The equalization probability of the Polya urn

We consider a Polya urn, started with b black and w white balls, where b>w. We compute the probability that there are ever the same number of black and white balls in the urn, and show that it is twice the probability of getting no more than w-1 heads in b+w-1 tosses of a fair coin

The marginalization paradox and the formal Bayes' law

It has recently been shown that the marginalization paradox (MP) can be resolved by interpreting improper inferences as probability limits. The key to the resolution is that probability limits need not satisfy the formal Bayes' law, which is used in the MP to deduce an inconsistency. In this paper, I explore the differences between probability limits and the more familiar pointwise limits, which do imply the formal Bayes' law, and show how these differences underlie some key differences in the interpretation of the MP.Comment: Presented at Maxent 2007, Saratoga Springs, NY, July 200

Transmission of Single HIV-1 Genomes and Dynamics of Early Immune Escape Revealed by Ultra-Deep Sequencing

We used ultra-deep sequencing to obtain tens of thousands of HIV-1 sequences from regions targeted by CD8+ T lymphocytes from longitudinal samples from three acutely infected subjects, and modeled viral evolution during the critical first weeks of infection. Previous studies suggested that a single virus established productive infection, but these conclusions were tempered because of limited sampling; now, we have greatly increased our confidence in this observation through modeling the observed earliest sample diversity based on vastly more extensive sampling. Conventional sequencing of HIV-1 from acute/early infection has shown different patterns of escape at different epitopes; we investigated the earliest escapes in exquisite detail. Over 3–6 weeks, ultradeep sequencing revealed that the virus explored an extraordinary array of potential escape routes in the process of evading the earliest CD8 T-lymphocyte responses – using 454 sequencing, we identified over 50 variant forms of each targeted epitope during early immune escape, while only 2–7 variants were detected in the same samples via conventional sequencing. In contrast to the diversity seen within epitopes, non-epitope regions, including the Envelope V3 region, which was sequenced as a control in each subject, displayed very low levels of variation. In early infection, in the regions sequenced, the consensus forms did not have a fitness advantage large enough to trigger reversion to consensus amino acids in the absence of immune pressure. In one subject, a genetic bottleneck was observed, with extensive diversity at the second time point narrowing to two dominant escape forms by the third time point, all within two months of infection. Traces of immune escape were observed in the earliest samples, suggesting that immune pressure is present and effective earlier than previously reported; quantifying the loss rate of the founder virus suggests a direct role for CD8 T-lymphocyte responses in viral containment after peak viremia. Dramatic shifts in the frequencies of epitope variants during the first weeks of infection revealed a complex interplay between viral fitness and immune escape

Prospective investigation of risk factors for prostate cancer in the UK Biobank cohort study

Prostate cancer is the most common cancer in British men but its aetiology is not well understood. We aimed to identify risk factors for prostate cancer in British males.We studied 219 335 men from the UK Biobank study who were free from cancer at baseline. Exposure data were collected at recruitment. Prostate cancer risk by the different exposures was estimated using multivariable-adjusted Cox proportional hazards models.In all, 4575 incident cases of prostate cancer occurred during 5.6 years of follow-up. Prostate cancer risk was positively associated with the following: black ethnicity (hazard ratio black vs white=2.61, 95% confidence interval=2.10-3.24); having ever had a prostate-specific antigen test (1.31, 1.23-1.40); being diagnosed with an enlarged prostate (1.54, 1.38-1.71); and having a family history of prostate cancer (1.94, 1.77-2.13). Conversely, Asian ethnicity (Asian vs white hazard ratio=0.62, 0.47-0.83), excess adiposity (body mass index (⩾35 vs <25 kg m-2=0.75, 0.64-0.88) and body fat (⩾30.1 vs <20.5%=0.81, 0.73-0.89)), cigarette smoking (current vs never smokers=0.85, 0.77-0.95), having diabetes (0.70, 0.62-0.80), and never having had children (0.89, 0.81-0.97) or sexual intercourse (0.53, 0.33-0.84) were related to a lower risk.In this new large British prospective study, we identified associations with already-established, putative and possible novel risk factors for being diagnosed with prostate cancer. Future research will examine associations by tumour characteristics

HIV-2 Genetic Evolution in Patients with Advanced Disease Is Faster than That in Matched HIV-1 Patients▿ †

The objective of this study was to estimate and compare the evolutionary rates of HIV-2 and HIV-1. Two HIV-2 data sets from patients with advanced disease were compared to matched HIV-1 data sets. The estimated mean evolutionary rate of HIV-2 was significantly higher than the estimated rate of HIV-1, both in the gp125 and in the V3 region of the env gene. In addition, the rate of synonymous substitutions in gp125 was significantly higher for HIV-2 than for HIV-1, possibly indicating a shorter generation time or higher mutation rate of HIV-2. Thus, the lower virulence of HIV-2 does not appear to translate into a lower rate of evolution

Expanded Breadth of the T-Cell Response to Mosaic Human Immunodeficiency Virus Type 1 Envelope DNA Vaccination▿ †

An effective AIDS vaccine must control highly diverse circulating strains of human immunodeficiency virus type 1 (HIV-1). Among HIV-1 gene products, the envelope (Env) protein contains variable as well as conserved regions. In this report, an informatic approach to the design of T-cell vaccines directed to HIV-1 Env M group global sequences was tested. Synthetic Env antigens were designed to express mosaics that maximize the inclusion of common potential T-cell epitope (PTE) 9-mers and minimize the inclusion of rare epitopes likely to elicit strain-specific responses. DNA vaccines were evaluated using intracellular cytokine staining in inbred mice with a standardized panel of highly conserved 15-mer PTE peptides. One-, two-, and three-mosaic sets that increased theoretical epitope coverage were developed. The breadth and magnitude of T-cell immunity stimulated by these vaccines were compared to those for natural strain Envs; additional comparisons were performed on mutant Envs, including gp160 or gp145 with or without V regions and gp41 deletions. Among them, the two- or three-mosaic Env sets elicited the optimal CD4 and CD8 responses. These responses were most evident in CD8 T cells; the three-mosaic set elicited responses to an average of eight peptide pools, compared to two pools for a set of three natural Envs. Synthetic mosaic HIV-1 antigens can therefore induce T-cell responses with expanded breadth and may facilitate the development of effective T-cell-based HIV-1 vaccines