Dealing with intractability: Recent Bayesian Monte Carlo methods for dealing with intractable likelihoods

Background
ABC methods for generative models
MC2 type methods
State-Space models, PMCMC
SMC2

Dealing with intractability: recent advances in
Bayesian Monte Carlo methods for intractable
likelihoods

N. CHOPIN1

CREST-ENSAE

1
joint work with S. BARTHELME, P.E. JACOB, & O.
PAPASPILIOPOULOS
N. CHOPIN Intractability 1/ 54

Background
MC2 type methods
SMC2

Outline

1 Background

2 ABC methods for generative models

3 MC2 type methods

4 State-Space models, PMCMC

5 SMC2


Background
MC2 type methods
SMC2

Tractable models
For a prototypic Bayesian model, deﬁned by (a) prior p(θ), and (b)
likelihood p(y |θ), a standard approach is to sample from the
posterior
p(θ|y ) ∝ p(θ)p(y |θ).
using the Metropolis-Hastings algorithm:
Metropolis-Hastings
From current point θn
1 Sample θp ∼ T (θn , dθp )
2 With probability 1 ∧ r , take θn+1 = θp , otherwise θn+1 = θn ,
where
p(θp )p(y |θp )T (θp , θn )
r=
p(θn )p(y |θn )T (θn , θp )

This generates a Markov chain which leaves the posterior invariant.

Background
MC2 type methods
SMC2

Intractable models
This generic approach cannot be applied in the following
situations:
1 The likelihood reads p(y |θ) = C (θ)hθ (y ), where C (θ) is an
intractable normalising constant; e.g. log-linear models, Ising
models.
2 The likelihood p(y |θ) is an intractable integral

p(y |θ) = p(y , x|θ) dx
X

of a tractable integrand; e.g. state-space models.
3 The likelihood is even more complicate, because it
corresponds to some generative process (scientiﬁc models).
Solutions to these problems involve auxiliary variables.

Background
MC2 type methods
SMC2

Outline

1 Background


3 MC2 type methods


5 SMC2


Background
MC2 type methods
SMC2

Example of a generative model: reaction times
Subject must choose between k alternatives. Evidence ej (t) in
favour of choice j follows a Brownian motion with drift:
τ dej (t) = mj dt + dWtj .
Decision is taken when one evidence “wins the race”; see plot.

Threshold for B

Threshold for A

Evidence for B

Evidence for A

0 50 100 150

time (ms)


Background
MC2 type methods
SMC2


ABC stands for “Approximate Bayesian Computation”. In such
algorithms, the auxiliary variable is an artiﬁcial dataset y ∼ p(y |θ).
Denote the actual dataset y . Consider the simple rejection
algorithm:
Basic ABC
Repeat
1 Sample θ ∼ p(θ).
2 Sample y ∼ p(y |θ).
3 Accept with probability Kε ( s(y ) − s(y ) ).

where Kε (x) = K (x/ε), K is a kernel function, and s is a vector of
“summary statistics”.


Background
MC2 type methods
SMC2

ABC target

This algorithm samples from:

πε (θ, y ) ∝ p(θ)p(y |θ)Kε ( s(y ) − s(y ) ).

and the marginal πε (θ) → p(θ|s(y )) as ε → 0.

If s is suﬃcient, then the limit is the true posterior
p(θ|s(y )) = p(θ|y ), but this is rarely possible unfortunately.


Background
MC2 type methods
SMC2

MCMC-ABC

One can instead derive a MCMC algorithm that sample from the
same distribution.
MCMC-ABC
From current point (θn , yn )
1 Sample θp ∼ T (θn , dθp ).
2 Sample y p ∼ p(y |θp ).
3 With probability 1 ∧ r , take (θn+1 , yn+1 ) = (θp , y p ), otherwise
(θn+1 , yn+1 ) = (θn , yn ), where

p(θp )Kε ( s(y p ) − s(y ) )T (θp , θn )
r=
p(θn )Kε ( s(yn ) − s(y ) )T (θn , θp )


Background
MC2 type methods
SMC2

Remarks on the KDE interpretation of ABC

Having sampled N pairs (θi , y i ) from p(θ)p(y |θ), choosing ε
essentially amounts to choosing the bandwidth of a KDE. There
are some speciﬁc aspects that may deserve some investigation
however:
1 The objective is to approximate a conditional density, that is
p(θ|s(y )). (But approximating p(s(y )) may be interesting
too.)
2 The marginal distribution of the simulated θ’s is known.
3 Could we use a bandwidth matrix instead?


Background
MC2 type methods
SMC2

Parametric interpretation of ABC

It would be great to take s(y ) = y . In that way, the ABC posterior
could be interpreted as the posterior distribution of the same
model, but corrupted with noise (of size ε). See the following
paper for a fast (EP) approximation of such an ABC posterior:

Barthelm´, S. and Chopin, N. (2011). ABC-EP: Expectation
e
Propagation for Likelihood-free Bayesian Computation, ICML
2011, L. Getoor and T. Scheﬀer (eds), 289-296. (see also
arXiv:1107.5959).


Background
MC2 type methods
SMC2

ABC: summary

We use ABC for very challenging models (generative/scientiﬁc
models). We pay a heavy price for this:
1 First level of approximation is p(θ|y ) ≈ p(θ|s(y ))
(althought not in ABC-EP).
2 Second level of approximation is p(θ|s(y )) ≈ πε (θ).
3 Huge CPU cost (but less in ABC-EP).
4 ABC-EP cannot be used in all situations.
In the rest of the talk, we will deal with milder problems, and we
will be able to avoid approximations.


Background
MC2 type methods
SMC2

Outline

1 Background


3 MC2 type methods


5 SMC2


Background
MC2 type methods
SMC2

Basic framework

Imagine a model such that

p(y |θ) = p(x, y |θ) dx

is intractable, but can be approximated by the following unbiased
MC estimate:
N
1 p(x j , y |θ)
p (y |θ) =
ˆ
N qθ (x j )
j=1

where the x j ’s are N points sampled from the (user-chosen)
proposal distribution qθ .


Background
MC2 type methods
SMC2

Naive question

Can we simply replace p(y |θ) by p (y |θ)? i.e.
ˆ

MC2
From current point θn (plus p (y |θn ) from previous iteration)
ˆ
2 Sample x 1:N ∼ qθp so as to compute p (y |θp ).
ˆ
3 With probability 1 ∧ r , set θn+1 = θp , otherwise θn+1 = θn
with
p(θp )ˆ(y |θp )T (θp , θn )
p
r= .
p(θn )ˆ(y |θn )T (θn , θp )
p


Background
MC2 type methods
SMC2

Answer: yes, and the algorithm is exact!

More precisely, this algorithm is a correct Metropolis step with
respect to the following extended distribution:
 
N N j , y |θ)
1 p(x
π(θ, x 1:N ) ∝ p(θ) qθ (x j )  
N qθ (x j )
j=1 j=1

which is such that the marginal distribution of θ is precisely the
true posterior distribution:

π(θ, x 1:N ) dx 1:N = p(θ|y ).


Background
MC2 type methods
SMC2

Outline

1 Background


3 MC2 type methods


5 SMC2


Background
MC2 type methods
SMC2

State Space Models
A system of equations
Hidden states (Markov): p(x1 |θ) = µθ (x1 ) and for t ≥ 1

p(xt+1 |x1:t , θ) = p(xt+1 |xt , θ) = fθ (xt+1 |xt )

Observations:

p(yt |y1:t−1 , x1:t−1 , θ) = p(yt |xt , θ) = gθ (yt |xt )

Parameter: θ ∈ Θ, prior p(θ). We observe y1:T = (y1 , . . . yT ),
T might be large (≈ 104 ). x and θ will also be of several
dimensions.
There are several interesting models for which fθ cannot be written
in closed form (but it can be simulated).

Background
MC2 type methods
SMC2

State Space Models

Some interesting distributions
Bayesian inference focuses on:

static: p(θ|y1:T ) dynamic: p(θ|y1:t ) , t ∈ 1 : T

Filtering (traditionally) focuses on:

∀t ∈ [1, T ] pθ (xt |y1:t )

Smoothing (traditionally) focuses on:

∀t ∈ [1, T ] pθ (xt |y1:T )


Background
MC2 type methods
SMC2

Examples

Population growth model
yt = xt + σw εt
log xt+1 = log xt + b0 + b1 (xt )b2 + σ ηt

θ = (b0 , b1 , b2 , σ , σW ).


Background
MC2 type methods
SMC2

Examples

Stochastic Volatility (L´vy-driven models)
e
Observations (“log returns”):
1/2
yt = µ + βvt + vt t ,t ≥ 1

Hidden states (“actual volatility” - integrated process):
k
1
vt+1 = (zt − zt+1 + ej )
λ
j=1


Background
MC2 type methods
SMC2

Examples

. . . where the process zt is the “spot volatility”:
k
zt+1 = e −λ zt + e −λ(t+1−cj ) ej
j=1

iid iid
k ∼ Poi λξ 2 /ω 2 c1:k ∼ U(t, t + 1) ei:k ∼ Exp ξ/ω 2
The parameter is θ ∈ (µ, β, ξ, ω 2 , λ), and xt = (vt , zt ) .
See the results


Background
MC2 type methods
SMC2

Why are those models challenging?

. . . It is eﬀectively impossible to compute the likelihood

p(y1:T |θ) = p(y1:T |x1:T , θ)p(x1:T |θ)dx1:T
XT

Similarly, all other inferential quantities are impossible to compute.


Background
MC2 type methods
SMC2

Problems with MCMC approaches

Metropolis-Hastings:
1 p(θ|y1:T ) cannot be evaluated point-wise (marginal MH)
2 p(x1:T , θ|y1:T ) are high-dimensional and it is hard to design
reasonable proposals
Gibbs sampler (updates states and parameters):
1 The hidden states x1:T are typically very correlated and it is
hard to update them eﬃciently in a block
2 Parameters and latent variables highly correlated
Common: they are not designed to recover the whole
sequence π(x1:t , θ | y1:t )


Background
MC2 type methods
SMC2

Particle ﬁlters

Consider the simpliﬁed problem of targeting

pθ (xt+1 |y1:t+1 )

This sequence of distributions is approximated by a sequence of
weighted particles which are properly weighted using importance
sampling, mutated/propagated according to the system dynamics,
and resampled to control the variance.
Below we give a pseudo-code version. Any operation involving the
superscript n must be understood as performed for n = 1 : Nx ,
where Nx is the total number of particles.


Background
MC2 type methods
SMC2

Step 1: At iteration t = 1,
n
(a) Sample x1 ∼ q1,θ (·).
(b) Compute and normalise weights
n n n
n µθ (x1 )gθ (y1 |x1 ) n w1,θ (x1 )
w1,θ (x1 ) = n , W1,θ = N
.
q1,θ (x1 ) i
i=1 w1,θ (x1 )

Step 2: At iteration t = 2 : T
n 1:Nx
(a) Sample the index at−1 ∼ M(Wt−1,θ ) of the ancestor
an
(b) Sample xtn ∼ qt,θ (·|xt−1 ).
t−1

(c) Compute and normalise weights
an an
n
at−1 fθ (xtn |xt−1 )gθ (yt |xtn )
t−1
wt,θ (xt−1 , xtn )
t−1

wt,θ (xt−1 , xtn ) = an
, n
Wt,θ = i
at−1 i
qt,θ (xtn |xt−1 )
t−1 Nx
i=1 wt,θ (xt−1 , xt )


Background
MC2 type methods
SMC2

Particle ﬁltering

time
Figure: Three weighted trajectories x1:t at time t.


Background
MC2 type methods
SMC2

Particle ﬁltering

time
Figure: Three proposed trajectories x1:t+1 at time t + 1.


Background
MC2 type methods
SMC2

Particle ﬁltering

time
Figure: Three reweighted trajectories x1:t+1 at time t + 1


Background
MC2 type methods
SMC2

Observations

(i) (i)
At each t, (wt , x1:t )Nx is a particle approximation of
i=1
pθ (xt |y1:t ).
Resampling to avoid degeneracy. If there were no interaction
between particles there would be typically polynomial or worse
increase in the variance of weights
Taking qθ = fθ simpliﬁes weights, but mainly yields a feasible
algorithm when fθ can only be simulated.


Background
MC2 type methods
SMC2

Unbiased likelihood estimator

A by-product of PF output is that
T Nx
ˆ 1 (i)
ZtN = wt
Nx
t=1 i=1

is an unbiased estimator of the likelihood Zt = p(y1:t |θ) for all t.
Whereas consistency of the estimator is immediate to check,
unbiasedness is subtle, see e.g Proposition 7.4.1 in Del Moral. The
variance of this estimator grows typically linealy with T (and not
exponentially) because of lack of independence.


Background
MC2 type methods
SMC2

PSMC

Breakthrough paper of Andrieu et al. (2011), based on the
unbiasedness of the PF estimate of the likelihood.
Marginal PMCMC
From current point θn (and current PF estimate p (y |θn )):
ˆ
2 Run a PF so as to obtain p (y |θp ), an unbiased estimate of
ˆ
p(y |θp ).
3 With probability 1 ∧ r , set θn+1 = θp , otherwise θn+1 = θn
with
p(θp )p(y |θp )T (θp , θn )
r=
p(θn )ˆ(y |θn )T (θn , θp )
p


Background
MC2 type methods
SMC2

Outline

1 Background


3 MC2 type methods


5 SMC2


Background
MC2 type methods
SMC2

Objectives

1 to derive sequentially

p(θ, x1:t |y1:t ), p(y1:t ), for all t ∈ {1, . . . , T }

2 to obtain a black box algorithm (automatic calibration).


Background
MC2 type methods
SMC2

Main tools of our approach

Particle ﬁlter algorithms for state-space models (this will be to
estimate the likelihood, for a ﬁxed θ).
Iterated Batch Importance Sampling for sequential Bayesian
inference for parameters (this will be the theoretical algorithm
we will try to approximate).
Both are sequential Monte Carlo (SMC) methods


Background
MC2 type methods
SMC2

IBIS

SMC method for particle approximation of the sequence p(θ | y1:t )
for t = 1 : T . PF is not going to work here by just pretending that
θ is a dynamic process with zero (or small) variance. Recall the
path degeneracy problem.
In the next slide we give the pseudo-code of the IBIS algorithm.
Operations with superscript m must be understood as operations
performed for all m ∈ 1 : Nθ , where Nθ is the total number of
θ-particles.


Background
MC2 type methods
SMC2

Sample θm from p(θ) and set ω m ← 1. Then, at time t = 1, . . . , T
(a) Compute the incremental weights and their weighted
average
Nθ
1
ut (θm ) = p(yt |y1:t−1 , θm ), Lt = Nθ
× ω m ut (θm ),
m
m=1 ω m=1
(b) Update the importance weights,
ω m ← ω m ut (θm ). (1)
˜
(c) If some degeneracy criterion is fulﬁlled, sample θm
independently from the mixture distribution
Nθ
1
Nθ
ω m Kt (θm , ·) .
m
m=1 ω m=1
Finally, replace the current weighted particle system:
˜
(θm , ω m ) ← (θm , 1).

Background
MC2 type methods
SMC2

Observations

Cost of lack of ergodicity in θ: the occasional MCMC move
Still, in regular problems resampling happens at diminishing
frequency (logarithmically)
Kt is an MCMC kernel invariant wrt π(θ | y1:t ). Its
parameters can be chosen using information from current
population of θ-particles
Lt is a MC estimator of the model evidence
Infeasible to implement for state-space models: intractable
incremental weights, and MCMC kernel


Background
MC2 type methods
SMC2

Our algorithm: SMC2

We provide a generic (black box) algorithm for recovering the
sequence of parameter posterior distributions, but as well ﬁltering,
smoothing and predictive.
We give next a pseudo-code; the code seems to only track the
parameter posteriors, but actually it does all other jobs.
Superﬁcially, it looks an approximation of IBIS, but in fact it does
not produce any systematic errors (unbiased MC).


Background
MC2 type methods
SMC2

Sample θm from p(θ) and set ω m ← 1. Then, at time
t = 1, . . . , T ,
(a) For each particle θm , perform iteration t of the PF: If
t = 1, sample independently x1 x ,m from ψ1,θm , and
1:N

compute
Nx
1 n,m
p (y1 |θm ) =
ˆ w1,θ (x1 );
Nx
n=1

If t > 1, sample xt1:Nx ,m , at−1x ,m from ψt,θm
1:N

1:Nx ,m 1:Nx ,m
conditional on x1:t−1 , a1:t−2 , and compute

Nx
1 an,m ,m
p (yt |y1:t−1 , θm ) =
ˆ wt,θ (xt−1 , xtn,m ).
t−1

Nx
n=1


Background
MC2 type methods
SMC2

(b) Update the importance weights,

ω m ← ω m p (yt |y1:t−1 , θm )
ˆ

(c) If some degeneracy criterion is fulﬁlled, sample
θm , x1:t x ,m , ˜1:t−1 independently from
˜ ˜1:N a1:Nx

Nθ
1
Nθ
ω m Kt θm , x1:t x ,m , a1:t−1 , ·
1:N 1:Nx ,m
m
m=1 ω m=1

Finally, replace current weighted particle system:

(θm , x1:t x ,m , a1:t−1 , ω m ) ← (θm , x1:t x ,m , ˜1:t−1 , 1)
1:N 1:Nx ,m ˜ ˜1:N a1:Nx ,m

Background
MC2 type methods
SMC2

Observations

It appears as approximation to IBIS. For Nx = ∞ it is IBIS.
However, no approximation is done whatsoever. This
algorithm really samples from p(θ|y1:t ) and all other
distributions of interest. One would expect an increase of MC
variance over IBIS.
The validity of algorithm is essentially based on two results: i)
the particles are weighted due to unbiasedness of PF estimator
of likelihood; ii) the MCMC kernel is appropriately constructed
to maintain invariance wrt to an expanded distribution which
admits those of interest as marginals; it is a Particle MCMC
kernel.
The algorithm does not suﬀer from the path degeneracy
problem due to the MCMC updates

Background
MC2 type methods
SMC2

The MCMC step
˜ ˜ ˜
(a) Sample θ from proposal kernel, θ ∼ T (θ, d θ).
˜
(b) Run a new PF for θ: sample independently
1:Nx 1:Nx
(˜1:t , ˜1:t−1 ) from ψt,θ , and compute
x a ˜
ˆt (θ, x 1:Nx , ˜1:Nx ).
Z ˜˜ a
1:t 1:t−1
(c) Accept the move with probability
˜ ˆ ˜ ˜1:N a1:Nx ˜
p(θ)Zt (θ, x1:t x , ˜1:t−1 )T (θ, θ)
1∧ .
ˆ ˜
p(θ)Zt (θ, x 1:Nx , a1:Nx )T (θ, θ)
1:t 1:t−1

It can be shown that this is a standard Hastings-Metropolis kernel
with proposal
˜ ˜1:N a1:N ˜ 1:N
a1:N
qθ (θ, x1:t x , ˜1:t x ) = T (θ, θ)ψt,θ (˜1:t x , ˜1:t x )
˜ x

1:N 1:Nx
invariant wrt to an extended distribution πt (θ, x1:t x , a1:t−1 ).

Background
MC2 type methods
SMC2

Some advantages of the algorithm

Immediate estimates of ﬁltering and predictive distributions
Immediate and sequential estimator of model evidence
Easy recovery of smoothing distributions
Principled framework for automatic calibration of Nx
Population Monte Carlo advantages


Background
MC2 type methods
SMC2

Numerical illustrations: SV

1.0 800

700
8 0.8
600
Squared observations

Acceptance rates
6 0.6 500

Nx
400
4 0.4
300

2 0.2 200

100
0 0.0

200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000
Time Iterations Iterations

(a) (b) (c)

Figure: Squared observations (synthetic data set), acceptance rates, and
illustration of the automatic increase of Nx .

See the model


Background
MC2 type methods
SMC2


T = 250 T = 500 T = 750 T = 1000
8

6
Density

4

2

0
−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0
µ

Figure: Concentration of the posterior distribution for parameter µ.


Background
MC2 type methods
SMC2


Multifactor model

k1 k2
1/2
yt = µ+βvt +vt t +ρ1 e1,j +ρ2 e2,j −ξ(w ρ1 λ1 +(1−w )ρ2 λ2 )
j=1 j=1

where vt = v1,t + v2,t , and (vi , zi )i=1,2 are following the same
dynamics with parameters (wi ξ, wi ω 2 , λi ) and w1 = w ,
w2 = 1 − w .


Background
MC2 type methods
SMC2


Evidence compared to the one factor model
variable
20 Multi factor without leverage
4 Multi factor with leverage
Squared observations

15

2

10

0

5

−2

100 200 300 400 500 600 700 100 200 300 400 500 600 700
Time Iterations

(a) (b)

Figure: S&P500 squared observations, and log-evidence comparison
between models (relative to the one-factor model).

Background
MC2 type methods
SMC2

Final Remarks

A powerful framework
A generic algorithm for sequential estimation and state
inference in state space models: only requirements are to be
able (a) to simulate the Markov transition fθ (xt |xt−1 ), and (b)
to evaluate the likelihood term gθ (yt |xt ).
The article is available on arXiv and our web pages
A package is available at:
http://code.google.com/p/py-smc2/.


Background
MC2 type methods
SMC2

Appendix


Background
MC2 type methods
SMC2

Why does it work? - Intuition for t = 1

At time t = 1, the algorithm generates variables θm from the prior
p(θ), and for each θm , the algorithm generates vectors x1 x ,m of
1:N
1:N
particles, from ψ1,θm (x1 x ).


Background
MC2 type methods
SMC2

Thus, the sampling space is Θ × X Nx , and the actual “particles” of
the algorithm are Nθ independent and identically distributed copies
1:N
of the random variable (θ, x1 x ), with density:
Nx
1:N n
p(θ)ψ1,θ (x1 x ) = p(θ) q1,θ (x1 ).
n=1


Background
MC2 type methods
SMC2

Then, these particles are assigned importance weights
corresponding to the incremental weight function
ˆ 1:N −1 Nx n
Z1 (θ, x1 x ) = Nx n=1 w1,θ (x1 ).

This means that, at iteration 1, the target distribution of the
algorithm should be deﬁned as:

1:N 1:N
ˆ 1:N
Z1 (θ, x1 x )
π1 (θ, x1 x ) = p(θ)ψ1,θ (x1 x ) × ,
p(y1 )

where the normalising constant p(y1 ) is easily deduced from the
ˆ 1:N
property that Z1 (θ, x1 x ) is an unbiased estimator of p(y1 |θ).


Background
MC2 type methods
SMC2

Direct substitutions yield
Nx n Nx n
1:N p(θ) i µθ (x1 )gθ (y1 |x1 )
1
π1 (θ, x1 x ) = q1,θ (x1 ) n
p(y1 ) Nx q1,θ (x1 )
i=1 n=1
 
N  Nx
1 x p(θ) n n i

= µθ (x1 )gθ (y1 |x1 ) q1,θ (x1 )
Nx p(y1 )  
n=1 i=1,i=n

and noting that, for the triplet (θ, x1 , y1 ) of random variables,
p(θ)µθ (x1 )gθ (y1 |x1 ) = p(θ, x1 , y1 ) = p(y1 )p(θ|y1 )p(x1 |y1 , θ)
one ﬁnally gets that:
 
Nx Nx
1:N p(θ|y1 ) n

i

π1 (θ, x1 x ) = p(x1 |y1 , θ) q1,θ (x1 ) .
Nx  
n=1 i=1,i=n


Background
MC2 type methods
SMC2

By a simple induction, one sees that the target density πt at
iteration t ≥ 2 should be deﬁned as:
ˆ 1:N 1:Nx
Zt (θ, x1:t x , a1:t−1 )
1:N 1:Nx 1:N 1:Nx
πt (θ, x1:t x , a1:t−1 ) = p(θ)ψt,θ (x1:t x , a1:t−1 ) ×
p(y1:t )

and the following Proposition


Background
MC2 type methods
SMC2

Proposition

The probability density πt may be written as:
1:N 1:Nx
πt (θ, x1:t x , a1:t−1 ) = p(θ|y1:t )
 
N
 N 
1 x p(xn |θ, y1:t )  x
 

1:t i
× t−1
q1,θ (x1 )
Nx Nx  
n=1  i=1
 n 

i=ht (1)
 
 t
 Nx


 i
as−1 i 
i as−1
× Ws−1,θ qs,θ (xs |xs−1 )

s=2 i=1 

 n

i=ht (s)


Dealing with intractability: Recent Bayesian Monte Carlo methods for dealing with intractable likelihoods

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Dealing with intractability: Recent Bayesian Monte Carlo methods for dealing with intractable likelihoods