# Intelligent Agents and Active Inference¶

### Agents¶

• In the previous lessons we assumed that a data set was given.
• In this lesson we consider agents. An agent is a system that interacts with its environment through both sensors and actuators.
• Crucially, by acting onto the environment, the agent is able to affect the data that it will sense in the future.
• As an example, by changing the direction where I look, I can affect the sensory data that will be sensed by my retina.
• With this definition of an agent, (biological) organisms are agents, and so are robots, self-driving cars, etc.
• In an engineering context, we are particularly interesting in agents that behave with a purpose (with a goal in mind), e.g., to drive a car or to design a speech recognition algorithm.
• In this lesson, we will describe how goal-directed behavior by biological (and synthetic) agents can also be interpreted as minimization of a free energy functional $F[q]$.

### Illustrative Example: Steering a cart to a parking spot¶

• In this example, we consider a cart that can move in a 1D space. At each time step the cart can be steered a bit to the left or right by a controller (the "agent"). The agent's knowledge about the cart's process dynamics (equations of motion) are known up to some additive Gaussian process noise. The agent also makes noisy observations of the position and velocity of the cart. Your challenge is to design an agent that steers the car to the zero position. (The agent should be specified as a probabilistic model and the control signal should be formulated as a Bayesian inference task).

• Solution at the end of this lesson.

### Karl Friston and the Free Energy Principle¶

• We begin with a motivating example that requires "intelligent" goal-directed decision making: assume that you are an owl and that you're hungry. What are you going to do?

• Have a look at Prof. Karl Friston's answer in this video segment by on the cost function for intelligent behavior. (Do watch the video!)

• Friston argues that intelligent decision making (behavior, action making) by an agent requires minimization of a functional of beliefs.

• Friston further argues (later in the lecture and his papers) that this functional is a (variational) free energy (to be defined below), thus linking decision making to Bayesian inference.

• In fact, Friston's Free Energy Principle (FEP) claims that all biological self-organizing processes (including brain processes) can be described as Free Energy minimization in a probabilistic model.

• This includes perception, learning, attention mechanisms, recall, action and decision making, etc.
• Taking inspiration from FEP, if we want to develop synthetic "intelligent" agents, we have (only) two issues to consider:

1. The specification of the FE functional (includes specification of generative model and constraints on the approximate posterior).
2. How to minimize the FE functional?

### What Makes a Good Agent?¶

• What should the agent's model be modeling? This question was (already) answered by Conant and Ashby (1970) as the good regulator theorem: every good regulator of a system must be a model of that system.

• From Conant and Ashby's paper (this statement was later finessed by Friston (2013)):

The theory has the interesting corollary that the living brain, insofar as it is successful and efficient as a regulator for survival, must proceed, in learning, by the formation of a model (or models) of its environment."

### Active Inference Agents¶

• We will follow the idea that an agent needs to hold a generative model for its environment, which is observed through sensory channels. The environmental dynamics can be affected through actions onto the environment.

• Agents that follow the FEP and infer actions by inference in a generative model of the environment are engaged in a process called active inference.

• Technically, an active inference-based agent comprises:

1. A free energy functional $F[q] = \mathbb{E}_q\left[ \log\frac{q(z)}{p(x,z)}\right]$, where
• $p(x,z) = \prod_k p(x_k,z_k|z_{k-1})$ is a generative model with observations $\{x_k\}$, latent variables $\{z_k\} = \left\{ \{s_k\}, \{u_k\}, \{\theta_k\}\right\}$ and $k$ is a time index.
• $q(z)$ is a recognition model.
2. A recipe to minimize the free energy $F[q]$
• Let's draw a diagram to show the interactions between an active inference agent and its environment.

• In the model above, the hidden variables $\{z_k\}$ of the agent comprise internal states $\{s_k\}$, control variables $\{u_k\}$ (which are "observed" by the environment as actions $\{a_k\}$), and parameters $\{\theta_k\}$.

• In neuroscience/psychology parlance,

• behavior (movement) is inference for the control signals ($u$)
• perception is inference for the internal states ($s$).
• learning is inference for the parameters ($\theta$)
• We also assume that the agent interacts with an environment, which we represent by a dynamic model $$(y_t,\tilde{s}_t) = R_t\left( a_t,\tilde{s}_{t-1}\right)$$ where $a_t$ are actions , $y_t$ are outcomes and $\tilde{s}_t$ holds the environmental states.

• In the above equations, $u_t$ and $x_t$ are owned by the agent model, whereas $a_t$ and $y_t$ are variables in the environment model.

• The agent can push actions $a_t$ onto the environment and measure responses $y_t$, but has no access to the environmental states $\tilde{s}_t$.

• Interactions between the agent and environment are described by \begin{align*} a_t &\sim q(u_t) \\ x_t &= y_t \end{align*} iow, actions are drawn from the posterior over control signals.

• Note that this system implies a recursive dependency since the agent's future observations depend on the agent's current (and past) actions: $$x_{t+1} = x_{t+1} \left( a_{t+1} \right) = x_{t+1} \left( a_{t+1} \left( u_{t+1}\left( x_t \left( a_t \left( \cdots \right) \right) \right)\right) \right)$$
• $\Rightarrow$ As a result, the agent actively engages in selecting its own data set!

### Goal-directed Behavior¶

• Biological agents select their observations by controling their environment. Perception (and learning) serve to improve this data selection process by updating beliefs about the state of the world.

• This process begs the question: if a (biological) agent seeks out observations, then which observations is the agent interested in? I.o.w., does the agent have a goal "in mind" when it engages in active data selection?

• Yes! Agents set preferences for future observations by setting prior distributions on future observations!

• E.g., a self-driving agent in a car expects to observe no collisions.
• Thus, the generative model for an active inference agent at time $t$ includes variables at future time steps and can be run forward to make predictions (beliefs) about future observations $x_{t+1:T}$.

### Active Inference Agent Model specification¶

• We assume that agents live in a dynamic environment and consider the following generative model for the agent (omitting parameters $\theta$), and assuming the current time is $t$: \begin{align*} p^\prime(x,s,u) &= p(s_{t-1}) \prod_{k=t}^{t+T} \underbrace{p(x_k|s_k) \cdot p(s_k | s_{k-1}, u_k)}_{\text{internal dynamics}} \cdot\underbrace{p(u_k)}_{\substack{\text{control}\\ \text{prior}}} \end{align*}
• Note that the generative model includes future time steps.
• In order to infer goal-driven (i.e., purposeful) behavior, we now add prior beliefs $\tilde{p}(x)$ about desired future observations, leading to an extended agent model: \begin{align*} p(x,s,u) &= \frac{p^\prime(x,s,u) \tilde{p}(x)}{\int_x p^\prime(x,s,u) \tilde{p}(x) \mathrm{d}x} \\ &\propto \underbrace{p(s_{t-1}) \prod_{k=t}^{t+T} p(x_k|s_k) p(s_k | s_{k-1}, u_k) p(u_k)}_{\text{original generative model}} \underbrace{\tilde{p}(x_k)}_{\substack{\text{extension}\\\text{"goal prior"}}} \end{align*}

• $\tilde{p}(x)$ encodes priors beliefs by the agent about future observations.
• Goal-directed behavior follows from inference for controls (actions) at $t$, based on expectations (encoded by priors) about future ($>t$) observations.

• $\Rightarrow$ Actions fulfill expectations about the future!

### FFG for Active Inference Agent Model¶

• After selecting an action $a_t$ and making an observation $y_t$, the FFG for the extended generative model is given by the following FFG:

• The (brown) dashed box is the agent's Markov blanket. Given the states on the Markov blanket, the internal states of the agent are independent of the state of the world.

### How to minimize FE: Online Active Inference¶

• Online active inference proceeds by iteratively executing three stages: (1) act-execute-observe, (2) infer the next control/action, (3) slide forward

In [5]:
using Pkg;Pkg.activate("probprog/workspace");Pkg.instantiate()
IJulia.clear_output();


### The Cart Park Problem Revisited¶

Here we solve the cart parking problem as stated at the beginning of this lesson. We first specify a generative model for the agent's environment (which is the observed noisy position of the cart) and then constrain future observations by a prior distribution that is located on the target parking spot. Next, we schedule a message passing-based inference algorithm for the next action. This is followed by executing the "Act-execute-observe --> infer --> slide" procedure to infer a sequence of consecutive actions. Finally, the position of the cart over time is plotted. Note that the cart convergees onto the target spot.

In [6]:
using PyPlot, LinearAlgebra, ForneyLab
# Load helper functions. Feel free to explore these
include("ai_agent/environment_1d.jl")
include("ai_agent/helpers_1d.jl")
include("ai_agent/agent_1d.jl")

# Internal model perameters
gamma   = 100.0 # Transition precision
phi     = 10.0 # Observation precision
upsilon = 1.0 # Control prior variance
sigma   = 1.0 # Goal prior variance

# Build internal model
fg = FactorGraph()

o = Vector{Variable}(undef, T) # Observed states
s = Vector{Variable}(undef, T) # internal states
u = Vector{Variable}(undef, T) # Control states

@RV s_t_min ~ GaussianMeanVariance(placeholder(:m_s_t_min),
placeholder(:v_s_t_min)) # Prior  state
u_t = placeholder(:u_t)
@RV u[1] ~ GaussianMeanVariance(u_t, tiny)
@RV s[1] ~ GaussianMeanPrecision(s_t_min + u[1], gamma)
@RV o[1] ~ GaussianMeanPrecision(s[1], phi)
placeholder(o[1], :o_t)

s_k_min = s[1]
for k=2:T
@RV u[k] ~ GaussianMeanVariance(0.0, upsilon) # Control prior
@RV s[k] ~ GaussianMeanPrecision(s_k_min + u[k], gamma) # State transition model
@RV o[k] ~ GaussianMeanPrecision(s[k], phi) # Observation model
GaussianMeanVariance(o[k],
placeholder(:m_o, var_id=:m_o_*k, index=k-1),
placeholder(:v_o, var_id=:v_o_*k, index=k-1)) # Goal prior
s_k_min = s[k]
end

# Schedule message passing algorithm
algo = messagePassingAlgorithm(u[2]) # Infer internal states
source_code = algorithmSourceCode(algo)
eval(Meta.parse(source_code)) # Loads the step!() function for inference

s_0 = 2.0 # Initial State

N = 20 # Total simulation time

(execute, observe)  = initializeWorld() # Let there be a world
(infer, act, slide) = initializeAgent() # Let there be an agent

# Step through action-perception loop
u_hat = Vector{Float64}(undef, N) # Actions
o_hat = Vector{Float64}(undef, N) # Observations
for t=1:N
u_hat[t] = act() # Evoke an action from the agent
execute(u_hat[t]) # The action influences hidden external states
o_hat[t] = observe() # Observe the current environmental outcome (update p)
infer(u_hat[t], o_hat[t]) # Infer beliefs from current model state (update q)
slide() # Prepare for next iteration
end

# Plot active inference results
plotTrajectory(u_hat, o_hat)
;


## OPTIONAL SLIDES¶

### Specification of Free Energy¶

• Consider the agent's inference task at time step $t$, right after having selected an action $a_t$ and having made an observation $y_t$.

• As usual, we record actions and observations by substituting the values into the generative model(in the Act-Execute-Observe phase): \begin{align*} p(x,s,u) &\propto \underbrace{p(x_t=y_t|s_t)}_{\text{observation}} p(s_t|s_{t-1},u_t) p(s_{t-1}) \underbrace{p(u_t=a_t)}_{\text{action}} \\ & \quad \cdot \underbrace{\prod_{k=t+1}^{t+T} p(x_k|s_k) p(s_k | s_{k-1}, u_k) p(u_k) p^+(x_k)}_{\text{future}} \end{align*}

• Note that (future) $x$ is also a latent variable and hence we include $x$ in the recognition model.

• This leads to the following free energy functional \begin{align*} F[q] &\propto \sum_{x,s,u} q(x,s,u) \log \frac{q(x,s,u)}{p(x,s,u)} \end{align*}

### FE Decompositions¶

• Lots of interesting FE decompositions are possible again. For instance \begin{align*} F[q] &\propto \sum_{x,s,u} q(x,s,u) \log \frac{q(x,s,u)}{p(x,s,u)} \\ &= \sum_{u} q(u) \underbrace{\sum_{x,s} q(x,s|u)\log \frac{q(x,s|u)}{p(x,s|u)}}_{F_u[q]} + \underbrace{\sum_{u} q(u) \log \frac{q(u)}{p(u)}}_{\text{complexity}} \end{align*} breaks the FE into a complexity term and a term $F_u[q]$ that is conditioned on the policy $u$.

• It can be shown (exercise) that the optimal posterior for the policy is now given by $$q^*(u) \propto p(u) \exp \left( -F^*_u \right)$$

• Let's consider a break-up $x=(x_t,x_{>t})$ with $x_{>t} = (x_{t+1},\ldots,x_{t+T})$ that recognizes the distinction between already observed and future data. Then \begin{align*} F_u[q] &= \underbrace{-\log p(x_t)}_{\substack{-\log(\text{evidence}) \\ \text{(surprise)}}} + \underbrace{\sum_{x,s} q(x_{>t},s|u)\log \frac{q(x_{>t},s|u)}{p(x_{>t},s|u)}}_{\substack{\text{divergence}\\ \text{(inference costs)}}}\,. \end{align*}

• The inference costs (divergence term) can be further decomposed to \begin{align*} \underbrace{-\sum_{x} q(x_{>t}) \log p(x_{>t})}_{\substack{\text{expected surprise} \\ \text{(goal-directed, pragmatic costs)}}} + \underbrace{\sum_{x,s} q(x_{>t},s|u) \log \frac{q(x_{>t},s|u)}{p(s|x_{>t},u)}}_{\text{epistemic costs}} \end{align*}

• Minimizing goal-directed costs selects actions that (expect to) fullfil the priors over future observations. Minimization of epistemic ("knowledge seeking") costs leads to actions that maximize information gain about the environmental dynamics. This can be seen by further decomposition of the epistemic costs into \begin{align*} &\sum_{x,s} q(x_>t,s|u) \log \frac{q(s|u)}{p(s|x_{>t},u)} + \sum_{x,s} q(x_{>t},s|u) \log q(x_{>t}|s,u) \\ \approx &\underbrace{\sum_{x,s} q(x_>t,s|u) \log \frac{q(s|u)}{q(s|x_{>t},u)}}_{-\text{mutual information}} - \underbrace{\mathbb{E}_{q(s|u)}\left[ H\left[ q(x_{>t}|s,u)\right]\right]}_{\text{ambiguity}} \end{align*} where we used the approximation $q(s|x_{>t},u) \approx p(s|x_{>t},u)$ to illuminate the link to the mutual information.

• Minimizing FE leads (approximately) to mutual information maximization between internal states $s$ and observations $x$. In other words, FEM leads to actions that aim to seek out observations that are maximally informative about the hidden causes of these observations.

• Ambiguous states have uncertain mappings to observations. Minimizing FE leads to actions that try to avoid ambiguous states.

• In short, if the generative model includes variables that represent (yet) unobserved future observations, then action selection by FEM leads to a very sophisticated behavioral strategy that is maximally consistent with

• Bayesian notions of model complexity
• evidence from past observations
• goal-directed imperatives by priors on future observations
• epistemic (knowledge seeking) value maximization, both in terms of MI maximization and avoidance of ambiguous states
• All these imperatives are simultaneously represented and automatically balanced against each other in a single time-varying cost function (Free Energy) that needs no tuning parameters.

• (Just to be sure, you don't need to memorize these derivations nor are you expected to derive them on-the-spot. We present these decompositions only to provide insight into the multitude of forces that underlie FEM-based action selection.)

### Free energy distribution in FFG¶

In [7]:
open("../../styles/aipstyle.html") do f