Measuring and making DNA is central to modern biology and biomedicine.

Here, we show that marginal variance tends to increase along the causal order for generically sampled additive noise models.

Here, we show that marginal variance tends to increase along the causal order for generically sampled additive noise models.

Our work does not encourage unethical aspects of machine learning technologies.

Uncertainty quantification has received increasing attention in machine learning in the recent past. In particular, a distinction between aleatoric and epistemic uncertainty has been found useful in this regard. The latter refers to the learner's (lack of) knowledge and appears to be especially difficult to measure and quantify. In this paper, we analyse a recent proposal based on the idea of a second-order learner, which yields predictions in the form of distributions over probability distributions. While standard (first-order) learners can be trained to predict accurate probabilities, namely by minimising suitable loss functions on sample data, we show that loss minimisation does not work for second-order predictors: The loss functions proposed for inducing such predictors do not incentivise the learner to represent its epistemic uncertainty in a faithful way.

Inverse problems seek to estimate unknown parameters using noisy observations or measurements. One of the main challenges is that they are often ill-posed.

Furthermore, in continuous state-action settings these methods must make coarse approximations to be computationally tractable (e.g.

Learning and generating various types of data based on conditional diffusion models has been a research hotspot in recent years. Although conditional diffusion models have made considerable progress in improving acceleration algorithms and enhancing generation quality, the lack of non-asymptotic properties has hindered theoretical research. To address this gap, we focus on a conditional diffusion model within the domains of classification and regression (CARD), which aims to learn the original distribution with given input x (denoted as Y|X). It innovatively integrates a pre-trained model f_ϕ(x) into the original diffusion model framework, allowing it to precisely capture the original conditional distribution given f (expressed as Y|f_ϕ(x)). Remarkably, when f_ϕ(x) performs satisfactorily, Y|f_ϕ(x) closely approximates Y|X. Theoretically, we deduce the stochastic differential equations of CARD and establish its generalized form predicated on the Fokker-Planck equation, thereby erecting a firm theoretical foundation for analysis. Mainly under the Lipschitz assumptions, we utilize the second-order Wasserstein distance to demonstrate the upper error bound between the original and the generated conditional distributions. Additionally, by appending assumptions such as light-tailedness to the original distribution, we derive the convergence upper bound between the true value analogous to the score function and the corresponding network-estimated value.