Faculty of Life Sciences Frequentist and Bayesian statistics Claus Ekstrøm E-mail: ekstrom@life.ku.dk Outline 1 Frequentists and Bayesians • What is a probability? • Interpretation of results / inference 2 Comparisons 3 Markov chain Monte Carlo Slide 2— PhD (Aug 23rd 2011) — Frequentist and Bayesian statistics What is a probability? Two schools in statistics: frequentists and Bayesians . Slide 3— PhD (Aug 23rd 2011) — Frequentist and Bayesian statistics
Frequentist school School of Jerzy Neyman, Egon Pearson and Ronald Fischer. Slide 4— PhD (Aug 23rd 2011) — Frequentist and Bayesian statistics Bayesian school “School” of Thomas Bayes P ( D | H ) · P ( H ) P ( H | D ) = � P ( D | H ) · P ( H ) dH Slide 5— PhD (Aug 23rd 2011) — Frequentist and Bayesian statistics Frequentists Frequentists talk about probabilities in relation to experiments with a random component . Relative frequency of an event, A , is defined as P ( A ) = number of outcomes consistent with A number of experiments The probability of event A is the limiting relative frequency. 1.0 0.8 Relative frequency 0.6 0.4 0.2 0.0 0 20 40 60 80 100 n Slide 6— PhD (Aug 23rd 2011) — Frequentist and Bayesian statistics
Frequentists — 2 The definition restricts the things we can add probabilities to: What is the probability of there being life on Mars 100 billion years ago? We assume that there is an unknown but fixed underlying parameter, θ , for a population (i.e., the mean height on Danish men). Random variation (environmental factors, measurement errors, ...) means that each observation does not result in the true value. Slide 7— PhD (Aug 23rd 2011) — Frequentist and Bayesian statistics The meta-experiment idea Frequentists think of meta-experiments and consider the current dataset as a single realization from all possible datasets. Slide 8— PhD (Aug 23rd 2011) — Frequentist and Bayesian statistics The meta-experiment idea Frequentists think of meta-experiments and consider the current dataset as a single realization from all possible datasets. 167.2 cm Slide 8— PhD (Aug 23rd 2011) — Frequentist and Bayesian statistics
The meta-experiment idea Frequentists think of meta-experiments and consider the current dataset as a single realization from all possible datasets. 167.2 cm 175.5 cm Slide 8— PhD (Aug 23rd 2011) — Frequentist and Bayesian statistics The meta-experiment idea Frequentists think of meta-experiments and consider the current dataset as a single realization from all possible datasets. 167.2 cm 175.5 cm 187.7 cm Slide 8— PhD (Aug 23rd 2011) — Frequentist and Bayesian statistics The meta-experiment idea Frequentists think of meta-experiments and consider the current dataset as a single realization from all possible datasets. 167.2 cm 175.5 cm 187.7 cm 182.0 cm Slide 8— PhD (Aug 23rd 2011) — Frequentist and Bayesian statistics
Confidence intervals Thus a frequentist believes that a population mean is real, but unknown, and unknowable, and can only be estimated from the data. Knowing the distribution for the sample mean, he constructs a confidence interval, centered at the sample mean. • Either the true mean is in the interval or it is not. Can’t say there’s a 95% probability (long-run fraction having this characteristic) that the true mean is in this interval, because it’s either already in, or it’s not. • Reason: true mean is fixed value, which doesn’t have a distribution. • The sample mean does have a distribution! Thus must use statements like “95% of similar intervals would contain the true mean, if each interval were constructed from a di ff erent random sample like this one.” Slide 9— PhD (Aug 23rd 2011) — Frequentist and Bayesian statistics Maximum likelihood How will the frequentist estimate the parameter? Slide 10— PhD (Aug 23rd 2011) — Frequentist and Bayesian statistics Maximum likelihood How will the frequentist estimate the parameter? Answer: maximum likelihood. Slide 10— PhD (Aug 23rd 2011) — Frequentist and Bayesian statistics
Maximum likelihood How will the frequentist estimate the parameter? Answer: maximum likelihood. Basic idea Our best estimate of the parameter(s) are the one(s) that make our observed data most likely. We know what we have observed so far (our data). Our best “guess” would therefore be to select parameters that make our observations most likely. Binomial distribution: � n � P ( Y = y ) = p y (1 − p ) n − y y Slide 10— PhD (Aug 23rd 2011) — Frequentist and Bayesian statistics Bayesians Each investigator is entitled to his/hers personal belief ... the prior information. No fixed values for parameters but a distribution . Thumb tack pin pointing down: All distributions are subjective. Yours is as good as mine. 3.0 Can still talk about the mean 2.5 — but it is the mean of my distribution. 2.0 Prior distribution 1.5 In many cases trying to circumvent by using vague 1.0 priors. 0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Theta Slide 11— PhD (Aug 23rd 2011) — Frequentist and Bayesian statistics Credibility intervals Bayesians have an altogether di ff erent world-view. They say that only the data are real. The population mean is an abstraction, and as such some values are more believable than others based on the data and their prior beliefs. Slide 12— PhD (Aug 23rd 2011) — Frequentist and Bayesian statistics
Credibility intervals Bayesians have an altogether di ff erent world-view. They say that only the data are real. The population mean is an abstraction, and as such some values are more believable than others based on the data and their prior beliefs. The Bayesian constructs a credibility interval, centered near the sample mean, but tempered by “prior” beliefs concerning the mean. Now the Bayesian can say what the frequentist cannot: “There is a 95% probability (degree of believability) that this interval contains the mean.” Slide 12— PhD (Aug 23rd 2011) — Frequentist and Bayesian statistics Comparison Advantages Disadvantages Frequentist Objective Confidence intervals (not quite the desi- red) Calculations Bayesian Credibility intervals Subjective (usually the desired) Complex models Calculations Slide 13— PhD (Aug 23rd 2011) — Frequentist and Bayesian statistics In summary • A frequentist is a person whose long-run ambition is to be wrong 5% of the time. • A Bayesian is one who, vaguely expecting a horse, and catching a glimpse of a donkey, strongly believes he has seen a mule. Slide 14— PhD (Aug 23rd 2011) — Frequentist and Bayesian statistics
In summary • A frequentist is a person whose long-run ambition is to be wrong 5% of the time. • A Bayesian is one who, vaguely expecting a horse, and catching a glimpse of a donkey, strongly believes he has seen a mule. A frequentist uses impeccable logic to answer the wrong question, while a Bayesean answers the right question by making assumptions that nobody can fully believe in. P. G. Hamer Slide 14— PhD (Aug 23rd 2011) — Frequentist and Bayesian statistics Jury duty Slide 15— PhD (Aug 23rd 2011) — Frequentist and Bayesian statistics Example: speed of light What is the speed of light in vacuum “really”? Results (m/s) 299792459.2 299792460.0 299792456.3 299792458.1 299792459.5 Slide 16— PhD (Aug 23rd 2011) — Frequentist and Bayesian statistics
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