*Ben Brubaker, Daniel Bump, and Solomon Friedberg*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691150659
- eISBN:
- 9781400838998
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691150659.003.0014
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter introduces the theorem that says Statement E implies Statement D, first by fixing a nodal signature η and presenting a “cut and paste” virtual resotope. It then presents the proof, ...
More

This chapter introduces the theorem that says Statement E implies Statement D, first by fixing a nodal signature η and presenting a “cut and paste” virtual resotope. It then presents the proof, whereby α ∈, CPsubscript Greek small letter eta(c₀, · · ·,csubscript d) and let σ = θ(α,η). The chapter proceeds by extending the function GΓ from the set of decorated Γ-accordions to the free abelian group by linearity. Also the involution on decorated accordions induces an isomorphism. The relevant equation is obtained using the principle of inclusion-exclusion.Less

This chapter introduces the theorem that says Statement E implies Statement D, first by fixing a nodal signature η and presenting a “cut and paste” virtual resotope. It then presents the proof, whereby α ∈, CPsubscript Greek small letter eta(*c*₀, · · ·,*c*subscript *d*) and let σ = θ(α,η). The chapter proceeds by extending the function *G*Γ from the set of decorated Γ-accordions to the free abelian group by linearity. Also the involution on decorated accordions induces an isomorphism. The relevant equation is obtained using the principle of inclusion-exclusion.

*Thomas S. Richardson, Robin J. Evans, and James M. Robins*

- Published in print:
- 2011
- Published Online:
- January 2012
- ISBN:
- 9780199694587
- eISBN:
- 9780191731921
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199694587.003.0019
- Subject:
- Mathematics, Probability / Statistics

We consider causal models involving three binary variables: a randomized assignment Z, an exposure measure X, and a final response Y. We focus particular attention on the situation in which there may ...
More

We consider causal models involving three binary variables: a randomized assignment Z, an exposure measure X, and a final response Y. We focus particular attention on the situation in which there may be confounding of X and Y, while at the same time measures of the effect of X on Y are of primary interest. In the case where Z has no effect on Y, other than through Z, this is the instrumental variable model. Many causal quantities of interest are only partially identified. We first show via an example that the resulting posteriors may be highly sensitive to the specification of the prior distribution over compliance types. To address this, we present several novel “transparent” re‐parametrizations of the likelihood that separate the identified and non‐ identified parts of the parameter. In addition, we develop parametrizations that are robust to model mis‐specification under the “intent‐to‐treat” null hypothesis that Z and Y are independent.Less

We consider causal models involving three binary variables: a randomized assignment *Z*, an exposure measure *X*, and a final response *Y*. We focus particular attention on the situation in which there may be confounding of *X* and *Y*, while at the same time measures of the effect of *X* on *Y* are of primary interest. In the case where *Z* has no effect on *Y*, other than through *Z*, this is the instrumental variable model. Many causal quantities of interest are only partially identified. We first show via an example that the resulting posteriors may be highly sensitive to the specification of the prior distribution over compliance types. To address this, we present several novel “transparent” re‐parametrizations of the likelihood that separate the identified and non‐ identified parts of the parameter. In addition, we develop parametrizations that are robust to model mis‐specification under the “intent‐to‐treat” null hypothesis that *Z* and *Y* are independent.

*IAN ANDERSON*

- Published in print:
- 2013
- Published Online:
- September 2013
- ISBN:
- 9780199656592
- eISBN:
- 9780191748059
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199656592.003.0014
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics, History of Mathematics

This chapter outlines the historical development of the study of combinatorial problems concerning finite sets, beginning with the inclusion–exclusion principle of de Moivre in the early 18th ...
More

This chapter outlines the historical development of the study of combinatorial problems concerning finite sets, beginning with the inclusion–exclusion principle of de Moivre in the early 18th century, and finishing with the 20th-century development of a unified body of theory relating to the intersections, unions, and orderings of collections of finite sets.Less

This chapter outlines the historical development of the study of combinatorial problems concerning finite sets, beginning with the inclusion–exclusion principle of de Moivre in the early 18th century, and finishing with the 20th-century development of a unified body of theory relating to the intersections, unions, and orderings of collections of finite sets.