Glossary

Acceptance Ratio: The proportion of proposals that are added to the chain in the acceptance step. For a Gaussian target, the optimal ratio is 0.234.
Approximate Bayesian Computation (ABC): A likelihood-free method for inference: data are simulated with parameters chosen with the prior, and simulations too dissimilar to the observed data are discarded (Sisson et al., 2019; Turner & Van Zandt, 2012). In the workshop we used the abc package (Csilléry et al., 2012).
Autocorrelated Bayesian Sampler (ABS): A model in which judgements come from autocorrelated MCMC samples rather than independent ones, giving a single account of probability judgements, estimates, choices, confidence judgements, confidence intervals, and response times (Zhu et al., 2024).
Anchoring: Psychological phenomenon by which estimates are pulled towards values seen before them, even if these cues are irrelevant (Strack & Mussweiler, 1997; Tversky & Kahneman, 1974). In sampling accounts this is explained by the MCMC chain starting from the cued value and few samples being drawn (Lieder et al., 2018). If co-occurs with its opposite, repulsion (Spicer et al., 2022).
Autocorrelated Proposals: One of the qualitative features that compose the available MCMC algorithms in the samplr package, which can be used to avoid random-walk behaviour. See RECHMC
Autocorrelation: When each sample (or response) resembles the ones before it rather than being independent. A key property of MCMC is that its samples will be autocorrelated (Brooks et al., 2011).
Bayesian inference: Combining a prior (existing expectations or biases) with a likelihood (from the incoming stimulus) to obtain a posterior (the resulting perception or judgment), via Bayes’ rule. Computing it exactly is usually intractable, which is what motivates sampling.
Drift Diffusion Model (DDM): Models a choice and its response time as noisy evidence accumulating like a biased random walk until it crosses a decision threshold (Ratcliff et al., 2016; Ratcliff & Rouder, 1998).

Erlang distribution: The distribution of the sum of \(k\) independent exponential variables, with mean \(1/\lambda\) each. In ABS, the response time is made of some non-decision time plus a value from an Erlang decision where \(k\) is the number of samples drawn to make the decision (Zhu et al., 2024).
Gradient-based proposals.: One of the qualitative features that compose the available MCMC algorithms in the samplr package, which can be used to make more efficient proposals. See HMC
Hamiltonian Monte Carlo (HMC): MCMC algorithm that uses the gradient of the distribution to make efficient proposals (Betancourt, 2018; Neal, 2011).

Markov chain Monte Carlo (MCMC): An autocorrelated sampling approach that can sample from a difficult posterior via a biased random walk, visiting each region as often as its probability warrants (Brooks et al., 2011).
Metropolis-coupled MCMC (MC³): This MCMC algorithm runs several chains at different temperatures at once and lets them swap, so the cold chain borrows exploration from the hot ones; the sampler used in the ABS (Geyer, 1991).
Metropolis–Hastings (MH): The basic MCMC algorithm: in each iteration a move is proposed, then accepted or rejected based on its density. When a proposal is rejected, the previous sample is added to the chain.
Multiple Chains: One of the qualitative features that compose the available MCMC algorithms in the samplr package, which can be used to tackle multimodality. See MC\(^3\)
Proposal width: In the typical implementation of MH and MC³, proposals are made by adding random normal noise to the previous sample. This refers to the width of the normal distribution that generates proposals. When this is small, exploration is slow; when it is large the acceptance ratio decreases.
Recycled Hamiltonian Monte Carlo (RECHMC): To avoid random-walk behaviour, instead the momentum parameter in HMC is “recycled”, so that successive momentum parameters are autocorrelated (Horowitz, 1991). This gives the sampler a sense of trajectory over multiple iterations.
Repulsion: The opposite of anchoring: an estimate is pushed away from a nearby preceding value, as often seen in perceptual tasks. Sampling accounts capture anchoring and repulsion together (Spicer et al., 2022).
Resource rationality: The idea that cognition makes optimal use of limited resources; e.g. relying on only a few samples is rational once the cost of sampling is counted Lieder & Griffiths (2020).
samplr: The R package introduced in this workshop, implementing these sampling algorithms as cognitive models (Castillo et al., 2025).
Speed–accuracy trade-off: Responding faster lowers accuracy and vice versa; in accumulation models this is set by the decision threshold (Ratcliff & Rouder, 1998).
Temperature: How flat or peaked a chain treats the distribution. Hot chains roam; cold chains hug the posterior; in MC³ different chains run at different temperatures (Geyer, 1991).

Sampling algorithms and their qualitative features

Algorithm	`samplr` function	M. Chains	Gradients	Autoc. Proposals
Metropolis Hastings (MH) (Hastings, 1970)	`sampler_mh()`	❌	❌	❌
Metropolis-coupled MCMC (MC\(^3\)) (Geyer, 1991)	`sampler_mc3()`	✅	❌	❌
Hamiltonian Monte Carlo (HMC) (Neal, 2011)	`sampler_hmc()`	❌	✅	❌
RECMH*	`sampler_mh(alpha=)`	❌	❌	✅
MCHMC	`sampler_mchmc()`	✅	✅	❌
RECMC3*	`sampler_mc3(alpha=)`	✅	❌	✅
RECHMC (Horowitz, 1991)	`sampler_rec()`	❌	✅	✅
MCREC	`sampler_mcrec()`	✅	✅	✅

* Use these algorithms by setting the alpha parameter to a value other than 0 in the sampler_mh() and sampler_mc3() functions respectively.

References

Betancourt, M. (2018). A Conceptual Introduction to Hamiltonian Monte Carlo (1701.02434). http://arxiv.org/abs/1701.02434 (Pre-published)

Brooks, S., Andrew Gelman, Jones, G., & Meng, X.-L. (Eds.). (2011). Handbook for Markov chain Monte Carlo. Taylor & Francis.

Castillo, L., Li, Y.-X., & Sanborn, A. N. (2025). The samplr package: A tool for modeling human cognition with sampling algorithms. https://doi.org/10.31234/osf.io/ax8hm_v2

Csilléry, K., François, O., & Blum, M. G. B. (2012). Abc: An R package for approximate Bayesian computation (ABC). Methods in Ecology and Evolution, 3(3), 475–479. https://doi.org/10.1111/j.2041-210X.2011.00179.x

Geyer, C. J. (1991). Markov Chain Monte Carlo Maximum Likelihood. Proc. 23rd Symposium on the Interface Interface Foundation, Fairfax Station, 1991, 156–163.

Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57(1), 97–109. https://doi.org/10.1093/biomet/57.1.97

Horowitz, A. M. (1991). A Generalized Guided Monte Carlo Algorithm. Physics Letters B, 268(2), 247–252. https://doi.org/10.1016/0370-2693(91)90812-5

Lieder, F., & Griffiths, T. L. (2020). Resource-rational analysis: Understanding human cognition as the optimal use of limited computational resources. Behavioral and Brain Sciences, 43, e1. https://doi.org/10.1017/S0140525X1900061X

Lieder, F., Griffiths, T. L., M. Huys, Q. J., & Goodman, N. D. (2018). The anchoring bias reflects rational use of cognitive resources. Psychonomic Bulletin & Review, 25(1), 322–349. https://doi.org/10.3758/s13423-017-1286-8

Neal, R. M. (2011). MCMC Using Hamiltonian Dynamics. In S. Brooks, A. Gelman, G. Jones, & X.-L. Meng (Eds.), Handbook of Markov Chain Monte Carlo (1st ed., pp. 113–162). Chapman and Hall/CRC. https://doi.org/10.1201/b10905

Ratcliff, R., & Rouder, J. N. (1998). Modeling Response Times for Two-Choice Decisions. Psychological Science, 9(5), 347–356. https://doi.org/10.1111/1467-9280.00067

Ratcliff, R., Smith, P. L., Brown, S. D., & McKoon, G. (2016). Diffusion Decision Model: Current Issues and History. Trends in Cognitive Sciences, 20(4), 260–281. https://doi.org/10.1016/j.tics.2016.01.007

Sisson, S. A., Fan, Y., & Beaumont, M. A. (Eds.). (2019). Handbook of approximate Bayesian computation. CRC Press, Taylor & Francis Group.

Spicer, J., Zhu, J.-Q., Chater, N., & Sanborn, A. N. (2022). Perceptual and Cognitive Judgments Show Both Anchoring and Repulsion. Psychological Science, 33(9), 1395–1407. https://doi.org/10.1177/09567976221089599

Strack, F., & Mussweiler, T. (1997). Explaining the enigmatic anchoring effect: Mechanisms of selective accessibility. Journal of Personality and Social Psychology, 73(3), 437–446. https://doi.org/10.1037/0022-3514.73.3.437

Turner, B. M., & Van Zandt, T. (2012). A tutorial on approximate Bayesian computation. Journal of Mathematical Psychology, 56(2), 69–85. https://doi.org/10.1016/j.jmp.2012.02.005

Tversky, A., & Kahneman, D. (1974). Judgment under Uncertainty: Heuristics and Biases: Biases in judgments reveal some heuristics of thinking under uncertainty. Science, 185(4157), 1124–1131. https://doi.org/10.1126/science.185.4157.1124

Vul, E., Goodman, N., Griffiths, T. L., & Tenenbaum, J. B. (2014). One and Done? Optimal Decisions From Very Few Samples. Cognitive Science, 38(4), 599–637. https://doi.org/10.1111/cogs.12101

Zhu, J.-Q., Sundh, J., Spicer, J., Chater, N., & Sanborn, A. N. (2024). The autocorrelated Bayesian sampler: A rational process for probability judgments, estimates, confidence intervals, choices, confidence judgments, and response times. Psychological Review, 131(2), 456–493. https://doi.org/10.1037/rev0000427