Neural dynamics as sampling: a model for stochastic computation in recurrent networks of spiking neurons

The proposed neural network model implements MCMC sampling of a given distribution in a spiking neural network, both for discrete and continuous time.

“We propose a neural network model and show by a rigorous theoretical analysis that its neural activity implements MCMC sampling of a given distribution, both for the case of discrete and continuous time.”

• Also stated in: § Introduction · plain

  “Formally speaking, our model implements Markov chain Monte Carlo (MCMC) sampling in a spiking neural network.”

The proposed family of network models can be shown to exactly sample from any arbitrary member of a well-defined class of probability distributions via their inherent network dynamics.

“we propose here a family of network models, that can be shown to exactly sample from any arbitrary member of a well-defined class of probability distributions via their inherent network dynamics.”

If a subset of variables are observed (clamped), the remaining network samples from the conditional distribution of the remaining variables given the observables.

“If a subset of the variables are observed, modelled as the corresponding neurons being “clamped” to the observed values, the remaining network samples from the conditional distribution of the remaining variables given the observables.”

Each stochastic state transition operator leaves the given distribution invariant.

“It can be shown that each of these stochastic state transition operators leaves the given distribution invariant, i.e., satisfies equation (1).”

In contrast to prior approaches, the model incorporates irreversible dynamics (i.e., no detailed balance) allowing for finite time PSPs and refractory mechanisms.

“In contrast to prior approaches however, our model incorporates irreversible dynamics (i.e., no detailed balance) allowing for finite time PSPs and refractory mechanisms.”

For a Boltzmann distribution, the neural computability condition is satisfied by neurons with the standard membrane potential (6).

“For the Boltzmann distribution, condition (4) is satisfied by neurons with the standard membrane potential(6)”

• AssumptionThe authors state this as an assumption: The membrane potential of a neuron at time t equals the log-odds of the corresponding binary variable being active (the neural computability condition).

Spiking network models with standard membrane potential can emulate any Boltzmann machine.

“As shown below, these spiking network models can emulate any Boltzmann machine (BM) [36].”

• cites [36] DH Ackley, 1985, A learning algorithm for boltzmann machines., Cogn Sci sourceNot checked yet

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The neural sampling process has the Markov property only with regard to the auxiliary variables.

“The neural sampling process has the Markov property only with regard to these more informative auxiliary variables .”

Under a certain computability assumption, a network of spiking neurons can sample from a given joint distribution over binary variables using its inherent stochastic dynamics.

“We will show that under a certain computability assumption on a network consisting of spiking neurons can sample from using its inherent stochastic dynamics.”

• AssumptionThe authors state this as an assumption: The membrane potential of a neuron at time t equals the log-odds of the corresponding binary variable being active (the neural computability condition).

Once stochastic dynamics converge to invariant distribution, the averaged firing rate of neuron i is proportional to the desired marginal probability.

“Then, as soon as the stochastic dynamics of has converged to its invariant distribution, the averaged firing rate of neuron is proportional to the following desired marginal probability”

The simulated network features a sparse, irregular spike response.

“The network features a sparse, irregular spike response with average firing rate of .”

Statistics computed from the model match experimentally observed statistics well.

“In general, the statistics computed from the model match experimentally observed statistics well.”

To represent the correct distribution over x conditioned on others, the neuron continuously adapts its instantaneous firing rate.

“To represent the correct distribution over conditioned on , the neuron continuously adapts its instantaneous firing rate.”

For any refractory function, there always exists a continuous, monotonic function which satisfies equation (11).

“One can show that, for any as above, there always exists a continuous, monotonic function which satisfies this equation (see Lemma 0 in Methods).”

The standard activation function is the solution of equation (11) for the absolute refractory function.

“It is worth noticing that the standard activation function is the solution of equation (11) for the absolute refractory function, i.e., for and for .”

If the sigmoidal activation function is replaced by a different activation function, one can still prove that sampling is "locally correct" as specified in equation (12).

“Nevertheless, if the sigmoidal activation function is replaced by a different activation function , one can still prove that the sampling is “locally correct”, as specified in equation (12) below.”

Networks consisting of neurons with relative refractory mechanism approximate the desired target distributions faithfully over a large range of distribution parameters.

“In the Methods section, we present further simulation results showing that the proposed networks consisting of neurons with relative refractory mechanism approximate the desired target distributions faithfully over a large range of distribution parameters.”

The unique invariant distribution of the relative refractory transition operator gives rise to the correct marginal distribution over x.

“In Lemma 0 in Methods it is proven that the unique invariant distribution of , denoted as , gives rise to the correct marginal distribution over , i.e.”

If several neurons in the network change their states in a short interval of time, the joint distribution over x is in general not the desired one.

“If however, several neurons in the network change their states in a short interval of time, the joint distribution over is in general not the desired one, i.e., , where denotes the invariant distribution of .”

In simulations, stochastic deviations of estimated probabilities due to finite samples are small and comparable to systematic deviations due to only locally correct computation of neurons with relative refractory mechanism.

“Stochastic deviations of the estimated probabilities due to the finite number of samples are quite small (typical errors ) and are comparable to systematic deviations due to the only locally correct computation of neurons with relative refractory mechanism.”

Simulation results indicate that the error of the approximation to the desired Boltzmann distributions introduced by neural sampling with relative refractory mechanism is rather minute.

“In the Methods section, we present simulation results that indicate that the error of the approximation to the desired Boltzmann distributions introduced by neural sampling with relative refractory mechanism is rather minute.”

For the relative refractory model, one can no longer prove that a network of such neurons samples from the desired distribution p.

“The introduction of this more complex neuron model comes at the price that one can no longer prove that a network of such neurons samples from the desired distribution .”

The neural sampling approximation error is orders of magnitude below the one introduced by a fully factorized distribution.

“It is shown that the neural sampling approximation error is orders of magnitudes below the one introduced by a fully factorized distribution (which amounts to assuming correct marginal distributions and independent neurons).”

The sampling process is expressed in the neural network model by a sparse, asynchronous and irregular spike response.

“One sees that the sampling process is expressed in this neural network model by a sparse, asynchronous and irregular spike response.”

In simulations, the onset of a sensory stimulus reduces the variability of firing activity, which represents a conditional distribution rather than the prior distribution after stimulus onset.

“Using this admittedly quite simplistic model of the input, we observed in simulations that our network model exhibits the following property: The onset of a sensory stimulus reduces the variability of the firing activity, which represents (after stimulus onset) a conditional distribution, rather than the prior distribution (see the difference between panels D and E of Figure 5.”

More realistic alpha-shaped, additive EPSPs are compatible with the functionality of the proposed network model.

“Furthermore, we have presented simulation results suggesting that more realistic alpha-shaped, additive EPSPs are compatible with the functionality of the proposed network model.”

The distribution of dominance durations in the model can be approximated quite well by a Gamma distribution.

“This distribution can be approximated quite well by a Gamma distribution, which also provides a good fit to experimental data (see the discussion in [18]).”

• AssumptionThe authors state this as an assumption: The orientation of the population vector corresponds to the dominant orientation of the percept, and its distance from the origin encodes the strength of this percept.

With ambiguous input, the network spends most of the time in network states corresponding to one of the two simultaneously presented input orientations and virtually no time on orientations in between.

“One clearly sees that the network spends most of the time in network states that correspond to one of the two simultaneously presented input orientations ( and ), and virtually no time on orientations in between.”

• AssumptionThe authors state this as an assumption: The neurons in the network represent neurons in the visual system that have roughly the same or neighboring receptive field, and each neuron receives visual input from either the left or the right eye.
• AssumptionThe authors state this as an assumption: The orientation of the population vector corresponds to the dominant orientation of the percept, and its distance from the origin encodes the strength of this percept.

In computer simulations, the two temporal processes considered (refractory processes and postsynaptic potentials) can approximately be decoupled.

“We have gone one first step in this direction by showing that in computer simulations the two temporal processes that we have considered (refractory processes and postsynaptic potentials) can approximately be decoupled.”

The simulated firing statistics are in accordance with experimentally observed data.

“The simulated data is in accordance with experimentally observed data.”

A focus of network activity on the two orientations given by the external input can be achieved in this model despite only two neurons being clamped for each of them.

“It is furthermore notable, that a focus of the network activity on the two orientations that are given by the external input can be achieved in this model, in spite of the fact that only two of the neurons were clamped for each of them.”

• AssumptionThe authors state this as an assumption: The neurons in the network represent neurons in the visual system that have roughly the same or neighboring receptive field, and each neuron receives visual input from either the left or the right eye.

The average firing rate when sampling from the posterior distribution is only slightly higher than the average firing rate of spontaneous activity.

“It is worth mentioning that the average firing rate when sampling from the posterior distribution is only slightly higher than the average firing rate of spontaneous activity ( and respectively), which is reminiscent of related experimental data [7].”

is an invariant distribution of .

“Lemma 8. is an invariant distribution of , i.e., it is a solution to the invariant Fokker-Planck equation:”

• AssumptionThe authors state this as an assumption: The state space of is for .

Due to the structured recurrent weights, the experimentally measured tuning curves were reasonably close to the tuning curves used for external stimulation.

“Due to the structured recurrent weights, the experimentally measured tuning curves were found to be reasonably close to the tuning curves used for external stimulation.”

• AssumptionThe authors state this as an assumption: The variables are observed with iid. uniformly in if and otherwise.

For Boltzmann distributions with intermediate weights (), there are substantial statistical dependencies between the random variables.

“distributions with intermediate weights () show substantial statistical dependencies between RVs.”

The absolute refractory model provides the best results due to the theoretical guarantee to sample from the correct distribution.

“The absolute refractory model provides the best results as we expected due to the theoretical guarantee to sample from the correct distribution (the non-zero Kullback-Leibler divergence is caused by the estimation from a finite number of samples).”

For any continuous, non-negative function with for , there exists a unique function with property (16).

“Lemma 9. Let be a continuous, non-negative function with for . There exists a unique function with the following property : (16)”

• AssumptionThe authors state this as an assumption: The state space of is for .

Relative refractory models are much more accurate approximations than fully factorized distributions if substantial statistical dependencies between the random variables are present.

“These relative refractory models are characterized by the theory to be “locally correct” and turn out to be much more accurate approximations than fully factorized distributions if substantial statistical dependencies between the RVs are present (i.e., , ).”

Most of the neurons responded in a highly irregular manner with a CV .

“Most of the neurons responded in a highly irregular manner with a CV .”

Networks consisting of neurons with relative refractory mechanisms, with only "locally" correct sampling, also provide a close fit to the true distribution compared to a fully factorized approximation.

“Networks consisting of neurons with relative refractory mechanisms, with only “locally” correct sampling, also provide a close fit to the true distribution (see inset) compared to a fully factorized approximation (assuming correct marginals and independent variables).”

Applying an ambiguous cue led to a bimodal posterior distribution.

“This led to a bimodal posterior distribution as shown in panel (E).”

• AssumptionThe authors state this as an assumption: The variables are observed with iid. uniformly in if and otherwise.

Sampling networks with more realistic, alpha-shaped, additive PSPs still fit the true distribution reasonably well.

“Furthermore, it can be seen that sampling networks with more realistic, alpha-shaped, additive PSPs still fit the true distribution reasonably well.”

The models with relative refractory mechanism provide faithful approximations for all values of the hyperparameter considered here.

“The models with relative refractory mechanism provide faithful approximations for all values of the hyperparameter considered here.”

Networks with absolute refractory mechanism provide the best approximation.

“Networks with absolute refractory mechanism provide the best approximation (as expected from theoretical guarantees).”

The operator leaves the conditional distribution invariant with .

“Lemma 7. The operator leaves the conditional distribution invariant with , i.e.:”

• AssumptionThe authors state this as an assumption: The state space of is for .

Systematic deviations due to the relative refractory mechanism are on the same order as the effect of estimating from finite samples.

“Systematic deviations due to the relative refractory mechanism are on the same order as the effect of estimating from finite samples (as can be seen, e.g., from a comparison with the absolute refractory model which has 0 systematic error).”

The Kullback-Leibler divergence to the true distribution is clearly higher for additive, alpha-shaped PSPs compared to renewed, rectangular PSPs.

“The Kullback-Leibler divergence to the true distribution is clearly higher compared to the case of renewed, rectangular PSPs.”

For very large weights (), the probability mass of the distributions is concentrated on very few states (usually 90% on less than 10 out of the states).

“For very large weights (), the probability mass of the distributions is concentrated on very few states (usually 90% on less than 10 out of the states).”

For all the invariant distribution of the operator fulfills .

“Lemma 10. For all the invariant distribution of the operator fulfills .”

• AssumptionThe authors state this as an assumption: The state space of is for .

The sampling network represented the bimodal distribution by encoding either global percept separately: the trace of network states roamed in one mode for multiple steps before quickly crossing the state space towards the opposite percept.

“The sampling network represented this distribution by encoding either global perception separately: The trace of network states roamed in one mode for multiple steps before quickly crossing the state space towards the opposite percept.”

• AssumptionThe authors state this as an assumption: The variables are observed with iid. uniformly in if and otherwise.

A late recovery of the refractory function is beneficial for the approximation quality of the model as it is closer to an absolute refractory mechanism.

“As expected, a late recovery of the refractory function is beneficial for the approximation quality of the model as it is closer to an absolute refractory mechanism.”

The paper introduces a different approach based on non-reversible Markov chains that reflects inherent temporal processes of spiking neuronal activity through a suitable choice of random variables.

“we introduce a different approach based on non-reversible Markov chains that is able to reflect inherent temporal processes of spiking neuronal activity through a suitable choice of random variables.”

Common methods for MCMC sampling in distributed systems, such as Gibbs sampling, are inconsistent with the dynamics of spiking neurons.

“Since common methods for MCMC sampling in distributed systems, such as Gibbs sampling, are inconsistent with the dynamics of spiking neurons, we introduce a different approach based on non-reversible Markov chains that is able to reflect inherent temporal processes of spiking neuronal activity through a suitable choice of random variables.”

Under some conditions, the stochastic firing activity of networks of spiking neurons can be interpreted as probabilistic inference via Markov chain Monte Carlo (MCMC) sampling.

“Here we create such a link and show that under some conditions the stochastic firing activity of networks of spiking neurons can be interpreted as probabilistic inference via Markov chain Monte Carlo (MCMC) sampling.”

The binary vector defined by network activity at time t does not fully capture the internal state of the stochastic system; additional non-binary auxiliary variables are needed.

“An obvious consequence of this definition is that the binary vector that is defined by the activity of at time does not fully capture the internal state of this stochastic system. Rather, one needs to take into account additional non-binary variables , where the value of at time specifies when within the time interval the neuron has fired (if it has fired within this time interval, thereby causing”

Experimental data suggests that neurons, synapses, and neural systems are inherently stochastic, especially in vivo.

“Experimental data suggests that neurons, synapses, and neural systems are inherently stochastic [1], especially in vivo, and therefore seem less suitable for implementing deterministic computations.”

• cites [1] ET Rolls, 2010, The Noisy Brain: Stochastic Dynamics as a Principle of Brain Function.Not checked yet

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External stimuli only modulate the highly stochastic spontaneous firing activity of cortical networks of neurons.

“several experimental studies arrive at the conclusion that external stimuli only modulate the highly stochastic spontaneous firing activity of cortical networks of neurons [7], [8].”

• cites [7] J Fiser, 2004, Small modulation of ongoing cortical dynamics by sensory input during natural vision., Nature sourceNot checked yet

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• cites [8] DL Ringach, 2009, Spontaneous and driven cortical activity: implications for computation., Curr Opin Neurobiol sourceNot checked yet

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The model is more biologically realistic than Boltzmann machine sampling approaches because it incorporates inherent temporal dynamics and spike-based communication of a network of spiking neurons.

“It is similar to sampling approaches which have already been applied extensively, e.g., in Boltzmann machines, however our model is more biologically realistic as it incorporates aspects of the inherent temporal dynamics and spike-based communication of a network of spiking neurons.”

Any composition or mixture of these operators also leaves the distribution invariant.

“This implies that any composition or mixture of these operators also leaves invariant, see, e.g., [38].”

• cites [38] C Andrieu, 2003, An introduction to MCMC for machine learning., Mach Learn sourceNot checked yet

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Previous network models that implement sampling in neural networks are mostly based on Gibbs (or general Metropolis-Hastings) sampling.

“Previous network models that implement sampling in neural networks are mostly based on a special sampling algorithm called Gibbs (or general Metropolis-Hastings) sampling [9], [17], [18], [37].”

• cites [9] S Geman, 1984, Stochastic relaxation, gibbs distributions, and the bayesian restoration of images., IEEE Trans Pattern Anal Mach Intell sourceNot checked yet

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• cites [17] R Sundareswara, 2008, Perceptual multistability predicted by search model for bayesian decisions., J Vis sourceNot checked yet

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• cites [18] SJ Gershman, 2009, Perceptual multistability as Markov chain Monte Carlo inference.Not checked yet

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The local computation performed by such a neuron can be considered correct.

“Hence the “local” computation performed by such neuron can be considered as correct.”

The continuous-time stochastically spiking neural network model's states represent correct samples from the desired probability distribution at any time.

“Here we propose a stochastically spiking neural network model in continuous time, whose states still represent correct samples from the desired probability distribution at any time .”

The irreducibility and aperiodicity of A ensures that p is the unique invariant distribution of the Markov chain defined by A, i.e., irrespective of initial network state successive application explores the whole state space non-periodically.

“The statement that (which is composed of the operators ) is irreducible and aperiodic ensures that is the unique invariant distribution of the Markov chain defined by , i.e., that irrespective of the initial network state the successive application of explores the whole state space in a non-periodic manner.”

By clamping observed neurons, the network can sample from the posterior distribution over remaining variables given observed values.

“For example, can be used to sample from the posterior distribution over given . One just needs to clamp those neurons to the corresponding observed values.”

The network can carry out probabilistic inference for the distribution p.

“Furthermore, the network can carry out probabilistic inference for this distribution.”

After sufficient burn-in (infinite limit), the dynamics given by transition operator A produce samples from distribution p.

“This theorem guarantees that after a sufficient “burn-in” time (more precisely in the limit of an infinite “burn-in” time), the dynamics of the network, which are given by the transition operator , produce samples from the distribution .”

A neuron whose dynamics is described by the relative refractory operator samples from the correct distribution if it receives a static input from other neurons, i.e., as long as its membrane potential is constant.

“This means that a neuron whose dynamics is described by samples from the correct distribution if it receives a static input from the other neurons in the network, i.e., as long as its membrane potential is constant.”

A large number of different probabilistic inferences can be carried out efficiently in parallel by readout neurons that receive spike input from different subsets of neurons in the network.

“Thus, a large number of different probabilistic inferences can be carried out efficiently in parallel by readout neurons that receive spike input from different subsets of neurons in the network .”

The spiking network represents the target probability distribution with high accuracy.

“The tight match between both distributions suggests that the spiking network represents the target probability distribution with high accuracy.”

Rather realistic neural dynamics can support computational functions rigorously formalized as inference via sampling.

“The main point of the modelling study is to show that rather realistic neural dynamics can support computational functions rigorously formalized as inference via sampling.”

On the basis of overall network activity, it is indistinguishable whether the network carries out an inference task or freely samples from its prior distribution.

“Thus on the basis of the overall network activity it is indistinguishable whether the network carries out an inference task or freely samples from its prior distribution.”

For the continuous-time relative refractory sampler, each neuron locally performs the correct computation under the assumption of static input from the remaining neurons.

“It is shown that each neuron “locally” performs the correct computation under the assumption of static input from the remaining neurons.”

For the continuous-time relative refractory sampler, one can no longer prove in general that the global network samples from the target distribution.

“However one can no longer prove in general that the global network samples from the target distribution .”

The perceptual system of humans and nonhuman primates does not produce a superposition of different possible percepts of an ambiguous stimulus, but switches between different self-consistent global percepts spontaneously.

“The most important finding is that the perceptual system of humans and nonhuman primates does not produce a superposition of different possible percepts of an ambiguous stimulus, but rather switches between different self-consistent global percepts in a spontaneous manner.”

The approach presented here can be interpreted as a biologically more realistic version of Gibbs sampling for a specific class of probability distributions by taking into account spike-based communication, finite duration PSPs and refractory mechanisms.

“The approach presented here can be interpreted as a biologically more realistic version of Gibbs sampling for a specific class of probability distributions by taking into account a spike-based communication, finite duration PSPs and refractory mechanisms.”

For all i, the operator T_i leaves the conditional distribution invariant.

“Lemma 2. For all the operator leaves the conditional distribution invariant.”

p* is the unique invariant distribution of operator T.

“Theorem 1. is the unique invariant distribution of operator .”

The time-invariant distribution of the continuous-time model gives rise to the desired marginal distribution over the binary variables.

“In the Methods section we prove that the resulting time invariant distribution, i.e., the distribution that solves , now denoted as it is not a function of time, gives rise to the desired marginal distribution over :(14)”

The model provides one possible neural implementation of the inference-by-sampling paradigm.

“The model, based on assumptions about the underlying probability distribution (formalized by the neural computability condition) as well as on certain assumptions regarding the underlying MCMC model, provides one possible neural implementation of the “inference-by-sampling paradigm” emerging in computational neuroscience.”

For all i, the operator T_i leaves the distribution p* invariant.

“Lemma 3. For all the operator leaves the distribution invariant.”

The experimentally observed stability of percepts on the time scale of seconds suggests that perception can be interpreted as probabilistic inference carried out by MCMC sampling producing successively correlated samples.

“Furthermore, the experimentally observed fact that percepts tend to be stable on the time scale of seconds suggests that perception can be interpreted as probabilistic inference that is carried out by MCMC sampling which produces successively correlated samples.”

A theoretically rigorous probabilistic inference process can be carried out by a network of spiking neurons with a spike raster similar to generic recorded data.

“Altogether this figure shows that a theoretically rigorous probabilistic inference process can be carried out by a network of spiking neurons with a spike raster that is similar to generic recorded data.”

In the limit of the discrete time step approaching zero, the discrete-time sampler becomes equal to the continuous-time system, which updates all units in parallel.

“In the limit of approaching , the probability that two or more neurons will fire at the same time approaches , and the discrete time sampler becomes equal to the continuous time system defined above, which updates all units in parallel.”

The continuous-time spiking model performs sampling from a given probability distribution.

“We have also presented a model of spiking dynamics in continuous time that performs sampling from a given probability distribution.”

The network's behavior with ambiguous input implements a sampling process from a bimodal conditional distribution.

“This implements a sampling process from a bimodal conditional distribution.”

• AssumptionThe authors state this as an assumption: The neurons in the network represent neurons in the visual system that have roughly the same or neighboring receptive field, and each neuron receives visual input from either the left or the right eye.
• AssumptionThe authors state this as an assumption: The orientation of the population vector corresponds to the dominant orientation of the percept, and its distance from the origin encodes the strength of this percept.

The previously described model for neural sampling could in principle fill the gap, providing a modelling framework consistent with the dynamics of networks of spiking neurons and understandable as probabilistic inference through MCMC sampling.

“We have demonstrated in a computer simulation that the previously described model for neural sampling could in principle fill this gap, providing a modelling framework that is on the one hand consistent with the dynamics of networks of spiking neurons, and which can on the other hand also be clearly understood from the perspective of probabilistic inference through MCMC sampling.”

The neuron model with relative refractory mechanism does not entail the correct overall invariant distribution .

“As stated before, the neuron model with relative refractory mechanism does not entail the correct overall invariant distribution .”

The theory for the neuron model with a relative refractory mechanism only shows that the sampling process is "locally correct", i.e., that it would yield correct conditional distributions for each individual neuron if the state of the remaining network stayed constant.

“In contrast, the theory for the neuron model with a relative refractory mechanism only shows that the sampling process is “locally correct”, i.e., that it would yield correct conditional distributions for each individual neuron if the state of the remaining network stayed constant.”

Neurons with high firing rates had a slightly lower CV due to the increased influence of the refractory mechanism.

“Neurons with high firing rates had a slightly lower CV due to the increased influence of the refractory mechanism”

The joint distribution over x and r in continuous time has marginal distribution over x equal to p*.

“Lemma 6. The distribution has the following marginal distribution:where .”

The structure of the CV-versus-average-ISI plot resembles data from recordings in behaving macaque monkeys, though at a lower average firing rate.

“The structure of this plot resembles, e.g., data from recordings in behaving macaque monkeys [68] (but note the lower average firing rate).”

• cites [68] W Softky, 1993, The highly irregular firing of cortical cells is inconsistent with temporal integration of random epsps., J Neurosci sourceNot checked yet

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For a given refractory function, there exists a unique function phi satisfying property (15).

“Lemma 4. Let be a tuple of non-negative real numbers, with and at least one element . This defines the refractory function via . There exists a unique function with the following property : (15)”

The stationary distribution of the sampling process with relative refractory mechanism only provides an approximation to the target distribution.

“Therefore, the stationary distribution of the sampling process with relative refractory mechanism only provides an approximation to the target distribution.”