gym-update

Gym-style API environment

A write up

Here's the most recent write up regarding the envoronment and algorithms applied to it.

Comments

• in general, if we record a transition up to "Done" or if we update as soon as we reach "Done", the info collected is very little. Done is reached after 1 or 2 transitions. specify a different condition

Environment dynamics

The functions used:

• $f_e(x^s, x^a) = \mathbb{E}[Y_e|X_e(1) = (x^s, x^a)]$: Causal mechanism determining probability of $Y_e = 1$ given $X_e(1)$. We will take $f_e(x^s, x^a) = (1 + \exp^{−x^s−x^a})^{−1}$
• $g^a_e(\rho, x^a) \in {g : [0, 1] \times \Omega \rightarrow \Omega }$: Intervention process on $X^a$ in response to a predictive score $\rho$ updating $X^a_e(0) \rightarrow X^a_e(1)$
• $\rho_e(x^s, x^a) \in {\rho_e : \Omega^s \times \Omega^a \rightarrow [0, 1]}$: Predictive score trained at epoch $e$

Additional information:

• At epoch $e$, the predictive score $\rho$ uses $X^a_e(0), X^s_e(0)$ and $Y_e$ as training data; previous epochs are ignored and $X^a_e(1), X^s_e(1)$ are not observed. The predictive score is computed at time $t=0$.
• We allow $\rho_e$ to be an arbitrary function, but generally presume it is an estimator of $\rho_e(x^s, x^a) \approx E [Y_e|X^s_e(0) = x^s, X^a_e(0) = x^a]= f_e(x^s, g^a_e(\rho_{e-1}, x^a)) \triangleq \tilde{f}_e(x^s, x^a) $
• $\forall e f_e = E[Y_e|X_e] = E[Y_e|X_e(1)]$: $Y_e$ depends on $X_e(1)$; that is, after any potential interventions
• a higher value $\rho$ means a larger intervention is made (we assume $g^a_e$ to be deterministic, but random valued functions may more accurately capture the uncertainty linked to real-world interventions)

Naive updating

By ‘naive’ updating it is meant that a new score $ρ_e$ is fitted in each epoch, and then used as a drop-in replacement of an existing score $ρ_{e−1}$. It leads to estimates $\rho_e(x^s, x^a)$ converging as $e \rightarrow \infty$ to a setting in which $\rho_e$ accurately estimates its own effect: conceptually, $\rho_e(x^s, x^a)$ estimates the probability of $Y$ after interventions have been made on the basis of $\rho_e(x^s, x^a)$ itself.

EPOCH 0
t=0

• observe a population of patients $(X_0^a(0),X_0^s(0))_{i=1}^N$

t=1

• there are no interventions, hence $X_0^a(1) = X_0^a(0)$
• the risk of observing $Y = 1$ depends only on covariates at $t1$ through $f_0$ and is $E[Y_0|X_0(0) = (x^s, x^a)] =f(x^s, x^a)$
• the score $\rho_0$ is therefore defined as $\rho_0(x^s, x^a) = f(x^s, x^a)$
• $Y_0$ is observed
• analyst decides a function $\rho_0$, which is retained into epoch 1. We will use initialized actions $\theta = (\theta^0, \theta^1, \theta^2)$

The model performance under non-intervention is equivalent to performance at epoch 0

EPOCH $>0$ t=0

• observe a new population of patients $(X_e^a(0),X_e^s(0))_{i=1}^N$
• analyst computes $\rho_0 (X^s_e(0), Xa_e(0))$

t=1

• $X^s_e(0)$ is not interventionable and becomes $X^s_e(1)$
• $\rho_0$ is used to inform interventions $g^a_e$ to change values $X^a_e(1) = g_e(\rho_{e-1}(x^s, x^a), x^a)$
• $E[Y_1]$ is determined by covariates $X^s_e(1), X^a_e(1)$
• the score $ρ_e$ is defined as $\rho_e(x^s, x^a) = f_e(x^s, g^a_e(\rho_{e-1}(x^s, x^a), xa)) \triangleq h(\rho_{e−1} (x^s, x^a))
• $Y_e$ is observed
• analyst decides a function $\rho_e$ using $X^s_e(1), X^a_e(1), Y_e$, which is retained into epoch $e+1$. We will use $\rho_e =(1 + exp^(−\theta^0 −x^s \theta^1 −x^a \beta^2 ))^{−1}$
  

Then the episodes repeat

state and action spaces:

Action space: 3D space $\in [-2, 2]$. Actions represent the coefficients thetas of a logistic regression that will be run on the dataset of patients

Observation space: aD space $\in [0, \infty)$. States represent values for the predictive score $f_e$

To install

• git clone https://github.com/claudia-viaro/gym-update.git
• cd gym-update
• !pip install gym-update
• import gym
• import gym_update
• env =gym.make('update-v0')

To change version

• change version to, e.g., 1.0.7 from setup.py file
• git clone https://github.com/claudia-viaro/gym-update.git
• cd gym-update
• python setup.py sdist bdist_wheel
• twine check dist/*
• twine upload --repository-url https://upload.pypi.org/legacy/ dist/*