Bayesian statistics for biologists and ecologists
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This talk on Bayesian statistics is tailored especially for biologists and ecologists. The main points are to highlight what the particular benefit to use Bayesian is in comparison to frequentist statistics and to understand the essence for practical application for scientific purposes.
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- Bayesian inference uses Bayes' theorem to update probabilities of hypotheses as new evidence becomes available. It is widely used in science, engineering, medicine and other fields.
- Bayes' theorem calculates the posterior probability (probability after seeing evidence) based on the prior probability (initial probability) and the likelihood function (probability of evidence under the hypothesis).
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The replication crisis: are P-values the problem and are Bayes factors the so...
Today’s posterior is tomorrow’s prior. Dennis Lindley
It has been claimed that science is undergoing a replication crisis and that when looking for culprits, the cult of significance is the chief suspect. It has also been claimed that Bayes factors might provide a solution.
In my opinion, these claims are misleading and part of the problem is our understanding
of the purpose and nature of replication, which has only recently been subject to formal
analysis.
What we are or should be interested in is truth. Replication is a coherence not a correspondence requirement and one that has a strong dependence on the size
of the replication study
.
Consideration of Bayes factors raises a puzzling question. Should the Bayes factor for a replication study be calculated as if it were the initial study? If the answer is yes, the approach is not fully Bayesian and furthermore the Bayes factors will be subject to
exactly the same replication ‘paradox’ as P-values. If the answer is no, then in what
sense can an initially found Bayes factor be replicated and what are the implications for how we should view replication of P-values?
A further issue is that little attention has been paid to false negatives and, by extension
to true negative values. Yet, as is well known from the theory of diagnostic tests, it is
meaningless to consider the performance of a test in terms of false positives alone.
I shall argue that we are in danger of confusing evidence with the conclusions we draw and that any reforms of scientific practice should concentrate on producing evidence
that is reliable as it can be qua evidence. There are many basic scientific practices in
need of reform. Pseudoreplication, for example, and the routine destruction of
information through dichotomisation are far more serious problems than many matters of inferential framing that seem to have excited statisticians.
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This document provides an overview of Bayesian inference:
- Bayesian inference uses Bayes' theorem to update probabilities of hypotheses as new evidence becomes available. It is widely used in science, engineering, medicine and other fields.
- Bayes' theorem calculates the posterior probability (probability after seeing evidence) based on the prior probability (initial probability) and the likelihood function (probability of evidence under the hypothesis).
- Common applications of Bayesian inference include artificial intelligence, expert systems, bioinformatics, and more. Advantages include incorporating prior information to improve predictions, while disadvantages include potential issues if prior information is incorrect.
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This document discusses standard deviation and related statistical concepts. It defines standard deviation as a measure of variability around the mean and explains how to calculate it from both ungrouped and grouped data. It also defines related terms like variance, standard error of the mean, and confidence limits of the mean. Standard deviation is calculated using the formula that sums the squared deviations from the mean, divided by n-1. Standard error is the standard deviation divided by the square root of the sample size, and confidence limits refer to ranges around the mean within which we can be certain the population mean falls.
Logistic regression
Logistic regression is a statistical model used to predict binary outcomes like disease presence/absence from several explanatory variables. It is similar to linear regression but for binary rather than continuous outcomes. The document provides an example analysis using logistic regression to predict risk of HHV8 infection from sexual behaviors and infections like HIV. The analysis found HIV and HSV2 history were associated with higher odds of HHV8 after adjusting for other variables, while gonorrhea history was not a significant independent predictor.
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Biostatistics
- Biostatistics refers to applying statistical methods to biological and medical problems. It is also called biometrics, which means biological measurement or measurement of life.
- There are two main types of statistics: descriptive statistics which organizes and summarizes data, and inferential statistics which allows conclusions to be made from the sample data.
- Data can be qualitative like gender or eye color, or quantitative which has numerical values like age, height, weight. Quantitative data can further be interval/ratio or discrete/continuous.
- Common measures of central tendency include the mean, median and mode. Measures of variability include range, standard deviation, variance and coefficient of variation.
- Correlation describes the relationship between two variables
Introduction to Maximum Likelihood Estimator
This document provides an overview of maximum likelihood estimation (MLE). It discusses key concepts like probability models, parameters, and the likelihood function. MLE aims to find the parameter values that make the observed data most likely. This can be done analytically by taking derivatives or numerically using optimization algorithms. Practical considerations like removing constants and using the log-likelihood are also covered. The document concludes by introducing the likelihood ratio test for comparing nested models.
Bayesian statistics
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Similar to Bayesian statistics for biologists and ecologists
Bayesian Statistics.pdf
This document provides an introduction to Bayesian statistics and MCMC methods. It begins by explaining the key concepts of Bayesian statistics, including the use of prior distributions and Bayes' theorem to obtain posterior distributions. It then gives examples of conjugate prior distributions that result in known posterior distributions for normal and binomial likelihoods. The document concludes by introducing MCMC methods as a way to fit more complex Bayesian models with unknown posterior distributions.
The replication crisis: are P-values the problem and are Bayes factors the so...
Today’s posterior is tomorrow’s prior. Dennis Lindley
It has been claimed that science is undergoing a replication crisis and that when looking for culprits, the cult of significance is the chief suspect. It has also been claimed that Bayes factors might provide a solution.
In my opinion, these claims are misleading and part of the problem is our understanding
of the purpose and nature of replication, which has only recently been subject to formal
analysis.
What we are or should be interested in is truth. Replication is a coherence not a correspondence requirement and one that has a strong dependence on the size
of the replication study
.
Consideration of Bayes factors raises a puzzling question. Should the Bayes factor for a replication study be calculated as if it were the initial study? If the answer is yes, the approach is not fully Bayesian and furthermore the Bayes factors will be subject to
exactly the same replication ‘paradox’ as P-values. If the answer is no, then in what
sense can an initially found Bayes factor be replicated and what are the implications for how we should view replication of P-values?
A further issue is that little attention has been paid to false negatives and, by extension
to true negative values. Yet, as is well known from the theory of diagnostic tests, it is
meaningless to consider the performance of a test in terms of false positives alone.
I shall argue that we are in danger of confusing evidence with the conclusions we draw and that any reforms of scientific practice should concentrate on producing evidence
that is reliable as it can be qua evidence. There are many basic scientific practices in
need of reform. Pseudoreplication, for example, and the routine destruction of
information through dichotomisation are far more serious problems than many matters of inferential framing that seem to have excited statisticians.
The replication crisis: are P-values the problem and are Bayes factors the so...
Today’s posterior is tomorrow’s prior. Dennis Lindley1 (P2)
It has been claimed that science is undergoing a replication crisis and that when looking for culprits, the cult of significance is the chief suspect. It has also been claimed that Bayes factors might provide a solution.
In my opinion, these claims are misleading and part of the problem is our understanding of the purpose and nature of replication, which has only recently been subject to formal analysis2. What we are or should be interested in is truth. Replication is a coherence not a correspondence requirement3 and one that has a strong dependence on the size of the replication study4.
Consideration of Bayes factors raises a puzzling question. Should the Bayes factor for a replication study be calculated as if it were the initial study? If the answer is yes, the approach is not fully Bayesian and furthermore the Bayes factors will be subject to exactly the same replication ‘paradox’ as P-values. If the answer is no, then in what sense can an initially found Bayes factor be replicated and what are the implications for how we should view replication of P-values?
A further issue is that little attention has been paid to false negatives and, by extension to true negative values. Yet, as is well known from the theory of diagnostic tests, it is meaningless to consider the performance of a test in terms of false positives alone.
I shall argue that we are in danger of confusing evidence with the conclusions we draw and that any reforms of scientific practice should concentrate on producing evidence that is reliable as it can be qua evidence. There are many basic scientific practices in need of reform. Pseudoreplication5, for example, and the routine destruction of information through dichotomisation6 are far more serious problems than many matters of inferential framing that seem to have excited statisticians.
References
1. Lindley DV. Bayesian statistics: A review. SIAM; 1972.
2. Devezer B, Navarro DJ, Vandekerckhove J, Ozge Buzbas E. The case for formal methodology in scientific reform. R Soc Open Sci. Mar 31 2021;8(3):200805. doi:10.1098/rsos.200805
3. Walker RCS. Theories of Truth. In: Hale B, Wright C, Miller A, eds. A Companion to the Philosophy of Language. John Wiley & Sons,; 2017:532-553:chap 21.
4. Senn SJ. A comment on replication, p-values and evidence by S.N.Goodman, Statistics in Medicine 1992; 11:875-879. Letter. Statistics in Medicine. 2002;21(16):2437-44.
5. Hurlbert SH. Pseudoreplication and the design of ecological field experiments. Ecological monographs. 1984;54(2):187-211.
6. Senn SJ. Being Efficient About Efficacy Estimation. Research. Statistics in Biopharmaceutical Research. 2013;5(3):204-210. doi:10.1080/19466315.2012.754726
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This document provides an introduction to probabilistic and stochastic models in machine learning. It discusses key concepts like probabilistic modeling, importance of probabilistic ML models, Bayesian inference, basics of probability theory, Bayes' rule, and examples of how to apply Bayes' theorem in machine learning. The document covers conditional probability, prior and posterior probability, and how a Naive Bayes classifier uses Bayes' theorem for classification tasks.
What should we expect from reproducibiliry
Is there really a reproducibility crisis and if so are P-values to blame? Choose any statistic you like and carry out two identical independent studies and report this statistic for each. In advance of collecting any data, you ought to expect that it is just as likely that statistic 1 will be smaller than statistic 2 as vice versa. Once you have seen statistic 1, things are not so simple but if they are not so simple, it is that you have other information in some form. However, it is at least instructive that you need to be careful in jumping to conclusions about what to expect from reproducibility. Furthermore, the forecasts of good Bayesians ought to obey a Martingale property. On average you should be in the future where you are now but, of course, your inferential random walk may lead to some peregrination before it homes in on “the truth”. But you certainly can’t generally expect that a probability will get smaller as you continue. P-values, like other statistics are a position not a movement. Although often claimed, there is no such things as a trend towards significance.
Using these and other philosophical considerations I shall try and establish what it is we want from reproducibility. I shall conclude that we statisticians should probably be paying more attention to checking that standard errors are being calculated appropriately and rather less to inferential framework.
Uncertainty
Conditional independence assumptions allow simpler probabilistic models to be constructed that can still accurately model real-world phenomena. Bayesian networks provide a systematic way to construct probabilistic models using conditional independence assumptions. Representing knowledge with probabilities provides a coherent framework for representing uncertainty, which is important for building intelligent systems that can reason effectively with incomplete information.
Naive Bayes.pptx
Naive Bayes is a simple and powerful algorithm for classification tasks in machine learning. It works well even with large datasets containing millions of records. Naive Bayes gives good results for natural language processing and text analysis. The Naive Bayes classifier uses Bayes' theorem to calculate the posterior probability of a class given the predictor values. It makes the assumption that predictor variables are independent of each other given the class.
Bayes' theorem
This presentation gives a brief introduction about Bayes' Theorem and its applications in Artificial intelligence
Replication Crises and the Statistics Wars: Hidden Controversies
D. Mayo presentation at the X-Phil conference on "Reproducibility and Replicabililty in Psychology and Experimental Philosophy", University College London (June 14, 2018)
Probability and basic statistics with R
Quantitative Data Analysis -
Part III: Probability and basic statistics-
Master in Global Environmental Change -
IE University
Warmup_New.pdf
This document provides an overview of the structure and topics covered in a course on machine learning and data science. The course will first focus on supervised learning techniques like linear regression and classification. The second half will cover unsupervised learning including clustering and dimension reduction. Optional deep learning topics may be included if time permits. Key aspects of data science like the data analysis process and probabilistic concepts such as Bayes' rule and conditional independence are also reviewed.
Statistical Inference as Severe Testing: Beyond Performance and Probabilism
A talk given by Deborah G Mayo
(Dept of Philosophy, Virginia Tech) to the Seminar in Advanced Research Methods at the Dept of Psychology, Princeton University on
November 14, 2023
TITLE: Statistical Inference as Severe Testing: Beyond Probabilism and Performance
ABSTRACT: I develop a statistical philosophy in which error probabilities of methods may be used to evaluate and control the stringency or severity of tests. A claim is severely tested to the extent it has been subjected to and passes a test that probably would have found flaws, were they present. The severe-testing requirement leads to reformulating statistical significance tests to avoid familiar criticisms and abuses. While high-profile failures of replication in the social and biological sciences stem from biasing selection effects—data dredging, multiple testing, optional stopping—some reforms and proposed alternatives to statistical significance tests conflict with the error control that is required to satisfy severity. I discuss recent arguments to redefine, abandon, or replace statistical significance.
4-ML-UNIT-IV-Bayesian Learning.pptx
1. Bayesian learning provides a probabilistic approach to inference based on probability distributions of quantities of interest together with observed data.
2. The maximum a posteriori (MAP) hypothesis is the most probable hypothesis given observed training data. Consistent learning algorithms that make no errors on training data will always output a MAP hypothesis under certain assumptions.
3. Bayesian learning can be used to characterize the behavior of learning algorithms like decision tree induction even when the algorithms do not explicitly manipulate probabilities.
Review Z Test Ci 1
1. The document discusses hypothesis testing using the z-test. It outlines the steps of hypothesis testing including stating hypotheses, setting the criterion, computing test statistics, comparing to the criterion, and making a decision.
2. Examples are provided to demonstrate a non-directional and directional z-test, including stating hypotheses, computing test statistics, comparing to criteria, and interpreting results.
3. Key concepts reviewed are the central limit theorem, type I and II errors, significance levels, rejection regions, p-values, and confidence intervals in hypothesis testing.
D. G. Mayo Columbia slides for Workshop on Probability &Learning
- The document discusses ongoing debates between frequentist and Bayesian approaches to statistical inference and the importance of resolving philosophical differences given issues like the replication crisis.
- It notes statisticians are not blameless for failing to agree on a unified methodology and that philosophical correctness is important for methodology.
- The document advocates an "error statistical" or "probativist" approach focused on assessing the severity or stringency of tests, beyond just probabilism or performance, to determine what claims are warranted by evidence. This involves considering a test's ability to detect flaws in claims.
Linear Regression
The document discusses linear regression machine learning models. It describes modeling data using a linear Gaussian distribution and estimating the model parameters using maximum likelihood or Bayesian methods. Bayesian estimation avoids the overfitting problems of maximum likelihood estimation by implicitly penalizing models that overfit the data. However, Bayesian methods can still be affected by using an incorrectly specified model.
Senn repligate
Stephen Senn slides:"‘Repligate’: reproducibility in statistical studies. What does it mean and in what sense does it matter?" presented May 23 at the session on "The Philosophy of Statistics: Bayesianism, Frequentism and the Nature of Inference"," at the 2015 APS Annual Convention in NYC
Meeting #1 Slides Phil 6334/Econ 6614 SP2019
Mayo slides Meeting #1 Slides Phil 6334/Econ 6614 SP2019: Current Debates on Statistical Inference and Modeling
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Describing and Interpreting an Immersive Learning Case with the Immersion Cub...
Current descriptions of immersive learning cases are often difficult or impossible to compare. This is due to a myriad of different options on what details to include, which aspects are relevant, and on the descriptive approaches employed. Also, these aspects often combine very specific details with more general guidelines or indicate intents and rationales without clarifying their implementation. In this paper we provide a method to describe immersive learning cases that is structured to enable comparisons, yet flexible enough to allow researchers and practitioners to decide which aspects to include. This method leverages a taxonomy that classifies educational aspects at three levels (uses, practices, and strategies) and then utilizes two frameworks, the Immersive Learning Brain and the Immersion Cube, to enable a structured description and interpretation of immersive learning cases. The method is then demonstrated on a published immersive learning case on training for wind turbine maintenance using virtual reality. Applying the method results in a structured artifact, the Immersive Learning Case Sheet, that tags the case with its proximal uses, practices, and strategies, and refines the free text case description to ensure that matching details are included. This contribution is thus a case description method in support of future comparative research of immersive learning cases. We then discuss how the resulting description and interpretation can be leveraged to change immersion learning cases, by enriching them (considering low-effort changes or additions) or innovating (exploring more challenging avenues of transformation). The method holds significant promise to support better-grounded research in immersive learning.
ESR spectroscopy in liquid food and beverages.pptx
With increasing population, people need to rely on packaged food stuffs. Packaging of food materials requires the preservation of food. There are various methods for the treatment of food to preserve them and irradiation treatment of food is one of them. It is the most common and the most harmless method for the food preservation as it does not alter the necessary micronutrients of food materials. Although irradiated food doesn’t cause any harm to the human health but still the quality assessment of food is required to provide consumers with necessary information about the food. ESR spectroscopy is the most sophisticated way to investigate the quality of the food and the free radicals induced during the processing of the food. ESR spin trapping technique is useful for the detection of highly unstable radicals in the food. The antioxidant capability of liquid food and beverages in mainly performed by spin trapping technique.
Applied Science: Thermodynamics, Laws & Methodology.pdf
When I was asked to give a companion lecture in support of ‘The Philosophy of Science’ (https://shorturl.at/4pUXz) I decided not to walk through the detail of the many methodologies in order of use. Instead, I chose to employ a long standing, and ongoing, scientific development as an exemplar. And so, I chose the ever evolving story of Thermodynamics as a scientific investigation at its best.
Conducted over a period of >200 years, Thermodynamics R&D, and application, benefitted from the highest levels of professionalism, collaboration, and technical thoroughness. New layers of application, methodology, and practice were made possible by the progressive advance of technology. In turn, this has seen measurement and modelling accuracy continually improved at a micro and macro level.
Perhaps most importantly, Thermodynamics rapidly became a primary tool in the advance of applied science/engineering/technology, spanning micro-tech, to aerospace and cosmology. I can think of no better a story to illustrate the breadth of scientific methodologies and applications at their best.
Phenomics assisted breeding in crop improvement
As the population is increasing and will reach about 9 billion upto 2050. Also due to climate change, it is difficult to meet the food requirement of such a large population. Facing the challenges presented by resource shortages, climate
change, and increasing global population, crop yield and quality need to be improved in a sustainable way over the coming decades. Genetic improvement by breeding is the best way to increase crop productivity. With the rapid progression of functional
genomics, an increasing number of crop genomes have been sequenced and dozens of genes influencing key agronomic traits have been identified. However, current genome sequence information has not been adequately exploited for understanding
the complex characteristics of multiple gene, owing to a lack of crop phenotypic data. Efficient, automatic, and accurate technologies and platforms that can capture phenotypic data that can
be linked to genomics information for crop improvement at all growth stages have become as important as genotyping. Thus,
high-throughput phenotyping has become the major bottleneck restricting crop breeding. Plant phenomics has been defined as the high-throughput, accurate acquisition and analysis of multi-dimensional phenotypes
during crop growing stages at the organism level, including the cell, tissue, organ, individual plant, plot, and field levels. With the rapid development of novel sensors, imaging technology,
and analysis methods, numerous infrastructure platforms have been developed for phenotyping.
20240520 Planning a Circuit Simulator in JavaScript.pptx
Evaporation step counter work. I have done a physical experiment.
(Work in progress.)
Equivariant neural networks and representation theory
Or: Beyond linear.
Abstract: Equivariant neural networks are neural networks that incorporate symmetries. The nonlinear activation functions in these networks result in interesting nonlinear equivariant maps between simple representations, and motivate the key player of this talk: piecewise linear representation theory.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
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The binding of cosmological structures by massless topological defects
Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
Unlocking the mysteries of reproduction: Exploring fecundity and gonadosomati...
The pygmy halfbeak Dermogenys colletei, is known for its viviparous nature, this presents an intriguing case of relatively low fecundity, raising questions about potential compensatory reproductive strategies employed by this species. Our study delves into the examination of fecundity and the Gonadosomatic Index (GSI) in the Pygmy Halfbeak, D. colletei (Meisner, 2001), an intriguing viviparous fish indigenous to Sarawak, Borneo. We hypothesize that the Pygmy halfbeak, D. colletei, may exhibit unique reproductive adaptations to offset its low fecundity, thus enhancing its survival and fitness. To address this, we conducted a comprehensive study utilizing 28 mature female specimens of D. colletei, carefully measuring fecundity and GSI to shed light on the reproductive adaptations of this species. Our findings reveal that D. colletei indeed exhibits low fecundity, with a mean of 16.76 ± 2.01, and a mean GSI of 12.83 ± 1.27, providing crucial insights into the reproductive mechanisms at play in this species. These results underscore the existence of unique reproductive strategies in D. colletei, enabling its adaptation and persistence in Borneo's diverse aquatic ecosystems, and call for further ecological research to elucidate these mechanisms. This study lends to a better understanding of viviparous fish in Borneo and contributes to the broader field of aquatic ecology, enhancing our knowledge of species adaptations to unique ecological challenges.
Sharlene Leurig - Enabling Onsite Water Use with Net Zero Water
Sharlene Leurig - Enabling Onsite Water Use with Net Zero WaterTexas Alliance of Groundwater Districts
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本公司拥有海外各大学样板无数，能完美还原。
1:1完美还原海外各大学毕业材料上的工艺：水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700
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2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
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8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
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Presented at June 6-7 Texas Alliance of Groundwater Districts Business Meeting
8.Isolation of pure cultures and preservation of cultures.pdf
Isolation of pure culture, its various method.
The debris of the ‘last major merger’ is dynamically young
The Milky Way’s (MW) inner stellar halo contains an [Fe/H]-rich component with highly eccentric orbits, often referred to as the
‘last major merger.’ Hypotheses for the origin of this component include Gaia-Sausage/Enceladus (GSE), where the progenitor
collided with the MW proto-disc 8–11 Gyr ago, and the Virgo Radial Merger (VRM), where the progenitor collided with the
MW disc within the last 3 Gyr. These two scenarios make different predictions about observable structure in local phase space,
because the morphology of debris depends on how long it has had to phase mix. The recently identified phase-space folds in Gaia
DR3 have positive caustic velocities, making them fundamentally different than the phase-mixed chevrons found in simulations
at late times. Roughly 20 per cent of the stars in the prograde local stellar halo are associated with the observed caustics. Based
on a simple phase-mixing model, the observed number of caustics are consistent with a merger that occurred 1–2 Gyr ago.
We also compare the observed phase-space distribution to FIRE-2 Latte simulations of GSE-like mergers, using a quantitative
measurement of phase mixing (2D causticality). The observed local phase-space distribution best matches the simulated data
1–2 Gyr after collision, and certainly not later than 3 Gyr. This is further evidence that the progenitor of the ‘last major merger’
did not collide with the MW proto-disc at early times, as is thought for the GSE, but instead collided with the MW disc within
the last few Gyr, consistent with the body of work surrounding the VRM.
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and the revised UWWTD (Urban Waste Water Treatment Directive)”
Micronuclei test.M.sc.zoology.fisheries.
Current Ms word generated power point presentation covers major details about the micronuclei test. It's significance and assays to conduct it. It is used to detect the micronuclei formation inside the cells of nearly every multicellular organism. It's formation takes place during chromosomal sepration at metaphase.
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Describing and Interpreting an Immersive Learning Case with the Immersion Cub...
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aziz sancar nobel prize winner: from mardin to nobel
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ESR spectroscopy in liquid food and beverages.pptx
ESR spectroscopy in liquid food and beverages.pptx
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Applied Science: Thermodynamics, Laws & Methodology.pdf
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Equivariant neural networks and representation theory
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The binding of cosmological structures by massless topological defects
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Unlocking the mysteries of reproduction: Exploring fecundity and gonadosomati...
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Sharlene Leurig - Enabling Onsite Water Use with Net Zero Water
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8.Isolation of pure cultures and preservation of cultures.pdf
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The debris of the ‘last major merger’ is dynamically young
Bayesian statistics for biologists and ecologists
- 1. Bayesian for us Masahiro Ryo masahiroryo@gmail.com Matthias Rillig Lab. Freie Universität Berlin Berlin-Brandenburg Institute of Advanced Biodiversity Research 2016.07.27
- 2. 1. Bayes' theorem 2. What is same? 3. What is “practically” different? 4. p-value and our intuition 5. Assumptions 6. By the way… 7. prior distribution 8. Summary 2 Outline Practical part Technical part
- 3. 3 1. Bayes' theorem P(A|B) = P(B|A) P(A) P(B) P(A): the probability of observing event A P(A|B): a conditional probability, the probability of observing event A given that B is true
- 4. 4 1. Bayes' theorem P(A|B) = P(B|A) P(A) P(B) P(A): the probability of observing event A P(A|B): a conditional probability, the probability of observing event A given that B is true
- 5. 5 2. What is same? 1. To assume a hypothesis • M under the treatment is larger than M under control (comparison) • Y is increased by increasing X (correlation) 2. To collect samples 3. To test the hypothesis population samples Statistical test
- 6. 6 3. What is “practically” different? - test Frequentists Bayesians “if p < 0.05 or not” “how probable” 1. Set a null hypothesis H0 2. Estimate probability of type I error (p-value) 3. Evaluate how rarely H0 occurs 4. Reject H0 and support H1 1. Set a hypothesis H1 2. Evaluate how possibly H1 occurs Bayesian is more direct and intuitive to test a hypothesis e.g., my hypothesis was supported with 80% probability.
- 7. 7 3. What is “practically” different? Frequentists Bayesians “how probable”“if p < 0.05 or not” y = β0 + β1 x Parameter β1 Probability 0.89 H0: β1 = 0 is rejected H1: β1 ≠ 0 is supported (and β1= 0.89) The parameter β1 is rarely <0, so H1 is greatly supported. Most probably β1=0.89 and can span between 0.28 and 1.52 with 95% probability.
- 8. 8 4. p-value and our intuition Importantly, p-value does not indicate how reliable your hypothesis is but how safely reject the corresponding null hypothesis. This does not fit our intuition. In most cases, we want to know “how reliable my hypothesis is?”
- 9. Frequentists 9 5. Assumptions Bayesians population Your samples Assuming repetitive sampling from the population to find the single true value for a parameter governing the population population Your samples Assuming the probability distribution of a parameter value, and its reliability is increased by increasing the sample size.
- 10. 0.3 1.3 β=0.8 β What you estimated from your samples Frequentists 10 5. Assumptions Bayesians 95% Confidence interval (CI) assumes that if you conduct same sampling design 20 times and estimate a 95%CI for each, only once a 95%CI would not cross the true value β. β 95% Credible interval (CI) assumes that the parameter β governing the population would be within the range of the 95% CI with 95% probability. β 0.3 1.3 Any value you estimated from your samples can be the parameter (does not matter if it is true or not)
- 11. 11 5. Assumptions Using same data, fitting a same probability distribution, estimating the parameters based on likelihood…. Then, while the concepts are different, do we eventually get identical results from frequentism and Bayesianism?
- 12. 12 5. Assumptions Answer: Yes, if we consider only in our scientific field, they are “most likely“ same. Recommendation: If your interest is the effect size, use Bayesian If it is to support hypothesis, use Frequentist
- 13. 13 6. By the way… The answer becomes NO when “prior distribution“ is explicitly assumed P(β|D) = P(D|β) P(β) P(D) Likelihood posterior distribution fixed value (practically neglectable) prior distribution
- 14. 14 6. By the way… Note that this is much less important for us... P(β|D) ∝ P(D|β) P(β) posterior distribution prior distributionLikelihood
- 15. 15 What I have explained was the following case: P(β|D) ∝ P(D|β) P(β) posterior distribution prior distributionLikelihood β 0.3 1.3 This was the outcome (estimated parameter value & range) β 0.3 1.3 This was estimated β −∞ +∞ This was like nothing 6. By the way…
- 16. 16 7. prior distribution P(β|D) ∝ P(D|β) P(β) prior distribution β −∞ +∞ What‘s this?
- 17. 17 7. prior distribution Prior distribution • This is the other key point of Bayesian statistics • But interestingly we don‘t care this aspect for our analysis (and if you google Bayesian, most articles explain this firstly: Suck!) • We can include a subjective assumption of the probability distribution of parameters (i.e., prior distribution) before analysis (but don‘t have to and indeed we don‘t)
- 18. 18 7. prior distribution An example • Question: to quantify mean and variability of our height — Probability districution of height: Gaussian — Shall we assume the parameters beforehand? μ= XX? ±σ=XX? This is called “prior distribution“ μ 170 180 175 ±σ 3 8 5 — We measure individuals and calculate the likelihoods — Multiply the posterior distribution by the likelihoods to obtain the posterior distribution
- 19. 19 7. prior distribution An example • In case of average μ P(β|D) ∝ P(D|β) P(β) μ 170 180 175 prior dist. (your assumption) μ 160 178 171 Likelihood (measurement) Posterior dist. (output: updated distribution after measurement) μ 174 166 179
- 20. 20 7. prior distribution 170 180 175 P(β|D) ∝ P(D|β) P(β) 160 178 171174 166 179 In our study μ 160 178 171 μ 171 160 178 β −∞ +∞ To avoid subjectivity
- 21. 21 8. Summary Recommendation: If your interest is the effect size, use Bayesian If it is to support hypothesis, use Frequentist β 0.3 1.3 Any value you estimated from your samples can be the parameter, following a probability distribution P(β|D) ∝ P(D|β) P(β) Prior information can be important in other field but this is not important for us: so, let‘s ignore.