiltration and exfiltration are obviously 0.

And this would be ideal for the agent: infiltration, exfiltration minimized. She keeps 100% of the bad stuff out, and keeps 100% of the good stuff in.

But that's not true in messy reality, so it's just an approximate Markov blanket. For example, the agent is leaking things out all of the time that she would rather keep (eg heat, private information).

The real Markov blanket is of course always:

{previous diagram but with connections E_t-->A_t+1 and A_t -->E_t+1}

But the agent is clearly separating herself in the real world, so we can approximate it with the simpler version. And we could measure that error statistically, too, if we wanted, using infiltration and exfiltration.

So that is our agent separates herself from the world to live.

{summary graphic}

Thanks to Ulisse Mini for the idea to write this.

Also, if you're interested in helping me finish this, DM me.

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