Bayesian adaptively updated Hamiltonian Monte Carlo with an application to high-dimensional BEKK GARCH models
-
Martin Burda
and
John M. Maheu
Abstract
Hamiltonian Monte Carlo (HMC) is a recent statistical procedure to sample from complex distributions. Distant proposal draws are taken in a sequence of steps following the Hamiltonian dynamics of the underlying parameter space, often yielding superior mixing properties of the resulting Markov chain. However, its performance can deteriorate sharply with the degree of irregularity of the underlying likelihood due to its lack of local adaptability in the parameter space. Riemann Manifold HMC (RMHMC), a locally adaptive version of HMC, alleviates this problem, but at a substantially increased computational cost that can become prohibitive in high-dimensional scenarios. In this paper we propose the Adaptively Updated HMC (AUHMC), an alternative inferential method based on HMC that is both fast and locally adaptive, combining the advantages of both HMC and RMHMC. The benefits become more pronounced with higher dimensionality of the parameter space and with the degree of irregularity of the underlying likelihood surface. We show that AUHMC satisfies detailed balance for a valid MCMC scheme and provide a comparison with RMHMC in terms of effective sample size, highlighting substantial efficiency gains of AUHMC. Simulation examples and an application of the BEKK GARCH model show the practical usefulness of the new posterior sampler.
- 1
Although not estimated, we expect our method could be extended to other innovation distributions such as multivariate Student-t with little modification.
- 2
There are notable exceptions, such as Girolami and Calderhead (2011) who also take the statistical perspective, but their paper focuses on RMHMC while here we elaborate on the statistical background to HMC.
- 3
In the physics literature, θ denotes the position (or state) variable and –ln π(θ) describes its potential energy, while γ is the momentum variable with kinetic energy γ′M–1γ/2, yielding the total energy H(θ, γ) of the system, up to a constant of proportionality. M is a constant, symmetric, positive-definite “mass” matrix which is often set as a scalar multiple of the identity matrix.
- 4
In the physics literature, the Hamiltonian dynamics describe the evolution of (θ, γ) that keeps the total energy H(θ, γ) constant.
We would like to thank Ben Calderhead, Mark Girolami, and Radford Neal for helpful discussions. We would also like to thank the participants of the Seminar on Bayesian Inference in Econometrics and Statistics 2011 at the Washington University in St. Louis, Meetings of the Midwest Econometrics Group 2011 at the University of Chicago Booth School of Business, Workshop on High-Dimensional Econometric Modelling 2010 at Cass Business School in London, UK, and MIT-Harvard and Brown University seminar audiences for their insightful comments and suggestions. Both Burda and Maheu thank SSHRC for financial support. This work was made possible by the facilities of the Shared Hierarchical Academic Research Computing Network (SHARCNET: www.sharcnet.ca) and Compute/Calcul Canada. We would like to thank the Editor, Bruce Mizrach, and two anonymous referees for their helpful comments and suggestions
Appendix A: Hamiltonian Monte Carlo
In this section we provide the stochastic background for HMC. This synthesis is based on previously published material, but unlike the bulk of literature presenting HMC in terms of the physical laws of motion based on preservation of total energy in the phase-space, we take a fully stochastic perspective familiar to the applied Bayesian econometrician.2 The HMC principle is thus presented in terms of the joint density over the augmented parameter space leading to a Metropolis acceptance probability update. We hope that our synthesis of the probabilistic perspective on HMC will provide useful insights for practitioners who wish to further explore the HMC principles.
HMC Principle
Consider a vector of parameters of interest 
distributed according to the posterior density π(θ). Let 
denote a vector of auxiliary parameters with 
distributed Gaussian with mean vector 0 and covariance matrix M, independent of θ. Denote the joint density of (θ, γ) by π(θ, γ). Then the negative of the logarithm of the joint density of (θ, γ) is given by the Hamiltonian equation3
Hamiltonian Monte Carlo (HMC) is formulated in the following three steps that we will describe in detail further below:
Draw an initial auxiliary parameter vector
Transition from (θr, γr) to
according to the Hamiltonian dynamics;Accept
with probability 
otherwise keep (θr, γr) as the next MC draw.
Step 1 provides a stochastic initialization of the system akin to a RW draw. This step is necessary in order to make the resulting Markov chain 
irreducible and aperiodic (Ishwaran 1999). In contrast to RW, this so-called refreshment move is performed on the auxiliary variable γ as opposed to the original parameter of interest θ, setting 
In terms of the HMC sampling algorithm, the initial refreshment draw of 
forms a Gibbs step on the parameter space of (θ, γ) accepted with probability 1. Since it only applies to γ, it will leave the target joint distribution of (θ, γ) invariant and subsequent steps can be performed conditional on 
(Neal 2010).
Step 2 constructs a sequence 
according to the Hamiltonian dynamics starting from the current state 
and setting the last member of the sequence as the HMC new state proposal 
The role of the Hamiltonian dynamics is to ensure that the M-H acceptance probability (2) for 
is kept close to 1. As will become clear shortly, this corresponds to maintaining the difference 
close to zero throughout the sequence 
This property of the transition from (θr, γr) to 
can be achieved by conceptualizing θ and γ as functions of continuous time t and specifying their evolution using the Hamiltonian dynamics equations4
for i=1,…,d. For any discrete time interval of duration s, (15) and (16) define a mapping Ts from the state of the system at time t to the state at time t+s. For practical applications of interest these differential equations (15) and (16) in general cannot be solved analytically and instead numerical methods are required. The Stormer-Verlet (or leapfrog) numerical integrator (Leimkuhler and Reich 2004) is one such popular method, discretizing the Hamiltonian dynamics as
for some small 
From this perspective, γ plays the role of an auxiliary variable that parametrizes (a functional of) π(θ, ‧) providing it with an additional degree of flexibility to maintain the acceptance probability close to one for every k. Even though 
can deviate substantially from 
resulting in favorable mixing for θ, the additional terms in γ in (14) compensate for this deviation maintaining the overall level of 
close to constant over k=1, …, L when used in accordance with (17)–(19), since 
and 
enter with the opposite signs in (15) and (16). In contrast, without the additional parametrization with γ, if only 
were to be used in the proposal mechanism as is the case in RW style samplers, the M-H acceptance probability would often drop to zero relatively quickly.
Step 3 applies a Metropolis correction to the proposal 
In continuous time, or for ε→0, (15) and (16) would keep 
exactly resulting in 
but for discrete ε>0, in general, 
necessitating the Metropolis step. A key feature of HMC is that the generic M-H acceptance probability (2) can be expressed in a simple tractable form using only the posterior density π(θ) and the auxiliary parameter Gaussian density ϕ(γ;0, M). The transition from 
to 
via the proposal sequence 
taken according to the discretized Hamiltonian dynamics (17)–(19) is a fully deterministic proposal, placing a Dirac delta probability mass 
on each 
conditional on 
The system (17)–(19) is time reversible and symmetric in (θ, γ), which implies that the forward and reverse transition probabilities 
and 
are equal: this simplifies the Metropolis-Hastings acceptance ratio in (2) to the Metropolis form 
From the definition of the Hamiltonian H(θ, γ) in (14) as the negative of the log-joint densities, the joint density of (θ, π) is given by
Hence, the Metropolis acceptance probability takes the form
The expression for 
shows, as noted above, that the HMC acceptance probability is given in terms of the difference of the Hamiltonian equations 
The closer can we keep this difference to zero, the closer the acceptance probability is to one. A key feature of the Hamiltonian dynamics (15) and (16) in Step 2 is that they maintain H(θ, γ) constant over the parameter space in continuous time conditional on 
obtained in Step 1, while their discretization (17)–(19) closely approximates this property for discrete time steps ε>0 with a global error of order ε2 corrected by the Metropolis update in Step 3.
The acceptance ratio can only be maintained at exactly one if the proposal trajectory evolution were continuous. However, due its discretization into individual steps, the acceptance probability always deviates from one due to discretization errors. The length of the proposal sequence can then be tuned using ε>0 and L to achieve a desired acceptance rate, analogously to the RW environment. The Hamiltonian dynamics approximately keeps the joint density π(θ, γ) of θ and γ constant, permitting changes in the marginal density π(θ). Due to this feature, the proposal sequence does not move along a “straight” trajectory in the parameter space Θ of θ but rather along a “curve.” This ensures that the proposal sequence does not travel “too far” into the tails and stays in regions with non-zero probability. The ESS is a useful diagnostic tool in this respect: proposals accepted too far from the current state would result in near-independent MCMC draws, bringing the ESS value close to the number of MCMC iterations, but we have not seen such phenomenon occur. Each proposal sequence in HMC and its extensions starts with a “refreshment” of the kinetic auxiliary variable γ newly drawn from N(0, M) where M is the mass matrix. This draw determines the direction in which the proposal sequence propagates through the parameter space. The stochastic nature of γ prevents the chain from getting stuck at the original point or too close to it.
RMHMC
The HMC features proposal dynamics that are based on the Hamiltonian equation of motion (14). The RMHMC is an extension of HMC that results from replacing the mass matrix M in the Hamiltonian equation (14) by the Fisher information matrix F(θ) of the underlying likelihood π(θ). This leads to the augmented Hamiltonian equation
The Hamiltonian equation (21) is non-separable in θ since its derivative with respect to γ
is a function of (γ, θ), and its derivative with respect to θ
is also a function of (γ, θ). Since both these derivatives also contain the Fisher information matrix F(θ) (and the latter one also its inverse F(θ)–1 and derivative ∂F(θ)/∂θIwith respect to θi for each i) while F(θ) is a function of θ, then F(θ), F(θ)–1, and ∂F(θ)/∂θI for each i have to be recomputed at each step during the proposal sequence to obtain the directional dynamics for the proposal given by the derivatives of the Hamiltonian (21). This feature renders RMHMC computationally intensive.
AUHMC Mechanism
The AUHMC is also an extension of HMC, but here the mass matrix in the Hamiltonian equation (14) is replaced by 
that is fixed constant for the entire leapfrog multi-step proposal sequence 
Since the derivative of the resulting Hamiltonian with respect to γ is only a function of γ and the derivative with respect to θ is only a function of θ, the proposal dynamics are not burdened by recomputing F(θ), which is only necessary to obtain at the final proposal point 
This alleviates much of the computational burden necessitated in RMHMC.
As the AUHMC principle is detailed in the main text, here we provide a heuristical description of its implementational algorithm. At the current values of the parameter MCMC draws (θr, γr), first “refresh” the momentum parameter γ by drawing a new value 
from the normal distribution with mean 0 and variance 
Then obtain the next step 
of the proposal sequence by taking a half-step in γ, full step in θ and a half-step in γ as given by the following equations, with k=0:
Take such next step L times in total, for k=0,…,L–1, arriving at 
which gives us the proposal 
Update the mass matrix 
with the new Fisher 
Repeat running the proposal sequence until convergence of 
on to a fixed point. Accept the final proposal 
as the next MCMC draw (θr+1, γ +1) with probability
where ϕ(‧) is the normal density function.
Appendix B: The AUHMC algorithm
Initialize current θ
for r=1 to R
{
initialize 
j=0
(j loop) do while
{
draw 
for j=0
and 
for j>0
j=j+1
(k loop) for k=1 to L
{
}
}
draw u~U[0, 1]
if (α*<u) then
else {θr+1=θ}
}
Appendix C: Proof of Lemma 1 and Theorem 1
Proof of Lemma 1
The AUHMC mapping is a special case of an implicit Runge-Kutta method (Leimkuhler and Reich, 150–151]. Hence, under our Assumptions 2 and 3, the proof of existence of a unique solution is given by Theorem 7.2 of Hairer, Norsett, and Wanner(1993, 206). Specifically, there exists a unique solution to the mapping Tk defined by (7)–(9) which can be obtained by iteration resulting in the repeated use of the triangle inequality that results from the Lipschitz condition satisfying a contraction mapping property.
Proof of Theorem 1
Recall that AUHMC constructs a distant proposal sequence 
in a sequence of k=1,…,L steps. For a given k (omitting the subscripts r denoting the MCMC steps), define the mapping 
of (θk, γk) into (θk+1, γk +1) as:
The coefficient notation for 
corresponds to the general setup of an implicit partitioned Runge-Kutta scheme of Leimkuhler and Reich (2004, 150–151). Here, all 
are equal to zero unless stated otherwise. Moreover, if in the summation sign the upper index is smaller than the lower index, then the corresponding coefficient 
or 
is equal to zero. The Hamiltonian 
for each k is given by
with
where the right-hand side is defined in (7), and 
and 
are implicitly determined in 
We will next state the definitions of an adjoint mapping (Leimkuhler and Reich 2004, 84).
Definition 1 The mapping
defined by
is called the adjoint mapping of
Equivalently, given
its adjoint is defined by
Given 
as defined above, its adjoint 
takes the form
We next proceed to symmetric compositions of mappings with their adjoints.
Definition 2 A mapping 
is called symmetric if 
i.e. 
The symmetry of 
then implies its time-reversibility (Leimkuhler and Reich 2004, 87). Knowing a mapping 
and its adjoint 
a symmetric mapping 
is obtained by composition (concatenation) of the two methods
even if neither 
nor 
are symmetric individually (Leimkuhler and Reich 2004, 84). The following Lemma provides a simple extension of this result.
Lemma 2. Given a symmetric mapping 
the mapping
is also symmetric.
Proof.
which satisfies the definition of a symmetric mapping. ■
Note that since the adjoint of the adjoint is the original mapping, i.e. 
Lemma 2 can be also equivalently stated as 
being symmetric.
For L even, let m=L/2, k=m and define the mapping
which, using (22), is symmetric. Then, let
and further
for m=1, …, L/2–1. The final composite mapping 
then takes the form
Symmetry of 
follows by repeated application of Lemma 2.
The mappings 
and 
are special cases of an implicit partitioned Runge-Kutta method (Leimkuhler and Reich 2004, 150–151] and thus the existence and uniqueness of their solutions follows from Lemma 1. The uniqueness of the solution to 
and 
for each k implies that there is a unique solution to 
Such solution is equivalent to the one given by AUHMC since the AUHMC fixed-point 
is identical to the fixed point 
of 
that solves 
Since, by Lemma 1, the solution to AUHMC is unique, AUHMC implements 
which is a symmetric and time reversible mapping, yielding the detailed balance condition of Theorem 1.
Equivalently, from the definition of 
it follows directly that reversing the momentum at 
and applying AUHMC solves 
which, due to symmetry of 
equals 
following the same proposal path back to (θr, γr) having negated the momentum again after the final step. This satisfies the definition of reversibility for AUHMC.
We can make an analogy between the pair of Euler B and A methods (Leimkuhler and Reich 2004, 84) and the pair of 
and 
In the former pair, the difference is in the point at which we evaluate directional derivatives (θk or θk+1). In the pair of 
and 
the difference is in the number of HMC steps needed to reach 
and 
which at the solution equal to θ0 and θL respectively, but the directional derivatives are always the same, taken with Mk evaluated at the endpoints of the proposal sequence. However, neither 
nor 
is symmetric on its own, and hence we need their concatenation to attain symmetry of the composite mapping.
At the implicit solution of ,
in analogy to the Euler-B method. Also, due to the symmetry of Mk in 
and 
from Assumption 1, at the solution of 
in analogy to the Euler-A method. These half-steps are performed by AUHMC during its proposal sequence.
M-H acceptance probability
The derivation of the M-H acceptance probability form is standard in the HMC literature and we merely adapt it to the AUHMC notation below. Denote by 
the proposal density and by 
the reverse proposal density. Given 
is constructed by the method of change of variables based on the sequence of steps given by the AUHMC mapping Tk for k=1, …, L. Since Tk is deterministic, placing the Dirac delta δ(‧,‧)=1 unit probability mass at each 
applying successive transformations Tk yields
where 
denotes the Jacobian matrix of the transformation Tk with respect to 
and 
for each k=1, …, L.
Denote by 
the reverse mapping obtained from Tk by reversing the signs in the Hamiltonian proposal dynamics. Then
with 
Conditional on 
satisfying Assumption 1, the leapfrog transformation defined by (7)–(9) satisfies
Then
for each k = 1, …, L and hence 
The ratio in the acceptance probability (2) then satisfies detailed balance in the Metropolis form
since all the Jacobian terms cancel out due to (26). By definition of the Hamiltonian equation in (3), the ratio in (27) is then equivalent to
Appendix D: Fisher information for the multivariate normal density
For the univariate case,
and for the multivariate case
where Dm is the duplication matrix (Magnus and Neudecker 2007). In our empirical application we used the numerical approximation to the diagonal of F(θ) instead of the full matrix for faster speed of the MC runs.
Appendix E: Summary statistics and results
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©2013 by Walter de Gruyter Berlin Boston
Articles in the same Issue
- Masthead
- Masthead
- Bayesian adaptively updated Hamiltonian Monte Carlo with an application to high-dimensional BEKK GARCH models
- Off-the-record target zones: theory with an application to Hong Kong’s currency board
- Nonlinear and nonparametric modeling approaches for probabilistic forecasting of the US gross national product
- Maximum likelihood estimation of continuous time stochastic volatility models with partially observed GARCH
- A value-at-risk analysis of carry trades using skew-GARCH models
- Income taxes and endogenous fluctuations: a generalization
Articles in the same Issue
- Masthead
- Masthead
- Bayesian adaptively updated Hamiltonian Monte Carlo with an application to high-dimensional BEKK GARCH models
- Off-the-record target zones: theory with an application to Hong Kong’s currency board
- Nonlinear and nonparametric modeling approaches for probabilistic forecasting of the US gross national product
- Maximum likelihood estimation of continuous time stochastic volatility models with partially observed GARCH
- A value-at-risk analysis of carry trades using skew-GARCH models
- Income taxes and endogenous fluctuations: a generalization
