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Isn't most or all of this avoided by explicitly using Bayes' theorem along with a correct formalization of the domain? E.g. for the mammogram: P(cancer) = 0.8
by devit 8y ago
Isn't most or all of this avoided by explicitly using Bayes' theorem along with a correct formalization of the domain?
E.g. for the mammogram:
P(cancer) = 0.8%
P(~cancer) = 1 - P(cancer) = 99.2%
P(positive_mammogram | cancer) = 90%
P(positive_mammogram | ~cancer) = 7%
P(cancer | positive_mammogram) = P(positive_mammogram | cancer) P(cancer) / (P(positive_mammogram | cancer) P(cancer) + P(positive_mammogram | ~cancer) (1 - P(cancer))) = 90% * 0.8% / (90% * 0.8% + 7% * 99.2%) = 9.39457%
- HillaryBriss 8y agoi think the answer is yes, it is avoided in the way that you demonstrate. i guess it's a matter of using an approach that reaches your audience. the book explains it with prose: Imagine 1,000 randomly selected women who choose to get mammograms. Eight of them (0.8%) have breast cancer. The mammogram correctly detects 90% of breast cancer cases, so about seven of the eight women will have their cancer discovered. However, there are 992 women without breast cancer, and 7% will get a false positive reading on their mammograms, giving us 70 women incorrectly told they have cancer. In total, we have 77 women with positive mammograms, 7 of whom actually have breast cancer. Only 9% of women with positive mammograms have breast cancer.
- vanderZwan 8y agoI know this is an example, but this assumes false positives and false negatives are equally likely, and I now am wondering if that is true in real life.
- janekm 8y agoIt doesn't: P(positive_mammogram | cancer) = 90% P(positive_mammogram | ~cancer) = 7%
- thaumasiotes 8y agoTo be a little clearer, the false negative rate is 10% and the false positive rate is 7%.
- montecarl 8y agoDirect application of Bayes theorem or more generally Bayesian inference (made easy now through probabilistic programming languages like stan or pymc3) can solve all of the issues that I read on this site. In particular, the issue raised here[1] is about comparing estimates when the sample sizes are very different. Hierarchical partial pooling, can be used to accurately compare groups when the samples sizes vary widely, even when some groups have zero observations. Here is an example talking about baseball batting averages[2]. [1] https://www.statisticsdonewrong.com/regression.html#little-extremes https://www.statisticsdonewrong.com/regression.html#little-e... [2] https://docs.pymc.io/notebooks/hierarchical_partial_pooling.html https://docs.pymc.io/notebooks/hierarchical_partial_pooling....
- AlexCoventry 8y ago> along with a correct formalization of the domain? That's the biggest issue. It's very unlikely that your model family actually contains the data-generating distribution.
- stared 8y agoYes, most of the problems (with conditional probability, statistical tests, significance, etc) disappear once you express it in a Bayesian way (it's not only Bayes' formula - it explicitly creating a Bayesian model). Basically, most of them boil down to: - mistakes that can be tackled if you write it down explicitly - hidden assumptions that can be discovered (and made explicit or modified) While there is some philosophical difference between frequentist and Bayesian probability (and for some reason, I know people moving only one way). "Frequentist probability is Bayesian probability, where priors are flat, hidden, and considered taboo". BTW: Frequentists vs. Bayesians https://xkcd.com/1132/ https://xkcd.com/1132/ (there is never too much of xkcd!)
- posix_me_less 8y agoThat is fascinating. Which one is correct? Is Bayesian using more assumptions than Frequentist, namely the fact that repeated queries that haven't been done yet will show that the machine's answer is NO most of the time?
- deleted 8y ago[deleted]
- dragonwriter 8y ago> Which one is correct? The Bayesian is correct to offer the bet. Who is correct about the sun exploding actually is irrelevant to that; only the conditional probability of the bet being collectable if the sun has exploded vs. that of it has not exploded is needed here. You would care about the probability that the sun actually had exploded if one of those weren't zero, but it is, so it doesn't matter.
- diffeomorphism 8y agoCould you give a brief explanation what or who "frequentists" and "Bayesians" are? Hearing it from the side of probability (measure spaces etc.) to me it sounds very much like "tomato" vs "tomato"?
- 8y ago
- callumt 8y ago> P(cancer | positive_mammogram) = P(positive_mammogram | cancer) Sorry if I've got this wrong, but does this line say that we're assuming the probability of someone having cancer if they've got a positive mammogram, is the same as the probability of them having a positive mammogram if they've got cancer? As far as I know, the first one would be the Positive Predictive Value (PPV) of the mammogram, the second one would be the Sensitivity of the mammogram. They're related but not usually the same. The PPV would change depending on the prevalence of the disease (go up as the prevalence goes up), but the sensitivity would remain the same.
- dfan 8y agoYou left out most of the equation. The whole equation as posted is just an application of Bayes' rule: P(A|B) = P(B|A)P(A)/P(B). Your excerpt leaves out everything after P(B|A).
- rpier001 8y agoIn short, not entirely. Bayes' theorem helps when you have a correct model. People don't have access to truth or correct models. Generally people applying Bayesian methods are using cookie cutter formulas. These can't protect you from the many many ways one can muck up their data especially if you want your decisions and density intervals to be close to "right".
- vanderZwan 8y ago> when you have a correct model I have a hunch that determining a correct model is like a science all on its own. Are there any good books/blog posts/etc on that?