SAMPLING FROM A RING DISTRIBUTION IN THREE DIMENSIONS
I will here demonstrate a variety of Markov chain sampling methods
using as an example a distribution that forms a ring in a space of
three dimensions, parameterized by variables called "x", "y", and "z".
For details on the various methods described, see mc-spec.doc. A
table comparing the performance of the methods on these examples is
included at the end of this section.
Specifying the distribution.
The "ring" distribution is specified by a 'dist-spec' command such as
the following (the "\" says the command continues on the next line):
> dist-spec glog \
"x^2/2 + y^2/2 + z^2/2 + (x+y+z)^2 + 10000/(1+x^2+y^2+z^2)"
Recall that the formula given to 'dist-spec' is the "energy function",
which is minus the log probability density, plus any constant. If the
energy function above consisted of only the first three terms, the
distribution would be multivariate normal, with x, y, and z being
independent, each having mean zero and variance one. The fourth term,
(x+y+z)^2, leaves the distribution still normal, but squashes it in
the direction where x, y, and z increase equally. The final term is
large near the origin, and from there decreases to zero symmetrically
in all directions. It has the effect of making a hole in the centre
of what would otherwise have been a normal distribution, leaving a
ring shape.
The examples below are assumed to start with a 'dist-spec' command of
the above form, except that "glog" is replaced with the name of the
log file used for the method being demonstrated.
Multivariate Metropolis updates.
We will first see how to sample from this distribution using a
variation of the Metropolis algorithm in which all three state
variables are changed simultaneously. In the proposal distribution
used, new values for the state variable are chosen independently, each
from a normal distribution with mean equal to the present value, and a
specified standard deviation.
The following command specifies that 50 Metropolis operations of this
sort should be done for each full iteration:
> mc-spec glog.met,1 repeat 50 metropolis 1 end
Here, "glob.met,1" is the name of the log file to use, which would
have been created using a 'dist-spec' command like the one above. The
"repeat 50 ... end" construction causes the enclosed operations to be
repeated the given number of times. This saves space in the log file,
compared to doing one metropolis operation for 50 times as many
iterations. This is appropriate when many such operations will be
needed to get to a substantially different point. A similar result
can be obtained using an extra argument to 'dist-mc' (see xxx-mc.doc),
but using "repeat" has the advantage that one can also easily look at
the rejection rate over the 50 repetitions.
We can now sample for 10000 Markov chain iterations (a total of 500000
metropolis updates) with the following command:
> dist-mc glog.met,1 10000
This takes about half a second on the test system (Ex-test-system.doc)
as will all the other sampling commands in this section.
We can look at what happened to the three state variables during these
10000 iterations with a command such as
> dist-plt t xyz glog.met,1 | plot
Or you might look at only the first 200 iterations:
> dist-plt t xyz glog.net,1 :200 | plot
Depending on your plot program's requirements, you might instead use
a command such as
> dist-tbl txyz glog.met,1 | plot
or you might have to plot the variables one at a time, with commands
such as
> dist-plt t x glog.met,1 | plot
From these plots, you can see that the chain quite rapidly reached the
equilibrium distribution - maybe even by the end of the first
iteration (ie, within the first 50 metropolis updates). Just to be
sure, however, let's discard the first 10 iterations as "burn-in".
We can now take a look at the "ring" distribution with commands like
> dist-plt x y glog.met,1 11: | plot-points
If "plot-points" plots points rather than lines, this will produce a
scatterplot of the distribution for "x" and "y" (with "z" ignored),
which will look like a flattened ring. The ring is actually circular,
but it's tilted with respect to the axes, so you'll be able to see it
as a circle only if you have a three-dimensional plotting program.
However, the 9990 points plotted are not independent, as can also be
seen from the plots of "x", "y", and "z" versus "t". The effect of
this can be seen more clearly by looking at only points up to
iteration 1000:
> dist-plt x y glog.met,1 11:1000 | plot-points
This plot will probably not show points distributed perfectly
uniformly around the ring. Instead, there will be clumps here or
there, which result from inadequate sampling. So to get a sample of
points that are a good representation of the distribution using this
chain, it is necessary to be run for quite a few iterations.
We can get a quantitative idea of how poor the sampling is with the
following command:
> dist-tbl x glog.met,1 11: | series mac 50
Number of realizations: 1 Total points: 9990
Sample mean: 0.403307 S.E. from correlations: 0.435419
Lag Autocorr. Cum. Corr.
1 0.951543 2.903085
2 0.908311 4.719707
3 0.866078 6.451863
4 0.826428 8.104720
5 0.790122 9.684965
6 0.755438 11.195842
7 0.722632 12.641105
8 0.691039 14.023184
9 0.661355 15.345894
10 0.631852 16.609599
11 0.604368 17.818335
12 0.579568 18.977472
13 0.555770 20.089013
14 0.533468 21.155949
15 0.510170 22.176290
16 0.488388 23.153066
17 0.468346 24.089758
18 0.449704 24.989165
19 0.429849 25.848863
20 0.413143 26.675149
21 0.396623 27.468396
22 0.381805 28.232006
23 0.366428 28.964862
24 0.351633 29.668128
25 0.336837 30.341802
26 0.323512 30.988826
27 0.308738 31.606303
28 0.294569 32.195441
29 0.282383 32.760207
30 0.270926 33.302059
31 0.260250 33.822559
32 0.248232 34.319023
33 0.235130 34.789283
34 0.221089 35.231461
35 0.209061 35.649584
36 0.198239 36.046063
37 0.189368 36.424798
38 0.181272 36.787342
39 0.174378 37.136097
40 0.167159 37.470416
41 0.159599 37.789614
42 0.152854 38.095322
43 0.145695 38.386712
44 0.139347 38.665405
45 0.131296 38.927997
46 0.124033 39.176064
47 0.118028 39.412119
48 0.111391 39.634901
49 0.106981 39.848864
50 0.103838 40.056539
The 'dist-tbl' command above outputs a list of values for "x" for
iterations from 11 on. The 'series' command with options "mac" finds
the mean of these numbers, their autocorrelations (out to lag 50
here), and the cumulative correlations. The cumulative correlation at
the earliest lag past which the autocorrelations are about zero
indicates the factor by which sampling is made inefficient by the
correlations (here, about 40); see Ex-dist-n.doc for more details.
From symmetry, we know that the true mean for "x" is zero. The
estimate of 0.403307 above is consistent with this, in view of the
estimated standard error of +-0.435419. (Note that differences from
the true value of up to about twice the standard error are plausible.)
We can also get estimates using 'dist-est', but as discussed in
Ex-dist-n.doc, the standard errors it produces do not account for
autocorrelation.
We might try to improve the efficiency of sampling by changing the
standard deviation of the Metropolis proposal distribution - which is
also known as the "stepsize" for the operation. One indication of
whether the stepsize is appropriate is the rejection rate for the
Metropolis operations, which can be viewed with a command such as
> dist-plt t r glog.met,1 | plot
Here, the rejection rate is about 0.75, which is acceptable. Very low
or very high rejection rates are usually an indication that sampling
would work better with a different stepsize.
Although the stepsize of 1 that was used above appears to be OK, we
could try a smaller stepsize with the following commands (following a
'dist-spec' command):
> mc-spec glog.met,0.2 repeat 50 metropolis 0.2 end
> dist-mc glob.met,0.2 10000
Or we might try a larger stepsize:
> mc-spec glog.met,5 repeat 50 metropolis 5 end
> dist-mc glob.met,5 10000
If enough iterations are done, the same estimates should be obtained
all these chains, but some stepsizes will produce a more efficient
chain than others. By examining plots of how the state variables
change, and looking at the autocorrelations with 'series', one can
conclude that sampling is much less efficient with a stepsize of 0.2
than with a stepsize of 1 (about ten times less efficient, based on
cumulative correlations). With a stepsize of 5, the sampling is about
as good as (or even better than) with a stepsize of 1, even though the
rejection rate is quite high. This is a phenomenon that occurs only
in problems with an effective dimensionality of three or less - for
higher-dimensional problems, a rejection rate close to one is
generally an indication of poor sampling.
Single-variable Metropolis updates.
We can also try sampling using Metropolis updates that change only one
variable at a time. This is done using "met-1" operations, specified
as follows:
> mc-spec glog.met1,1 repeat 18 met-1 1 end
As with "metropolis" operations, we specify a "stepsize", which is the
standard deviation for proposed change to a variable. Each "met-1"
operation tries to change each variable in turn, accepting or
rejecting the change based on the change in energy as a result of
making the proposed change to just that variable. Since there are
three state variables for this distribution, a single "met-1"
operation must therefore calculate the energy three times, and hence
takes about three times as long as a "metropolis" operation. To
facilitate comparisons, the repeat count is corresponding less in this
specification.
As before, can now sample for 10000 iterations using a 'dist-mc' command:
> dist-mc glog.met1,1 10000
You can see how well this method samples in the same ways as discussed
above. You could also try sampling using "met-1" with a stepsize of
0.2 and 5. You should see that the rejection rate with "met-1" is
lower than with "metropolis" operations using the same stepsize.
Nevertheless, sampling from this distribution seems to be less
efficient with "met-1" than with "metropolis". This is not always so,
however. For distributions where at least some of the variables are
close to being independent, updating one variable at a time can be
more efficient. It is also sometimes possible to save computation
time when recomputing the energy after a change to just one variable,
though that possibility is not presently exploited by this software.
Single-variable slice sampling.
Variables can also be updated one at a time is using single-variable
slice sampling, which is described in my tech report on "Markov chain
Monte Carlo methods based on `slicing' the distribution" (available
from my web page), or in the published paper, called "Slice sampling".
Several variations on this procedure are implemented in this software.
The method in which the slice is found by "stepping out" can be done
as follows:
> mc-spec glog.slc1,1 repeat 4 slice-1 1 end
> dist-mc glog.slc1,1 10000
This does single-variable slice sampling using an initial interval of
size 1, which is extended in steps of the same size until both ends
are outside the slice. The "doubling" procedure is also implemented,
but is not illustrated here.
The "e" quantity records the average number of energy function
evaluations done in the slice sampling updates for one iteration.
We can find the average of this quantity over all iterations with
a command such as
> dist-tbl e glog.slc1,1 | series m
Number of realizations: 1 Total points: 10000
Sample mean: 5.381167
Note that 5.381167 is the average number of evaluations for updating
one variable, not for updating all three of them.
As with the Metropolis methods, performance varies with the stepsize
chosen. However, one advantage of single-variable slice sampling is
that it is a bit less sensitive to the choice of stepsize than the
single-variable Metropolis algorithm.
Multivariate slice sampling.
We can also use variations of slice sampling in which all variables
are updated simultaneously (as described in my "Slice sampling"
paper). The simplest such scheme randomly places a hyperrectangle
containing the current point, picks points randomly from it, and
shrinks it when the point chosen is outside the slice, until a point
inside the slice is finally found. This can be done with an mc-spec
command such as the following:
> mc-spec glog.slc,5 repeat 13 slice 5 end
This works about as well as multivariate Metropolis with a stepsize of
1 or 5.
One can also specify that shrinkage is to occur only in the coordinate
direction where the product of the energy gradient and the dimension
of the hyperrectangle is greatest in magnitude. The following command
does this, with the number of repetitions set so that the time per
iteration is about the same:
> mc-spec glog.slcg,5 repeat 6 slice -g 5 end
For this problem, using the gradient information with -g (or -G,
another variant) gives little or no advantage, after accounting for
the extra time needed to compute the gradient. However, for problems
where variables have greatly differing scales (not compensated for by
differing stepsizes), the -g and -G options can be very beneficial.
One can also try multivariate slice sampling with Gaussian "crumbs"
rather than hyperrectangles:
> mc-spec glog.sgau,5 repeat 13 slice-gaussian -e 5 end
The -e option results in the Gaussian distribution being shrunk on the
basis of the energy of the rejected trial point.
Sampling with Hamiltonian dynamics.
It is possible to sample much more efficiently by suppressing the
random walk behaviour that the methods above all exhibit. One way of
doing this is by adding "momentum" variables, which will keep the
state moving in the same direction for an extended period of time.
The original "position" variables along with these "momentum"
variables can be updated by applying Hamiltonian dynamics for some
period of fictitious time, implemented by performing some number of
"leapfrog" steps with a specified stepsize. To produce an ergodic
Markov chain, the momentum should also be changed using "heatbath"
updates, but it should not be changed too quickly, as this will
re-introduce random walk behaviour. For a detailed discussion of this
"stochastic dynamics" method, see my book, Bayesian Learning for
Neural Networks, or my review paper, Probabilistic Inference Using
Markov Chain Monte Carlo Methods, or may later review, MCMC Using
Hamiltonian Dynamics.
We can try out this dynamical method as follows:
> mc-spec glog.dyn,0.3 repeat 40 heatbath 0.98 dynamic 1 0.3 end
> dist-mc glog.dyn,0.3 10000
The argument of "heatbath" is the factor by which to multiply the old
momentum variables (after which noise is added). A value of 1-d
results in random walks being suppressed for around 1/d iterations.
If you now look at the state variables with a command such as
> dist-plt t xyz glog.dyn,0.3 | plot
you will see that they are initially far from their stationary
distribution, but after about 50 iterations they settle down, and from
there on the chain samples very well. The initial behaviour of the
chain can be understood by looking at what is happening to the the
"kinetic energy", which is half the sum of squares of the momentum
variables:
> dist-plt t K glog.dyn,0.3 | plot
Initially, the kinetic energy (as well as the "potential" energy,
which is what is specified in 'dist-spec') is very large. It is only
slowly dissipated, as a result of the "heatbath" updates of the
momentum variables. Eventually, however, the kinetic energy reaches
its equilibrium distribution (around a value of 3/2), and the chain
samples from approximately the desired distribution.
This method is not exact, however, because the Hamiltonian dynamics is
simulated inexactly, biasing the results. The hybrid Monte Carlo
(also called Hamiltonian Monte Carlo) method eliminates this bias by
using an acceptance test.
Hybrid/Hamiltonian Monte Carlo.
Several variations of the hybrid/Hamiltonian Monte Carlo method are
supported by this software. In the "standard" method, each iteration
starts by picking completely new values for the momentum variables
with the "heatbath" operation. Hamiltonian dynamics is then simulated
for some number of leapfrog steps, using some specified stepsize, and
the end-point of the simulated trajectory is accepted or rejected
based on the change in the total energy.
The following commands apply the standard hybrid Monte Carlo method,
using trajectories of 45 leapfrog steps, done with a stepsize of 0.3.
To avoid problems with large initial energies, a few Metropolis
updates are done at the beginning.
> mc-spec glog.hmc,0.3 repeat 50 metropolis 1
> dist-mc glog.hmc,0.3 1
> mc-spec glog.hmc,0.3 heatbath hybrid 45 0.3
> dist-mc glog.hmc,0.3 10000
The length of the trajectory should be chosen based on the number of
steps for which we want to suppress random walks - longer for more
difficult problems where it takes many steps to get from one end of
the distribution to the other.
As with the Metropolis methods, you can check the rejection rate with
a command such as the following (the 2: causes the first iteration,
which consisted of Metropolis updates, to be excluded):
> dist-tbl r glog.hmc,0.3 2: | series m
Number of realizations: 1 Total points: 9999
Mean: 0.039004
It is also useful to look at the changes in energy on which the
decisions to accept or reject were made:
> dist-plt t D glog.hmc,0.3 2: | plot-points
If most of these points are much greater than one, the rejection rate
will be high. With a stepsize of 0.3, the change in energy is seldom
greater than 0.5, but if the stepsize is increased to 0.8 much larger
changes are often seen, and with a stepsize of 1, almost no
trajectories are accepted.
We can see how well the chain is sampling by plotting the state
variables, as described above, and by looking at the autocorrelations,
with a command such as:
> dist-tbl x glog.hmc,0.3 10: | series mac 10
Number of realizations: 1 Total points: 9991
Sample mean: -0.023125 S.E. from correlations: 0.126909
Lag Autocorr. Cum. Corr.
1 0.538886 2.077772
2 0.282877 2.643526
3 0.146972 2.937470
4 0.066028 3.069527
5 0.033698 3.136924
6 0.023167 3.183258
7 0.020455 3.224167
8 0.029720 3.283607
9 0.025383 3.334374
10 0.019557 3.373488
From the cumulative correlations, we can estimate that estimating the
expectation of "x" using points from this chain is a factor of only
about 3.4 less efficient than using independent points. This is about
ten times better than the best of the chains described above.
On the other hand, we have seen that some care is needed to pick an
appropriate stepsize for hybrid Monte Carlo, and it is also often
necessary to start at a reasonably good point, as was done here by
doing a few Metropolis updates at the beginning.
The "persistent" form of hybrid Monte Carlo (described in Bayesian
Learning for Neural Networks) is also implemented. Here are the
appropriate commands:
> mc-spec glog.phmc,0.18 repeat 50 metropolis 1
> dist-mc glog.phmc,0.18 1
> mc-spec glog.phmc,0.18 repeat 35 heatbath 0.99 hybrid 1 0.18 negate end
> dist-mc glog.phmc,0.18 10000
The use of a "heatbath" operation with a non-zero "decay" causes the
momentum to change only slowly. Because of this, even though only a
single leapfrog update is done, random walk behaviour will still be
suppressed. The "negate" operation negates the momentum, which
normally undoes a negation at the end of the "hybrid" operation. If
the "hybrid" update was rejected, however, the first negation will not
have been done, and movement will be reversed, undermining the random
walk suppression. To avoid this, a smaller stepsize is used, to keep
the rejection rate very small. Performance is a bit worse than for
the standard hybrid Monte Carlo method.
Reflective slice sampling.
Another way of avoiding random walks is to apply slice sampling to all
variables at once, and to sample from within the multivariate slice by
a sort of dynamical method that proceeds at a constant speed while
reflecting off the boundaries of the slice. We would rather not
compute these boundaries exactly, however. Instead, we can proceed in
steps of some size and when we cross the boundary reflect at either
the last inside point or the first outside point. These reflection
operations are based on the gradient of the energy at that point.
Slice sampling with reflection from the last inside point is done as
follows:
> mc-spec glog.slci,0.3 heatbath slice-inside 35 0.3
> dist-mc glog.slci,0.3 10000
The "heatbath" operation picks a random momentum vector, which sets
the speed of motion and the initial direction. We then find the slice
and take 35 steps within the slice with that velocity, using a
stepsize of 0.3 (ie, the change in the state at each step is 0.3 times
the momentum vector). If a step takes us from inside the slice to
outside the slice, we backtrack, and change the momentum vector by
reflection based on the gradient. It is necessary to check that this
reflection would also occur in the reversed trajectory; if not, we
must reject by negating the momentum, causing further steps to retrace
the trajectory. Rejections will be less frequent if the stepsize is
small.
Slice sampling with outside reflection is done as follows:
> mc-spec glog.slco,0.3 heatbath slice-outside 45 0.3
> dist-mc glog.slco,0.3 10000
This method operates as described for inside reflection, except that
when a step ends at a point outside the slice, we stay there, but
reflect based on the gradient at that point. If the final point is
inside the slice, we accept it. Otherwise we reject, and return to
the point from which the trajectory started.
Both inside and outside reflection work reasonably well for this
distribution - not quite as good as hybrid Monte Carlo, but better
than the other methods.
Overrelaxed slice sampling.
Random walks can also be suppressed for some distributions using
single-variable slice sampling with "overrelaxation", in which
variables are changed from their current point to the corresponding
point on the "other side" of their conditional distribution. This can
be done with the following commands:
> mc-spec glog.slcv,0.3 slice-over 8 0.1 0.3
> dist-mc glog.slcv,0.3 10000
The first argument of "slice-over" is the number of "refinements" used
to more accurately pin down the end-points of the slice (using
bisection). More refinements result in an update that is closer to an
ideal overrelaxed update, and also reduces the probability of
rejection (caused by the overrelaxed point being outside the slice).
The second argument specifies the probability of doing an ordinary
single-variable slice sampling update rather than an overrelaxed one.
Occasional ordinary updates are necessary in order to ensure
ergodicity, but they should not be done too often if random walks are
to be suppressed. The final argument is the stepsize to use when
finding the endpoints of the slice initially.
For this distribution, overrelaxation does not work as well as the
dynamical methods or reflective slice sampling, but it is effective
for some other distributions.
Performance summary.
The following table summarizes the inefficiency factor for estimating
the expectation of the "x" variable using the various methods (the
autocorrelation time), determined from estimates of the cumulative
correlations up to when the autocorrelations are near zero. The
repetition counts for the methods were set so that they all required
about half a second of computation time for 10000 iterations on the
machine described in Ex-test-system.doc (between 0.42 and 0.52
seconds, except as noted below). Hence the autocorrelation time shown
below is a good guide to their relative efficiency.
METHOD STEPSIZE AUTOCORR. NOTES
TIME
metropolis 0.2 450
1 40
5 25
met-1 0.2 550
1 90
5 180
slice-1 0.2 125 Computation time was 0.58 seconds
1 80
5 100
slice 5 40
slice -g 5 30
slice -G 5 55
slice-gaussian 5 65
dynamic 0.3 3.0 This method is not exact
hybrid (HMC) 0.3 3.2
persistent HMC 0.18 8.9
slice-inside 0.3 11
slice-outside 0.3 4.6
slice-over 2 30
The relative performance of the different methods will of course be
somewhat different for different distributions, but large gains from
using hybrid Monte Carlo (HMC) or other methods that suppress random
walks are common.