Priors, Normal Models,

Computing Posteriorsst5219: Bayesian hierarchical modelling

lecture 2.4

Monte Carlo: requires knowing the distribution---often don’t

Importance Sampling: requires being able to sample from something vaguely like the posterior---often can’t

Markov chain Monte Carlo can almost always be used

An all purpose sampling tool

Discrete time stochastic process obeying the Markov property

Denote the distribution of t+1 given t as π(t+1 | t)

We consider on some part of ℝd

t+1 may have a stationary distribution, meaning that if s comes from the stationary distribution then so does t for s<t

Markov chains

The big idea: if you could set up a Markov chain that had a stationary distribution

equal to the posterior, you could sample just by simulating a Markov chain and

then using its trajectory as your sample from the posterior

Obviously if π(t+1 | t) = f( |data) then you would just be sampling the posterior

When Monte Carlo is not possible, it’s really easy to set up a Markov chain with this property using the Metropolis-Hastings algorithm

Metropolis et al (1953) J Chem Phys 21:1087--92 Hastings (1970) Biometrika 57:97--109

Markov chain Monte Carlo

Metropolis-Hastings algorithm

Board

work

Suppose you can calculate f( |data) only to a constant of proportionality

Eg f(data| )f( ) No problem: the constant of proportionality

cancels

1: constants

You can choose almost whatever you want for q(*→ ). A common choice is N(,Σ), with Σ an arbitrary (co)variance

Note that for 1-D, you haveand so the qs cancel

This cancelling of qs iscommon to all symmetric distributions around the current value

2: choice of q

Board work

If you can propose from the posterior itself, so that q(*→ )=f( |data), then α=1 and you always accept proposals

So Monte Carlo is a special case of MCMC

3: special case Monte Carlo

If using normal proposals, how to choose the standard deviation or bandwidth?

Aim for 20% to 40% of proposals accepted and you’re close to optimal

Too small: very slow movement Too big: very slow movement Goldilocks: fast movement

4: choice of bandwidth

If you know the marginal posterior of a parameter or block of parameter given the other parameters, you can just propose from that marginal

This gives α=1 This is called Gibbs sampling---nothing

special, just a good MCMC algorithm

5: special case Gibbs sampler

It is common to start with an arbitrary 0 To stop this biasing your estimates, usually

discard samples from a “burn in” period This lets the chain forget where you started

it If you start near posterior, short burn in ok If you start far from posterior, longer burn in

needed

6: burn in

Running multiple chains in parallel allows:◦ you to check convergence to the same

distribution even from initial values far from each other

◦ you can utilise X multiple processors (eg on a server) to get a sample X times as big in the same amount of time

Make sure you start with different seeds though (eg not set.seed(666) all the time)

7: multiple chains

If 2+ parameters are tightly correlated, then sampling one at a time will not work efficiently

Several options:◦ reparametrise the model so that the posteriors

are more orthogonal◦ use a multivariate proposal distribution that

accounts for the correlation

8: correlation of parameters

The cowboy approach is to look at a trace plot of the chain only (Butcher’s test)

More formal methods exist (see tutorial 2)

9: assessing convergence

As before logpost=function(cu){ cu$logp=dbinom(98,727,cu$p*cu$sigma,log=TRUE) +dbeta(cu$p,1,1,log=TRUE) +dbeta(cu$sigma,630,136,log=TRUE) cu}

New rejecter=function(cu){ reject=FALSE if(cu$p<0)reject=TRUE; if(cu$p>1)reject=TRUE if(cu$sigma<0)reject=TRUE;if(cu$sigma>1)reject=TRUE reject}

An example: H1N1 again

current=list(p=0.5,sigma=0.5) current=logposterior(current) NDRAWS=10000 dump=list(p=rep(0,NDRAWS),sigma=rep(0,NDRAWS)) for(iteration in 1:NDRAWS) { old=current current$p=rnorm(1,current$p,0.1) current$sigma=rnorm(1,current$sigma,0.1) REJECT=rejecter(current) if(!REJECT) { current=logposterior(current)

An example: H1N1 again

if(!REJECT) { current=logposterior(current) accept_prob=current$logp-old$logp lu=log(runif(1)) if(lu>accept_prob)REJECT=TRUE } if(REJECT)current=old dump$p[iteration]=current$p dump$sigma[iteration]=current$sigma }

An example: H1N1 again

Using that routine

Using that routine

Using that routine

The choice of bandwidth is arbitrary Asymptotically, doesn’t matter But in practice, need to choose right...

Bandwidths

Using bandwidths = 1

Using bandwidths = 1

Using bandwidths = 1

Using bandwidths = 0.01

Using bandwidths = 0.01

Using bandwidths = 0.01

Using bandwidths = 0.001

Using bandwidths = 0.001

Using bandwidths = 0.001

Same dataset, but now the non-informative prior:◦ p ~ U(0,1)◦σ ~ U(0,1)

Another example

Using bandwidths = 0.001

Using bandwidths = 0.01

Using bandwidths = 0.1

Using bandwidths = 1

Using bandwidths = 0.1

Example 2 Why does this not work? Tightly correlated posterior Plus weird shape Very hard to design a local movement rule

to encourage swift mixing through the joint posterior distribution

SummaryMonte Carlo:use whenever you can: but rarely are able

Importance sampling:if you can find a distribution quite close to the posterior

MCMC:• good general purpose tool• sometimes an art to get working effectively

Next week: everything you already know how to do, differentlyVersions of:• t-tests• regression• etc

After that:• hierarchical modelling• BUGS