analysis with r. introduction to bayesian data - … · · 2017-08-14bayesian data analysis in r?...
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Introduction to Bayesian Data Analysis with R.
?
Rasmus Bååth, Lund University@rabaath || [email protected]
http://www.sumsar.net
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Source: I borrowed these three examples from a presentation. But which presentation I can’t remember or find. I you know what presentation it could have been, please let me know and I will credit it here.
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What do these have in common?
● Complex problems● Large inherent uncertainty that needs to be
quantified.● Requires efficient integration of many
sources of information.● They all use Bayesian data analysis.
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Bayesian data analysis is a great tool!
… and R is a great tool for doing Bayesian data analysis.
But if you google “Bayesian” you get philosophy:
Subjective vs ObjectiveFrequentism vs Bayesianism
p-values vs subjective probabilities
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Bayesian data analysisWhat? Why? How?
?
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Overview of this tutorial
● What is Bayesian data analysis?
Prediction contest
● Why use Bayesian data analysis?
Exercises
● How to interpret and perform a Bayesian
data analysis in R?
More Exercises
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Why am I here?
● I use Bayesian methods in my research at Lund University where I also run a network for people interested in Bayes.
● I’m working on an R-package to make simple Bayesian analyses simple to run.
● I blog about Bayesian data analysis.
www.sumsar.net
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● It is when you use probability to represent uncertainty in all parts of a statistical model.
● A flexible extension of maximum likelihood.● Potentially the most information-efficient
method to fit a statistical model. (But potentially also the most computationally intensive method…)
What is Bayesian data analysis?
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Data
Generative model
Parameters
µ θ σρ
5, 2, 7, 8, 3, 9, 1, 2, ...
If we know the parameters
Bayesian models as generative models
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When we know the data.Data
Generative model
Parameters
µ θ σρ
5, 2, 7, 8, 3, 9, 1, 2, ...
Bayesian models as generative models
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How many fish are in the lake?
● An actual problem in Abundance estimation. Use in, for example, wildlife management.
● Also other uses, for example, to estimate how many DKK 1,000 bills are in circulation.
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How many fish are in the lake?
● The problem: We can’t catch them all.● But we can catch some of them...
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Mark and Re-capture
1. Catch a couple of fish.2. Mark them and throw them back.
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Mark and Re-capture
1. Catch a couple of fish.2. Mark them and throw them back.
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Mark and Re-capture
1. Catch a couple of fish.2. Mark them and throw them back.3. At a later point, catch a couple of fish again.4. Count how many are marked.
20 were marked and five out of the 20 that were caught the second time were marked.
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So, how many fish are in the lake?
● What are the probable number of fish in the lake?
● We have almost already described the solution! (If we know about Bayesian Data Analysis, that is...)
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Data
Generative model
Parameters
µ θ σρ
5, 2, 7, 8, 3, 9, 1, 2, ...
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Data
1. Mark 20 “fish”2. Sample 20 “fish”3. Count the no. marked fish
No. of Fish
5 marked fish
Parameters
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Data
1. Mark 20 “fish”2. Sample 20 “fish”3. Count the no. marked fish
No. of Fish
5 marked fish
Parameters
Uncertainty
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Data
1. Mark 20 “fish”2. Sample 20 “fish”3. Count the no. marked fish
No. of Fish
5 marked fish
Uncertainty
Uniform(0, 250)
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1. Draw a large random sample from the “prior” probability distribution on the parameters. Here, for example:no_fish: [63, 30, 167, 30, 164, 222, 225, 42, 122,…]
2. Plug in each draw into the generative model which generates a vector of “fake” data. For example:
fish-pick
fish = 63
4
fish-pick
fish = 30
13
fish-pick
fish = 167
5
fish-pick
fish = 30
15
…
One simple way of fitting the model
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fish-pick
fish = 63
4
fish-pick
fish = 30
13
fish-pick
fish = 167
5
fish-pick
fish = 30
15
…
3. Keep only those parameter values that generated the data that was actually observed.
…
4 = 5 ? 13 = 5 ? 5 = 5 ? 15 = 5 ?
One simple way of fitting the model
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fish-pick
fish = 63
4
fish-pick
fish = 30
13
fish-pick
fish = 167
5
fish-pick
fish = 30
15
…
3. Keep only those parameter values that generated the data that was actually observed.
4. The distribution of the retained parameters now represent the probability that the data was produced by a certain parameter value. For example:
…
[167, 135, 148, 90, 162, 88, 98, 110, 176, … ]
4 = 5 ? 13 = 5 ? 5 = 5 ? 15 = 5 ?
One simple way of fitting the model
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Time for a demonstration
The script that was “live coded” can be found here:http://rpubs.com/rasmusab/live_coding_user_2015_bayes_tutorial
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Prior
Posterior
Maximum likelihood estimate
50 % Credible Intervall
Posterior median
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fish-pick
n <><
5 o
P(100 <>< | 5 o ) ∝ P(100 <><) × P(5 o | 100 <><)
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fish-pick
n <><
5 o
P(100 <>< | 5 o ) = P(100 <><) × P(5 o | 100 <><)
Σ P(n <><) · P(5 o | n <><)
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fish-pick
Generative model
5 o n <><
Data D
P(Θ | D) = P(Θ) · P(D | Θ)
Σ P(Θ) · P(D | Θ)Bayes theorem
P(100 <>< | 5 o ) = P(100 <><) × P(5 o | 100 <><)
Σ P(n <><) · P(5 o | n <><)
Parameters Θ
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What have we done?● We have specified prior information
● A generative model● And have calculated the probability of
of different parameter values
P(Θ)
P(D | Θ)
P(Θ | D)
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What have we done?● In this example we used a capture-
recapture model with one parameter.● But the general method works on any
generative model and with any number of parameters.
● The specific computational method we used only works in rare cases...
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What is not Bayesian data analysis?
● A category of models● Subjective● Not necessarily the most computationally
efficient method of fitting a model.● Anything new.
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Inverse Probability
Bayes 1701–1761
Laplace 1749–1827
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Inverse Probability
Bayesians!
“Bayesian data analysis” is not the best of names...
Bayes 1701–1761
Laplace 1749–1827 Fisher 1890–1962
“Probabilistic modeling” would be better!
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UseR! 2015 prediction competition
http://bit.ly/1LuF64m20 minutes
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Why use Bayesian data analysis?
● You have great flexibility when building models, and can focus on that, rather than computational issues.
Why?
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“Marked fish get shy! It is half as likely to catch a marked fish compared to a fish that has not been marked.”
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Data
1. Mark 20 “fish”2. Sample 20 “fish”3. Count the no. marked fish
No. of Fish
5 marked fish
Parameters
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Data
1. Mark 20 “fish”2. Sample 20 “fish”, where there is a
50% chance to sample a marked fish compared to a unmarked fish.
3. Count the no. marked fish
No. of Fish
5 marked fish
Parameters
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Demo
The script that was “live coded” can be found here:http://rpubs.com/rasmusab/live_coding_user_2015_bayes_tutorial
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Why use Bayesian data analysis?
● You have great flexibility when building models, and can focus on that, rather than computational issues.
● You can include information sources in addition to the data, for example, expert opinion.
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“There has always been plenty of fish in the lake. Around 200, I would say!”
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Demo
The script that was “live coded” can be found here:http://rpubs.com/rasmusab/live_coding_user_2015_bayes_tutorial
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“If you’re not using a informative prior, you’re leaving money on the table.”
- Robert Weiss, UCLA, Los Angeles.
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Why use Bayesian data analysis?
● You have great flexibility when building models, and can focus on that, rather than computational issues.
● You can include information sources in addition to the data, for example, expert opinion.
● The result of a Bayesian analysis retains the uncertainty of the estimated parameters, which is very useful in decision analysis.
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draw_id no_fish
1 90
2 162
3 202
4 98
5 176
... ...
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draw_id no_fish no_fish * 100
1 90 9000
2 162 16200
3 202 20200
4 98 9800
5 176 17600
... ... ...
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“If there are less than 50 fish in the lake, they wont last the season. It will cost 10 000 kr to plant new fish
into the lake!
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draw_id no_fish catch 80 fish x 100 kr
1 90 8000
2 162 8000
3 202 8000
4 98 8000
5 71 7100
... ... ...
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draw_id no_fish catch 80 fish x 100 kr
fish left
repopulation cost
Profit
1 90 8000 10 10,000 -2000
2 162 8000 82 0 8000
3 202 8000 122 0 8000
4 98 8000 18 10,000 -2000
5 71 7100 0 10,000 -2900
... ... ... ... ... ...
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draw_id no_fish catch 80 fish x 100 kr
fish left
repopulation cost
Profit
1 90 8000 10 10,000 -2000
2 162 8000 82 0 8000
3 202 8000 122 0 8000
4 98 8000 18 10,000 -2000
5 71 7100 0 10,000 -2900
... ... ... ... ... ...
profit <- min(no_fish, 80) * 100 - (no_fish - 80 < 50) * 10000
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draw_id no_fish catch 80 fish x 100 kr
fish left
repopulation cost
Profit
1 90 8000 10 10,000 -2000
2 162 8000 82 0 8000
3 202 8000 122 0 8000
4 98 8000 18 10,000 -2000
5 71 7100 0 10,000 -2900
... ... ... ... ... ...
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draw_id no_fish catch 80 fish x 100 kr
fish left
repopulation cost
Profit
1 90 8000 10 10,000 -2000
2 162 8000 82 0 8000
3 202 8000 122 0 8000
4 98 8000 18 10,000 -2000
5 71 7100 0 10,000 -2900
... ... ... ... ... ...
> mean(profit)[1] -1013
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What’s the optimal catch quota?
Catch quota: 27 fishExpected profit: 2409 kr
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80vs
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Why use Bayesian data analysis?
● You have great flexibility when building models, and can focus on that, rather than computational issues.
● You can include information sources in addition to the data, for example, expert opinion.
● The result of a Bayesian analysis retains the uncertainty of the estimated parameters, which is very useful in decision analysis.
● You probably are already...
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t.test(y)
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t.test(y1, y2)
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lm(y ~ 1 + x)
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glm(y ~ 1 + x, family = “poisson”)
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Why not use Bayesian data analysis?
● Everything is just working fine as it is.
● I’m not that interested in uncertainty.
● It’s too computationally demanding.
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Bayesian A/B testing forSwedish Fish Incorporated
http://bit.ly/1SSCAaj
Exercise 1
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How to interpret and perform a Bayesian data analysis in R?
● Interpreting the result of an Bayesian data analysis is usually straight forward.
How?
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With 95% probability the support of the voters lie within this band.
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How to interpret and perform a Bayesian data analysis in R?
● Interpreting the result of an Bayesian data analysis is usually straight forward.
● But if you scratch the surface there is a lot of Bayesian jargon!
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Prior
Posterior
Maximum likelihood estimate
50 % Credible Intervall
Posterior median
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More Bayesian Jargon
● Priors○ Objective priors
○ Subjective priors
○ Informative priors
○ Improper priors
○ Conjugate priors
Expert opinionCompletely data
driven model
Bayesian models
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More Bayesian Jargon: Distributions!
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More Bayesian Jargon: Distributions!
● The usual suspects: The Normal
x ~ Normal(µ, σ) x <- rnorm(n_draw, mu, sd)
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More Bayesian Jargon: Distributions!
● The usual suspects: The Binomial
x ~ Binomial(p, n) x <- rbinom(n_draw, size, prob)
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More Bayesian Jargon: Distributions!
● The usual suspects: The Poisson
x ~ Poisson(λ) x <- rpois(n_draw, lambda)
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More Bayesian Jargon: Distributions!
● Less common beasts: The Beta
x ~ Beta( , ) x <- rbeta(n_draw, shape1, shape2)
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More Bayesian Jargon: Distributions!
● Less common beasts: The Gamma
x ~ Gamma(k, θ) x <- rgamma(n_draw, shape, scale)
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More Bayesian Jargon: Distributions!
● Less common beasts: The Hypergeometric
● Fisher's noncentral hypergeometric distribution
● When it comes to distributions, Wikipedia is your friend!
fish-pick
5 o n <><
fish-pick with shy marked fish
5 o n <><
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More Bayesian Jargon● Samples, samples, samples.
Prior samples:[63, 30, 167, 30, 164, 222, 225, 42, 122, … ]
Posterior samples: [167, 135, 148, 90, 162, 88, 98, 110, 176, … ]
● Methods to generate posterior samples:○ Approximate Bayesian Computation (ABC)○ Markov Chain Monte Carlo (MCMC)
■ Metropolis-Hastings■ Gibbs Sampling■ Hamiltonian monte carlo
● Other methods○ Conjugate models○ Laplace Approximation○ Etc. Etc. Etc.
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Faster Bayesian computation
● We have been doing approximate Bayesian computation, which is the most general and slowest method for fitting a Bayesian model.
● Faster methods have in common that:○ They require that the likelihood that the generative
model will generate any given data can be calculated.
fish-pick
5 o n <><
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Faster Bayesian computation
● We have been doing approximate Bayesian computation, which is the most general and slowest method for fitting a Bayesian model.
● Faster methods have in common that:○ They require that the likelihood that the generative
model will generate any given data can be calculated.
fish-pick
5 o
n <><P(5 o | n <><)
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Faster Bayesian computation
● We have been doing approximate Bayesian computation, which is the most general and slowest method for fitting a Bayesian model.
● Faster methods have in common that:○ They require that the likelihood that the generative
model will generate any given data can be calculated.
○ They explore the parameter space in a smarter way.○ What you get are samples as if you would have
done the analysis using approximate Bayesian computation.
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MCMC: The Metropolis-Hasting algorithm● The “classic” MCMC algorithm. Performs a
random walk in the parameter space, and will stay at a parameter value proportional to its posterior probability.
● A good R implementation can be found in the MCMCpack package as the functionMCMCmetrop1R(fun, theta.init, ...)
Source: https://theclevermachine.wordpress.com/tag/metropolis-hastings-sampling/
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Source: http://mbjoseph.github.io/blog/2013/09/08/metropolis/
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Source: http://mbjoseph.github.io/blog/2013/09/08/metropolis/
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● Similar to Metropolis, but moves by one parameter at a time.
● Can be much more efficient, but usually required custom built sampling schemes.
● Unless you use JAGS!
MCMC: Gibbs sampling and JAGS
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JAGS: Just Another Gibbs Sampler
● A cross-platform implementation of the BUGS language, an R-likeprobabilistic programminglanguage.
● It builds a custom Gibbs sampler for you.
● Created by Martyn Plummer, member of the R core group.
● Made to be called from R.
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JAGS: Just Another Gibbs Sampler
● JAGS is tailor made for building generative models.
● A minimal JAGS program:
JAGS codemodel {
n <- 30p ~ dunif(0, 1)x ~ dbinom(p, n)
}
R code
n <- 30p <- runif(1, 0, 1)x <- rbinom(1, p, n)
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JAGS codemodel {
n <- 30p ~ dunif(0, 1)x ~ dbinom(p, n)
}
jags.model(..., data = list())
R code
n <- 30p <- runif(1, 0, 1)x <- rbinom(1, p, n)
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JAGS codemodel {
n <- 30p ~ dunif(0, 1)x ~ dbinom(p, n)
}
jags.model(..., data = list(x = 10))
R code
n <- 30p <- runif(1, 0, 1)x <- rbinom(1, p, n)+ ABC step
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JAGS: Just Another Gibbs Sampler
● JAGS is declarative…
model {n <- 30p ~ dunif(0, 1)x ~ dbinom(p, n)
}
model {x ~ dbinom(p, n)n <- 30p ~ dunif(0, 1)
}
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JAGS: Just Another Gibbs Sampler
● JAGS is declarative…
model {n <- 30n <- n + 1p ~ dunif(0, 1)x ~ dbinom(p, n)x ~ dbinom(p, n)
}
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JAGS: Just Another Gibbs Sampler
● JAGS is (unfortunately not) vectorized.
JAGS code
model {x ~ dpois(m)
}
R code> m <- 1:5> x <- rpois(5, m)> x[1] 0 2 5 4 6
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JAGS: Just Another Gibbs Sampler
● JAGS is (unfortunately not) vectorized.
JAGS code
model {for(i in 1:length(m)) {
x ~ dpois(m[i])}
}
R code> m <- 1:5> x <- rpois(5, m)> x[1] 0 2 5 4 6
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JAGS: Just Another Gibbs Sampler
● JAGS is (unfortunately not) vectorized.
JAGS code
model {for(i in 1:length(m)) {
x[i] ~ dpois(m[i])}
}
R code> m <- 1:5> x <- rpois(5, m)> x[1] 0 2 5 4 6
✓
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Demo
The script that was “live coded” can be found here:http://rpubs.com/rasmusab/live_coding_user_2015_bayes_tutorial
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Bayesian computation with JAGS and farmer Jöns
http://bit.ly/1RGuK0X
Exercise 2
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Fitting Bayesian models in R
Pre-specified models and a general metropolis algorithm.
Gibbs sampling
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Fitting Bayesian models in R
Hamiltonian Monte Carlo
R-inla Integrated nested Laplace
approximation
Extra everything!
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Some things we have not covered
● Priors● Distributions● Decision analysis and “post-processing” of
posteriors.● Model selection and Bayes factors● Philosophy● Math
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To summarize Bayesian data analysis● What?
○ Bayesian data analysis is a flexible method to fit any type of statistical model.
○ Maximum likelihood is a special case of Bayesian model fitting.
● Why?○ Makes it possible to define highly custom models.○ Makes it possible to include information from many
sources, for example, data and expert knowledge.○ Quantifies and retains the uncertainty in parameter
estimates and predictions.● How?
○ R! Using ABC, MCMCpack, JAGS, STAN, R-inla, etc.
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Summer reading / listening
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"The theory of probabilities is at bottom nothing but common sense reduced to calculus; it enables us to appreciate with exactness that which accurate minds feel with a sort of instinct for which ofttimes they are unable to account."
Pierre-Simon LaplaceEssai philosophique sur les Probabilités (1814)
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Introduction to Bayesian Data Analysis with R.
!
Rasmus Bååth, Lund University@rabaath || [email protected]
http://www.sumsar.net