Towards Likelihood Free Inference

1
Towards Likelihood Free Inference

• Tony Pettitt
• QUT, Brisbane
• a.pettitt_at_qut.edu.au
• Joint work with Rob Reeves

2
Outline

1. Some problems with intractable likelihoods.
2. Monte Carlo methods and Inference.
3. Normalizing constant/partition function.
4. Likelihood free Markov chain Monte Carlo.
5. Approximating Hierarchical model
6. Indirect Inference and likelihood free MCMC
7. Conclusions.

3

• Stochastic models (Riley et al, 2003)
• Macroparasite within a host.
• Juvenile worm grows to adulthood in a cat.
• Host fights back with immunity.
• Number of Juveniles, Adults and amount of
  Immunity (all integer).
• evolve through time
  according to Markov process unknown parameters,
  eg
• Juvenile ? Adult rate of maturation
• Immunity changes with time
• Juveniles die due to Immunity
• Moment closure approximations for distribution
  of limited to
  restricted parameter values.

4

• Numerical computation of
  limited by small maximum values
• of J, A, I.
• Can simulate
  process easily.
• Data J at t0 and A at t (sacrifice of cat),
  replicated with several cats

Source Riley et al, 2003.
5

• Other stochastic process models include
• spatial stochastic expansion of species
• (Hamilton et al, 2005 Estoup et al, 2004)
• birth-death-mutation process for estimating
  transmission rate from TB genotyping
• (Tanaka et al, 2006)
• population genetic models, eg coalescent models
• (Marjoram et al 2003)
• Likelihood free Bayesian MCMC methods are often
  employed with quite precise priors.

6

• Normalizing constant/partition function problem.
• The algebraic form of the distribution for y is
  known but it is not normalized, eg Ising model
• For
  means neighbours (on a lattice, say). The
  normalizing constant involves in general a sum
  over terms.
• Write

7
N-S and E-W neighbourhood
8
Outline

1. Some problems with intractable likelihoods.
2. Monte Carlo methods and Inference.
3. Normalizing constant/partition function.
4. Likelihood free Markov chain Monte Carlo.
5. Approximating Hierarchical model
6. Indirect Inference and likelihood free MCMC
7. Conclusions.

9

• Monte Carlo methods and Inference.
• Intractable likelihood, instead use easily
  simulated values of y.
• Simulated method of moments (McFadden, 1989).
• Method of estimation comparing theoretical
  moments or frequencies with observed moments or
  frequencies.
• Can be implemented using a chi-squared
  goodness-fit-statistic, eg Riley et al, 2003.
  Data number of adult worms in cat at sacrifice.

10
Plot of goodness-of-fit statistic versus
parameter. Greedy Monte Carlo. Precision of
estimate?
Source Riley et al 2003.
11
Outline

1. Some problems with intractable likelihoods.
2. Monte Carlo methods and Inference.
3. Normalizing constant/partition function.
4. Likelihood free Markov chain Monte Carlo.
5. Approximating Hierarchical model
6. Indirect Inference and likelihood free MCMC
7. Conclusions.

12

• 3. Normalizing constant/partition function and
  MCMC
• (half-way to likelihood free inference)
• Here we assume (Møller, Pettitt, Reeves and
  Berthelsen, 2006)
• Key idea Importance sample estimate of
  given by
• Sample
• .

13

• Used off-line to estimate then
  carry out standard Metropolis-
• Hastings with interpolation over a
  grid of values.( eg Green
• and Richardson, 2002, in a Potts model).
• Standard Metropolis Hastings Simulating from
  target distribution
• Acceptance ratio for changing
• accepted with probability
  .
• Key Question Can be calculated
  on-line or avoided?

14

• On-line algorithm single auxiliary variable
  method.
• Introduce auxiliary variable x on same space as
  y and extend target distribution for the MCMC
• Key Question How to choose distribution of x
  so that
• removed from
• Now acceptance ratio is as
  a new pair proposed.
• Proposal becomes
  .
• Assume the factorisation
• Choose the proposal so that
• Then algebra ? cancellation of and
• does not depend on

15

• Note Need perfect or exact simulation from
  for the proposal.
• Key Question How to choose
  , the auxiliary variable distribution?
• The best choice

16
Choice (i)
17
Choice (ii)
Choice (ii)
18

• Choice (i)
• Fix , say at a good estimate of
  . Then
• so does not depend on only y and
  cancels in .
• Choice (ii)
• Eg Partially ordered Markov mesh model for Ising
  data
• Comment
• Both choices can suffer from getting stuck
  because
• can be very different from the ideal
  .

19
(No Transcript)
20
Source Møller et al, 2006
Single auxiliary method tends to get stuck Murray
et al (2006) offer suggestions involving multiple
auxiliary variables
21
Outline

1. Some problems with intractable likelihoods.
2. Monte Carlo methods and Inference.
3. Normalizing constant/partition function.
4. Likelihood free Markov chain Monte Carlo.
5. Approximating Hierarchical model
6. Indirect Inference and likelihood free MCMC
7. Conclusions.

22

• 4. Likelihood free MCMC
• Single Auxiliary Variable Method as almost
  Approximate Bayesian Computation (ABC)
• We wish to eliminate or
  equivalently , the likelihood from
  the M-H algorithm.
• Solution The distribution of x given y and
  puts all probability on y, the observed data,
• then
• with the likelihood
• This might work for discrete data, sample size
  small, and if the proposal
  were a very good approximation to
  .
• If sufficient statistics s(y) exist then

23
(No Transcript)
24

• Likelihood free methods, ABC- MCMC
• Change of notation, observed data (fixed),
  y is pseudo data or auxiliary data generated
  from the likelihood .
• Instead of , now have y close to
  in the sense of statistics s( ),
• distance
• ABC allows rather than
  equal to 0
• Target distribution for variables
• Standard M-H with proposals
• (Marjoram et al 2003 ABC MCMC)
• for acceptance of .
• Ideally e should be small but this leads to very
  small acceptance probabilities.

25

• Issues of implementing Metropolis-Hastings ABC
• (a) Tune for e to get reasonable acceptance
  probabilities
• (b) All satisfying
  (hard) accepted
• with equal probability
• rather than smoothly weighted by
  (soft).
• (c) Choose summary statistics carefully if no
  sufficient statistics

26

• Tune for e
• A solution is to allow e to vary as a
  parameter (Bortot et al, 2004). The target
  distribution is
• Run chain and post filter output for small
  values of e

27
Outline

1. Some problems with intractable likelihoods.
2. Monte Carlo methods and Inference.
3. Normalizing constant/partition function.
4. Likelihood free Markov chain Monte Carlo.
5. Approximating Hierarchical model
6. Indirect Inference and likelihood free MCMC
7. Conclusions.

28
Beaumont, Zhang and Balding (2002) use kernel
smoothing in ABC-MC
29
(No Transcript)
30
Approximating Hierarchical Model
31
(No Transcript)
32
(No Transcript)
33
(No Transcript)
34
(No Transcript)
35
(No Transcript)
36
(No Transcript)
37
(No Transcript)
38
(No Transcript)
39
Outline

1. Some problems with intractable likelihoods.
2. Monte Carlo methods and Inference.
3. Normalizing constant/partition function.
4. Likelihood free Markov chain Monte Carlo.
5. Approximating Hierarchical model
6. Indirect Inference and likelihood free MCMC
7. Conclusions.

40
(No Transcript)
41
(No Transcript)
42
(No Transcript)
43
(No Transcript)
44
(No Transcript)
45
(No Transcript)
46
(No Transcript)
47
Some points

• How could approximate posterior be made more
  precise?
• Use more parameters in approximating likelihood,
  the POMM? (Gouriéroux at al (1993), Heggland
  and Frigassi (2004) discuss this in the
  frequentist setting)
• More iterations for side chain exact
  calculation of approximate posterior?
• How to choose a good approximating likelihood?
• Relationship to summary statistics approach?

48
Outline

1. Some problems with intractable likelihoods.
2. Monte Carlo methods and Inference.
3. Normalizing constant/partition function.
4. Likelihood free Markov chain Monte Carlo.
5. Approximating Hierarchical model
6. Indirect Inference and likelihood free MCMC
7. Conclusions.

49
Conclusions

• For the normalizing constant problem we presented
  a single on-line M-H algorithm.
• We linked these ideas to ABC-MCMC and developed a
  hierarchical model (HM) to approximate the true
  posterior showed variance inflation.
• We showed that the approximating HM could be
  tempered
• swaps made to improve mixing using parallel
  chains,
• variance inflation effect corrected by
  smoothing posterior summaries from the tempered
  chains.
• We extended indirect inference to an HM to find a
  way of implementing the Metropolis Hastings
  algorithm which is likelihood free.
• We demonstrated the ideas with the
  Ising/autologistic model.
• Application to specific examples is on-going and
  requires refinement of general approaches.

50
Acknowledgements

• Support of the Australian Research Council
• Co-authors Rob Reeves, Jesper Møller, Kasper
  Berthelsen
• Discussions with Malcolm Faddy, Gareth Ridall,
  Chris Glasbey, Grant Hamilton