Odds

Summary

In probability theory, odds provide a measure of the likelihood of a particular outcome. They are calculated as the ratio of the number of events that produce that outcome to the number that do not. Odds are commonly used in gambling and statistics. Odds also have a simple relation with probability: the odds of an outcome are the ratio of the probability that the outcome occurs to the probability that the outcome does not occur. In mathematical terms, where p is the probability of the outcome: :\text{odds} = \frac{p}{1-p} where 1 – p is the probability that the outcome does not occur. Odds can be demonstrated by examining rolling a six-sided die. The odds of rolling a 6 is 1 to 5 (abbreviated 1:5). This is because there is 1 event (rolling a 6) that produces the specified outcome of "rolling a 6", and 5 events that do not (rolling a 1, 2, 3, 4 or 5). The odds of rolling either a 5 or 6 is 2:4. This is because there are 2 events (rolling a 5 or 6) tha

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Tumor control and radiobiological fingerprint after Gamma Knife radiosurgery for posterior fossa meningiomas: A series of 46 consecutive cases

Constantin Tuleasca

Introduction: Gamma Knife radiosurgery (GKR) can be a valuable treatment option for posterior cranial fossa meningiomas (PCFM). We retrospectively analyzed outcomes of GKR for PCFM.& nbsp;Methods: Were included forty-six patients with 47 PCFM. Primary endpoint was tumor control. Secondary endpoint was clinical improvement. Biologically effective dose (BED) was evaluated in relationship to primary and secondary outcomes. Mean marginal dose was 12.4 Gy (median 12, 12-14). Mean BED was 63.6 Gy (median 65, 49.1-88.3). Mean target volume (TV) was 2.21 cc (range 0.3-8.9 cc).& nbsp;Results: Overall tumor control rate was 93.6% (44/47) after mean follow-up of 47.8 months & PLUSMN; 28.46 months (median 45.5, range 6-108). Radiological progression-free survival at 5 years was 94%. Higher pretherapeutic TVs were predictive for higher likelihood of tumor progression (Odds ratio, OR 1.448, 95% confidence interval CI 1.001-2.093, p = 0.049). At last clinical follow-up, 28 patients (71.8%) remained stable, 10 (25.6%) improved and 1 patient (2.6%) worsened. Using logistic regression, the relationship between BED and clinical improvement was assessed (OR 0.903, standard error 0.59, coefficient 0.79-1.027, CI-0.10; 0.01; p = 0.14). The highest probability of clinical improvement corresponded to a range of BED values between 56 and 61 Gy.& nbsp;Conclusion: Primary GKR for PCFM is safe and effective. Higher pretherapeutic TV was predictor of volumetric progression. Highest probability of clinical improvement might correspond to a range of BED values between 56 and 61 Gy, although this was not statistically significant. The importance of BED should be further validated in larger cohorts, other anatomical locations and other pathologies.

2022

, , ,

Probabilistic programming is a powerful high-level paradigm for probabilistic modeling and inference. We present Odds, a small domain-specific language (DSL) for probabilistic programming, embedded in Scala. Odds provides first-class support for random variables and probabilistic choice, while reusing Scala's abstraction and modularity facilities for composing probabilistic computations and for executing deterministic program parts. Odds accurately represents possibly dependent random variables using a probability monad that models committed choice. This monadic representation of probabilistic models can be combined with a range of inference procedures. We present engines for exact inference, rejection sampling and importance sampling with look-ahead, but other types of solvers are conceivable as well. We evaluate Odds on several non-trivial probabilistic programs from the literature and we demonstrate how the basic probabilistic primitives can be used to build higher-level abstractions, such as rule-based logic programming facilities, using advanced Scala features.

ACM2013