In this paper, we propose polynomial forms to represent distributions of state variables over time for discrete-time stochastic dynamical systems. This problem arises in a variety of applications in areas ranging from biology to robotics. Our approach allows us to rigorously represent the probability distribution of state variables over time, and provide guaranteed bounds on the expectations, moments and probabilities of tail events involving the state variables. First, we recall ideas from interval arithmetic, and use them to rigorously represent the state variables at time t as a function of the initial state variables and noise symbols that model the random exogenous inputs encountered before time t. Next, we show how concentration of measure inequalities can be employed to prove rigorous bounds on the tail probabilities of these state variables. We demonstrate interesting applications that demonstrate how our approach can be useful in some situations to establish mathematically guaranteed bounds that are of a different nature from those obtained through simulations with pseudo-random numbers.

The idea is to first develop a prediction model without concern for the downstream risk profile or robustness guarantee, and then utilize calibration (or recalibration) methods to quantify the uncertainty of the prediction.

Finding models that satisfy these twoconditions ischallenging and current methods tendtotackle only one ofthe two. Exponential and implicit generative models have typically strong approximation properties (see e.g.

PSD studied probability have14] Notation.

We follow the derivation route in [7] except the coordinating weight part. According to Eq.(7), we update θ According to the chain rule, Eq.(15) can be written as: For the right part of Eq.(16), it follows that [ ( Figure 3: The Class Activation Map (CAM) [10] is used to perform visual ablation analysis on a chest x-ray image in Open-i dataset. The background color is blue, with red or yellow representing the disease location. The number on the top left corner of each image is the predicted probability for the corresponding disease. We visualize the domain adaptation performance on the transfer scenario NIH-CXR14, CheXpert, MIMIC-CXR to Open-i. The visualization sample in the Open-i is suffering from Atelecsis and Effusion disease.