How Markov Chain Monte Carlo Gets Posterior Distributions

Markov Chain Monte Carlo MCMC

A set of potent statistical techniques known as Markov Chain Monte Carlo (MCMC) approaches uses Markov chains to generate samples from a probability distribution. By averaging over the produced samples, this method is very useful for calculating predicted values.

The fundamental idea behind MCMC is building a Markov chain in which detailed balance, or reversibility, is a sufficient condition for a desired distribution to be its stationary distribution. A transition from one state to another should therefore have an equal chance, and vice versa. This target distribution is usually the posterior distribution of the parameters given the observed data in the context of Bayesian inference.

You can also read Barren Plateaus Quantum With Dissipative Computation & Noise

How MCMC Works: The Metropolis-Hastings (MH) Algorithm

The Metropolis-Hastings (MH) Algorithm: How MCMC Operates Rejection sampling can be used to reach the detailed balancing condition. A generic framework known as the Metropolis-Hastings (MH) method splits the transition distribution into two sections:

A proposal distribution q(θ’|θ) that, given the present state θ, proposes a new state θ’.
The proposed state is either accepted or rejected based on an acceptance/rejection step a(θ’, θ). To maintain the flexibility in selecting the proposal distribution, an adjustment is applied at this stage. The chain stays in its present state θ(i-1) unless the proposed state θ is accepted, in which case it becomes the next state in the chain. Since the chain has not yet reached equilibrium, a number of initial samples referred to as the burn-in period are usually deleted. Additionally, to lessen correlation between successive data, which can skew computations, thinning (preserving only every Lth sample) can be employed.

Random Walk Metropolis (RWM)

Metropolis of Random Walks (RWM) The Random Walk Metropolis (RWM) method is a condensed form of MH. q(θ’|θ) = q(θ|θ’), assuming a symmetric proposal distribution. A Gaussian proposal centred on the present state is a popular option for this. RWM functions as a random walk, which means that the direction and speed of exploration are not tailored to the target function, even if it is based on stochastic exploration adjusted for probability density. Proposals may be excessively conservative (slow convergence) or too daring (high rejection rates), which can result in delayed mixing and inefficiency, particularly in high-dimensional domains. RWM can work well for simpler situations despite its simplicity.

You can also read IBM Quantum Releases Qiskit SDK v2.1 for Quantum Advantage

Hamiltonian Monte Carlo (HMC)

Monte Carlo Hamiltonian (HMC) A sophisticated MCMC technique called Hamiltonian Monte Carlo (HMC), sometimes referred to as Hybrid Monte Carlo, was created to get beyond the drawbacks of naive random walks, especially in high-dimensional fields. By treating the target distribution as a potential energy and supplementing it with a fake momentum distribution that defines a kinetic energy, HMC uses a physical analogy. This enlarged phase space is then explored under the guidance of classical Hamiltonian dynamics.

Effective Exploration: HMC produces ideas that can deviate greatly from the existing state while still having a high acceptance rate, ideally close to 100%, by simulating this physical evolution using approximate numerical techniques (such as the leapfrog integrator). Its resilience to intricate distribution features is a result of its capacity to take advantage of the target function’s differential structure, or gradients.
Hyperparameters: The stepsize (ϵ) and path length (L), which together define the integration time, are two examples of hyperparameters that affect HMC performance.
Difficulties: HMC can become stuck in isolated energy minima and is not naturally suited for multi-modal distributions. Additionally, target distributions that contain discontinuities or points of null probability may present challenges since they erect “potential barriers” that prevent further investigation. When problems like high curvature impact numerical stability and lead to divergent trajectories, HMC provides strong diagnostic tools to identify them. Although it may not be effective, screening for low acceptance rates and performing a RWM step if discovered is one possible solution for null-points.

You can also read Parameterized Quantum circuits enhance QFFN-BERT converter

MCMC within Sequential Monte Carlo (SMC)

Sequential Monte Carlo (SMC) with MCMC It is possible to integrate MCMC techniques (such RWM or HMC) as “move steps” or Markov kernels into an SMC framework to combine the advantages of MCMC’s resilience with SMC’s capacity for online estimation and multi-modality handling.

Unlike other straightforward SMC resamplers, such as the Liu-West filter, which only keep the first two moments, approach maintains the full distribution.
It successfully resolves problems with multimodality capture and makes it possible for the algorithm to manage intricate situations in which normalcy assumptions are violated.
Since the particle cloud is already an approximation of the intended distribution, MCMC motions in SMC don’t need a burn-in period. Additionally, it eliminates the need to eliminate lag samples for formal correctness.
The particle cloud’s structure can also help with MCMC kernel tuning, like estimating the covariance of the target distribution for the HMC mass matrix.
Because all data must be kept, this combination increases the algorithm’s scalability and robustness at the expense of much higher likelihood evaluations and memory needs.

You can also read Quantum Mechanical Calculations of bio & chemical processes

The use of such sophisticated statistical techniques, such as Bayesian inference and different numerical techniques like sequential importance resampling, MCMC, and Gaussian rejection filtering, to quantum device calibration is highlighted in the news article “Advanced Statistical Methods Dramatically Improve Quantum System Characterization And Calibration.” which was published by Quantum News on July 13, 2025.

Research Focus: Ramoa, Santagati, and Wiebe’s work focusses on improving the precision and effectiveness of quantum device calibration, especially for IBMQ’s superconducting qubits.
Improved Performance: Without the need for further measurements, their work showed notable reductions in uncertainty, with factors of 10 in Hahn echo experiments and 3 in Ramsey studies.In certain cases, they got comparable results with 99.5% less data. This is important when data collection is costly or time-consuming.
Challenging Scenarios: The techniques are especially well-suited for describing intricate quantum systems, particularly those displaying open quantum dynamics, in which information may be difficult to obtain or complex.
Simulation and Hardware Experiments: The researchers used real IBMQ quantum hardware to evaluate their techniques for iterative phase estimation and precession dynamics, as well as to simulate these experiments.
- Hahn echo experiments (for T₂ estimate), which showed that learning was enhanced by sorting data according to descending evolution time. Estimating the energy relaxation time, or T₁.
- Ramsey experiments, which show quantum-enhanced estimation even with high decoherence (for detuning and coherence time estimation, T₂).
- Tested adaptive experimental design that sped up learning and reduced uncertainty through echoed Ramsey trials (Hahn-Ramsey sequences).
Addressing Multimodality: The study also tackled multi-parameter estimating issues that result in multi-modal likelihood functions (such as sum-of-cosines likelihoods), which are difficult for less complex approaches to handle. For these situations, Tempered Likelihood Estimation (TLE) with HMC move steps shown greater resilience.
Data Subsampling: Subsampling techniques were investigated in order to save processing expenses, particularly for probability assessments. Subsampling decreased assessments by 60% for Ramsey experiments without raising uncertainty. This is frequently used in conjunction with control variables to lessen the volatility that subsampling introduces. Energy-saving subsampling (where subsampling indices are incorporated in the target density and updated via Metropolis steps) and stochastic gradient HMC (which adds friction to control divergence from noisy gradients) were considered as potential solutions for HMC.