1. Priors, Normal Models, Computing Posteriors
st5219: Bayesian hierarchical modelling
lecture 2.4

2. An all purpose sampling tool
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

3. 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
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

4. Markov chain Monte Carlo
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:
Hastings (1970) Biometrika 57:

5. Metropolis-Hastings algorithm
Board work

6. 1: constants
Suppose you can calculate f( |data) only to a constant of proportionality
Eg f(data| )f( )
No problem: the constant of proportionality cancels

7. 2: choice of q
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 have and so the qs cancel
This cancelling of qs is common to all symmetric distributions around the current value
Board work

8. 3: special case Monte Carlo
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

9. 4: choice of bandwidth
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

10. 5: special case Gibbs sampler
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

11. 6: burn in 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

13. 7: multiple chains 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)

14. 8: correlation of parameters
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

15. 9: assessing convergence
The cowboy approach is to look at a trace plot of the chain only (Butcher’s test)
More formal methods exist (see tutorial 2)

16. An example: H1N1 again As before New
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}

17. 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)

18. 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

19. Using that routine

20. Using that routine

21. Using that routine

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

23. Using bandwidths = 1

24. Using bandwidths = 1

25. Using bandwidths = 1

26. Using bandwidths = 0.01

27. Using bandwidths = 0.01

28. Using bandwidths = 0.01

29. Using bandwidths = 0.001

30. Using bandwidths = 0.001

31. Using bandwidths = 0.001

32. Another example Same dataset, but now the non-informative prior:
p ~ U(0,1)
σ ~ U(0,1)

33. Using bandwidths = 0.001

34. Using bandwidths = 0.01

35. Using bandwidths = 0.1

36. Using bandwidths = 1

37. Using bandwidths = 0.1

38. 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

39. Summary Importance sampling: Monte Carlo: MCMC:
if you can find a distribution quite close to the posterior
Monte Carlo:
use whenever you can: but rarely are able
MCMC:
good general purpose tool
sometimes an art to get working effectively

40. Next week: everything you already know how to do, differently
Versions of:
t-tests
regression
etc
After that:
hierarchical modelling
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