Twoschoolsinstatistics: frequentists and Bayesians . Whatisaprobability? MarkovchainMonteCarlo Comparisons FrequentistsandBayesians Outline ClausEkstrøm FrequentistandBayesianstatistics

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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.

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Frequentist school

School of Jerzy Neyman, Egon Pearson and Ronald Fischer.

Bayesian school

“School” of Thomas Bayes

P(H|D) =_! P(D|H) · P(H) P(D|H) · P(H)dH

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.

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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.

The meta-experiment idea

Frequentists think of meta-experiments and consider the current dataset as a single realization from all possible datasets.

The meta-experiment idea

167.2 cm

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The meta-experiment idea

167.2 cm 175.5 cm

The meta-experiment idea

167.2 cm 175.5 cm 187.7 cm

The meta-experiment idea

167.2 cm 175.5 cm

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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 aconfidence 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 diﬀerent random sample like this one.”

Maximum likelihood

How will the frequentist estimate the parameter?

Maximum likelihood

Answer: maximum likelihood.

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Maximum likelihood

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:

P(Y = y ) =

"

n y

#

p^y(1− p)ⁿ^−y

Bayesians

Each investigator is entitled to his/hers personal belief ... the prior information. No fixed values for parameters but a distribution.

All distributions are subjective.

Yours is as good as mine.

Can still talk about the mean

— but it is the mean of my distribution.

In many cases trying to circumvent by using vague priors.

Thumb tack pin pointing down:

0.0 0.2 0.4 0.6 0.8 1.0

0.00.51.01.52.02.53.0

Theta

Prior distribution

Credibility intervals

Bayesians have an altogether diﬀerent 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.

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Credibility intervals

Bayesians have an altogether diﬀerent 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 acredibility 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.”

Comparison

Advantages Disadvantages

Frequentist Objective Confidence intervals (not quite the desired)

Calculations

Bayesian Credibility intervals (usually the desired)

Subjective Complex models Calculations

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.

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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

Jury duty

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