相关论文: A Proof of the Front-Door Adjustment Formula

Backdoor adjustment is a technique in causal inference for estimating interventional quantities from purely observational data. For example, in medical settings, backdoor adjustment can be used to control for confounding and estimate the…

The front-door criterion can be used to identify and compute causal effects despite the existence of unmeasured confounders between a treatment and outcome. However, the key assumptions -- (i) the existence of a variable (or set of…

In this paper, we study the inversion formula for recovering a function from its windowed Fourier transform. We give a rigorous proof for an inversion formula which is known in engineering. We show that the integral involved in the formula…

We present a simple, closed formula which gives all the primes in order. It is a simple product of integer floor and ceiling functions.

By changing variables in a suitable way and using dominated convergence methods, this note gives a short proof of Stirling's formula and its refinement.

In this paper, we develop an elementary proof of the change of variables in multiple integrals. Our proof is based on an induction argument. Assuming the formula for (m-1)-integrals, we define the integral over hypersurface in Rm, establish…

We study the topological $\mu$-calculus, based on both Cantor derivative and closure modalities, proving completeness, decidability and FMP over general topological spaces, as well as over $T_0$ and $T_D$ spaces. We also investigate…

In this paper, we present a novel approach to solving the American put options pricing model by hugely relying on a front-fixing Crank-Nicolson finite difference method. Since the American put option pricing model is a widely used financial…

In observational studies, adjusting for confounders is required if a treatment comparison is planned. A crude comparison of the primary endpoint without covariate adjustment will suffer from biases, and the addition of regression models…

Formal transformations somehow resembling the usual derivative are surprisingly common in computer science, with two notable examples being derivatives of regular expressions and derivatives of types. A newcomer to this list is the…

Causal effect estimation often succeeds cost-constrained sequential data collection. This work considers multivariate linear front-door models with arbitrary unobserved confounding on treatment and response. We optimize the experimental…

We present a simple inductive proof of the Lagrange Inversion Formula.

We derive some simple relations that demonstrate how the posterior convergence rate is related to two driving factors: a "penalized divergence" of the prior, which measures the ability of the prior distribution to propose a nonnegligible…

We give a half-page proof of the Lagrange-Good formula, using the Fourier representation of Dirac delta function.

We prove a discrete approximation of functionals with jumps and creases.

Alias analysis, which determines whether two expressions in a program may reference to the same object, has many potential applications in program construction and verification. We have developed a theory for alias analysis, the "alias…

The front-door criterion is an identification strategy for the intervention-specific mean outcome in settings where the standard back-door criterion fails due to unmeasured exposure-outcome confounders, but an intermediate variable exists…

This note explores the theoretical justification for some approximations of arithmetic forwards ($F_a$) with weighted averages of overnight (ON) forwards ($F_k$). The central equation presented in this analysis is: \begin{equation*}…

In this paper we give a mathematical proof of Dodgson algorithm [1]. Recently Zeilberger [2] gave a bijective proof. Our techniques are based on determinant properties and they are obtained by induction.

This work proposes a conformable fractional predictor-corrector algorithm for solving conformable fractional differential equations. Fractional calculus is finding applications in various scientific fields, but existing numerical methods…