Models Of Distribution

[Abbie Pilz]

Iassume by specifying a different distribution family I could increase the fit of my model. However, I am new to bayesian modelling and R, and I have no idea which type of distribution family would be suitable to represent my clearly bimodal outcome variable. Any suggestions would be greatly appreciated.

Does your dependent variable ALLAccuracies have a minimum possible value of 0? Does it have a maximum possible value (e.g. of 10) or can it be arbitrarily high? I ask because the range of your response data can help determine the best family to use.

Thanks! Was going to try that, but not actually none of my brm models will run anymore, and I have no idea why. Any input in this topic: Brms model doesn't run after custom family (hurdle_gamma()) specified

would be hugely appreciated

We propose a new prior distribution for classical (nonhierarchical) logistic regression models, constructed by first scaling all nonbinary variables to have mean 0 and standard deviation 0.5, and then placing independent Student-t prior distributions on the coefficients. As a default choice, we recommend the Cauchy distribution with center 0 and scale 2.5, which in the simplest setting is a longer-tailed version of the distribution attained by assuming one-half additional success and one-half additional failure in a logistic regression. Cross-validation on a corpus of datasets shows the Cauchy class of prior distributions to outperform existing implementations of Gaussian and Laplace priors.

We recommend this prior distribution as a default choice for routine applied use. It has the advantage of always giving answers, even when there is complete separation in logistic regression (a common problem, even when the sample size is large and the number of predictors is small), and also automatically applying more shrinkage to higher-order interactions. This can be useful in routine data analysis as well as in automated procedures such as chained equations for missing-data imputation.

We implement a procedure to fit generalized linear models in R with the Student-t prior distribution by incorporating an approximate EM algorithm into the usual iteratively weighted least squares. We illustrate with several applications, including a series of logistic regressions predicting voting preferences, a small bioassay experiment, and an imputation model for a public health data set.

Andrew Gelman. Aleks Jakulin. Maria Grazia Pittau. Yu-Sung Su. "A weakly informative default prior distribution for logistic and other regression models." Ann. Appl. Stat. 2 (4) 1360 - 1383, December 2008. -AOAS191

Keywords: Bayesian inference , generalized linear model , hierarchical model , least squares , Linear regression , logistic regression , multilevel model , noninformative prior distribution , weakly informative prior distribution

The bottom line success of your business hinges on its ability to execute the sales that will generate revenue. Developing a reliable means of reaching clients is a challenge for many sales and marketing managers. The way that you choose to communicate with clients and sell your products or services is reflected in your choice of sales distribution model. Unfortunately, picking the wrong strategy could impact your bottom line, requiring that you re-assess those missed opportunities for the sake of your long-term survival.

The word "distribution" might conjure up images of trucks loaded to the gills with freight, but distribution is also a sales and marketing issue. Your company's sales distribution model is the method by which it sells products and services to its target clients. In a perfect world, products and services would sell themselves. Unfortunately, this isn't reality, and you're going to have to put in some work to:

Before you can sell someone your products and services, an extensive amount of research needs to be done to understand their particular needs and how they prefer to do business. For example, some clients want more personalized service while others just want to place their orders and not be bothered. Some want to learn about products and place orders online, and others need customized materials and more prompting to place orders.

The two different types of distribution channels are Business-to-Business (B2B) and Business-to-Consumer (B2C) distribution. For the sake of this discussion, most sales and marketing departments struggle with finding the best strategies to connect with other business clients, or B2B marketing. The three main categories of sales distribution models are:

The type of distribution channel you choose will depend on several factors such as the type of products and services you have, your industry, and business model. In the past, distribution models were standardized by industry which left little to no wiggle room for entrepreneurs to innovate. Fortunately, this is no longer the case. For example, clothing manufacturers were limited to sales through department stores. Now, those same companies can choose the traditional model, sell to other retail outlets, and go directly to the consumer with online sales.

As a seller, you aren't limited to just one sales distribution strategy. In fact, choosing this path would be a mistake in most cases. Having different distribution strategies for your various clients is the best way to optimize the return on your efforts and investment. Even after you've chosen and implemented the different distribution models for your clients, it remains vital that you continue to assess your results so that you can better take advantage of missed opportunities in the future.

Every B2B company must make difficult decisions about where to best invest time and resources to achieve business goals. This is a tough balancing act that doesn't always find the scales tipping in the right direction. What happens, for example, when your strategy fails for any number of factors that might include poor timing, bad positioning, incorrect pricing, or ineffective promotion? There have been countless high-profile product and service failures (New Coke, The U.S. Football League, the $2 bill) and you'd probably like to keep yours off that list.

The good news is that you can quickly re-assess your strategy and make some much-needed changes provided you are willing to do the footwork and keep an open mind. Even if you are convinced that you did everything right and formed a solid ring of trust with a client, there is still a chance that they will say "no" or, worse, "ghost" you and simply go silent. For the sake of your business, it's vital that you take the next step and attempt to find out where you went astray.

If you can get into contact with one of the decision-makers at the company of your lost opportunity, ask them if they can spare a few minutes to help you improve your business. Either through a personal interview or an email exchange, whichever works best for them, ask them the following questions:

The answers to these questions could help you adjust some of the activities within your sales distribution strategy going forward. There is also the possibility that you have been using the wrong strategy altogether for a certain class of clients. The more open-minded you become to accepting feedback and re-assessing your failures, the more successful you will be at taking advantage of strong sales opportunities in the future.

One of the responsibilities of a chief revenue officer is to develop a successful sales distribution strategy. We have an eBook to help you define the Role of a Chief Revenue Officer in greater depth.

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In addition to this challenge of out-of-distribution generalization, underrepresentation of specific groups, conditions or hospitals also causes notable challenges of fairness and equity even when systems are deployed in datasets mirroring their training environment, with lower performance typically seen in rarer groups, conditions, individuals or their intersections. Previous work showed that a developed model may perform unexpectedly poorly on underrepresented populations or population subgroups in radiology12,13, histopathology14 and dermatology15. However, the issues of robustness to distribution shifts and statistical fairness have rarely been tackled together. Building a method that is robust across populations and subgroups, such that model performance does not degrade and benefits can be transferred when applied across groups, is a nontrivial task. This is because of data scarcity16, challenges in the acquisition strategies of evaluation datasets (for example, different imaging or screening protocols10,17,18) and the limitations of evaluation metrics10.

a, Samples generated by our conditional diffusion model for the different imaging modalities. b, Method overview. In the proposed approach, we first trained a diffusion model on both labeled and unlabeled data (if available). In a general setting, unlabeled data may consist of in-distribution or OOD data (for example, from an unseen hospital) for which expert labels are not available. Subsequently, we sampled synthetic images from the diffusion model according to particular specifications (for example, an image of a female individual with pulmonary edema). Finally, we trained a downstream diagnostic model on a combination of the real labeled images and the synthetic images sampled from the diffusion model. The dotted outlines represent synthetic data, while the dashed outlines represent unlabeled data.

To measure the performance of the different baselines and the proposed method, we used two sets of metrics: one set was more focused on diagnostic accuracy (that is, top-1 accuracy for histopathology, receiver operating characteristic (ROC)-area under the curve (AUC) for radiology and high-risk sensitivity for dermatology), while the second set was more geared toward fairness (see summary in Table 1). The performance metrics varied depending on the classification task performed for each modality (that is, binary versus multiclass versus multilabel) and considered label imbalance. High-risk sensitivity captured the true positive rate for the high-risk conditions and was deemed the most relevant for the diagnostic tool by expert dermatologists. For fairness, we looked at the performance gap (depending on the metric of interest) in the binary attribute setting and the difference between the worst and best subgroup performance for categorical attributes, for example, hospital ID and ethnicity. For continuous sensitive attributes, like age, we discretized them into appropriate buckets (Methods and Extended Data Table 1).

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