in probability, a factor \(\phi\) is a value you can assign to each distinct value in a discrete distribution which acts as the probability of that value occurring. They are considered parameters of the discrete distribution.

If you don’t have discrete variables, factors allow you to state \(p(x|y)\) in terms of a function \(\phi(x,y)\).

See also Rejection Sampling

factor operations

factor product

\begin{equation} \phi_{3} (x,y,z) = \phi_{1} (x,y) \cdot \phi_{2}(y,z) \end{equation}

factor marginalization

\begin{equation} \phi(x) = \sum_{y=Y} \phi(x,y) \end{equation}

Motivation

Multiple agents need to collaborate to achieve common goal.

Joint Utility Maximization: maximize the joint utility between various agents.

Possible Approaches

• Using a traditional MDP: an MDP considers “action” as a joint action between all agents (exponential blow up because the agent actions multiply)
• Local Optimization: share rewards/values among agents
• Local Optimization: search and maximize joint utility explicitly (no need to model the entire action space)

Problems with single Reward Sharing:

For a trajectory \(p\qty(\tau)\), the failure distribution is $p \qty(τ | τ ¬ ∈ψ)$—the probability of a particular trajectory given that its a failure:

\begin{equation} p \qty( \tau \mid \tau \not \in \psi) = \frac{\mathbb{1}\qty {\tau \not \in \psi} p\qty(\tau)}{ \int \mathbb{1}\qty {\tau \not \in \psi} p\qty(\tau) \dd{\tau}} \end{equation}

This bottom integral could be very difficult to compute; but the numerator may take a bit more work to compute!

So ultimately we can also give up and don’t normalize (and then use systems that allows us to draw samples from unnormalized probability densities:

take a bunch of failure trajectories, and cluster them; can possibly do it with STL systems

fairness through unawareness

procedural fairness, or fairness through unawareness is a fairness system

If you have no idea about the demographics of protected groups, you will make better decisions.

1. exclude sensitive features from datasets
2. exclude proxies of protected groups

Problem: deeply correlated information (such as stuff that people like) is hard to get rid of—individual features does nothing with respect to predicting gender, but taken in groups it can recover protected group information.

fairness through awareness

we only care about the outcome