Source-linked AI summary
Collective Animal Behavior from Bayesian Estimation and Probability Matching
Alfonso Pérez-Escudero, Gonzalo G. de Polavieja
TL;DR
The paper addresses how social decision rules in animal groups can be derived beyond empirical fits, given that animals decide under uncertainty using personal and social information. It models Bayesian estimation followed by probability matching, and reports good correspondence with three-spined stickleback data across three experimental settings, with s = 2.5 fitting all three.
Problem
Animal collective decision rules are understood mainly through empirical fits, while first-principles derivations receive less emphasis despite animals making decisions under uncertainty.
Source: Introduction · p. 2Method
The model combines personal and observed social information in Bayesian estimates, then has each animal choose a behavior with probability equal to its estimated probability of being best.
Source: Introduction · p. 2, Decision rule: Probability matching · p. 4Results
s = 2.5 gives a very good fit to the three-spined stickleback data in all three experimental settings.
Source: Discussion · p. 13Takeaways & Limitations
The model predicts that manipulating reliability parameters a and sk should test whether collective behavior changes as predicted.
Source: Discussion · p. 14The model excludes decision rules that depend on factors or constraints beyond estimation alone.
Source: Decision rule: Probability matching · p. 4
Abstract
from arXiv · showhide
Animals living in groups make movement decisions that depend, among other factors, on social interactions with other group members. Our present understanding of social rules in animal collectives is mainly based on empirical fits to observations, with less emphasis in obtaining first-principles approaches that allow their derivation. Here we show that patterns of collective decisions can be derived from the basic ability of animals to make probabilistic estimations in the presence of uncertainty. We build a decision-making model with two stages: Bayesian estimation and probabilistic matching. In the first stage, each animal makes a Bayesian estimation of which behavior is best to perform taking into account personal information about the environment and social information collected by observing the behaviors of other animals. In the probability matching stage, each animal chooses a behavior with a probability equal to the Bayesian-estimated probability that this behavior is the most appropriate one. This model derives very simple rules of interaction in animal collectives that depend only on two types of reliability parameters, one that each animal assigns to the other animals and another given by the quality of the non-social information. We test our model by obtaining theoretically a rich set of observed collective patterns of decisions in three-spined sticklebacks, Gasterosteus aculeatus, a shoaling fish species. The quantitative link shown between probabilistic estimation and collective rules of behavior allows a better contact with other fields such as foraging, mate selection, neurobiology and psychology, and gives predictions for experiments directly testing the relationship between estimation and collective behavior.
Introduction
Animals make decisions under uncertainty, and the behavior of conspecifics can provide ambiguous or misleading social information. The paper proposes that Bayesian estimation combining personal and social information can derive collective decision rules and connect collective behavior with other areas of animal research.
Introduction
- Probabilistic estimation is a general way to decide under uncertainty, with evidence of use in perception, learning, foraging, and mate selection.Uncertainty can arise from sensory ambiguity and limited processing capabilities.Source: Introduction · p. 2
- Social information is often ambiguous and potentially misleading, although animals use it in contexts including predator avoidance and mate choice.Conspecific behavior can sometimes directly reveal environmental characteristics, such as food-patch quality.Source: Introduction · p. 2
- A Bayesian model combines personal and social information according to reliability to estimate the environment, from which collective decision patterns emerge.The authors propose that group members use these estimates to make decisions through their perceptual systems.Source: Introduction · p. 2
- The model derives social rules that explain animal-group decision experiments and links collective behavior with foraging, neurobiology, and psychology.The authors present it as a complement to empirical approaches and a way to propose experiments relating estimation to collective motion.Source: Introduction · p. 2
Results
The results evaluate Bayesian-estimation and probability-matching models against stickleback choices across symmetric, modified-replica, and predator-present settings. They also examine dependency-aware predictions from prior decision sequences and whether fish follow replicas as much as real fish.
Estimation model
- Each animal estimates which option is best from non-social information and others’ behaviors, multiplying a non-social likelihood ratio by a social term built from behavioral reliability.The model assumes others’ behaviors are independent, an approximation that the authors report performs well relative to a model including correlations.Source: Estimation model · pp. 2–4
Decision rule: Probability matching
- Under probability matching, an animal chooses an option with probability equal to its estimated probability that the option is best, yielding the final rule without extra parameters.The paper uses this rule because a deterministic threshold would make animals facing the same stimuli behave identically, unlike the experiments considered.Source: Decision rule: Probability matching · p. 4
Symmetric set-up
- The symmetric choice rule is sigmoidal in the difference in fish counts, steepening with reliability; sequential choices reproduce the observed U-shaped distributions and their group-size dependence.Lower reliability can produce bell- or M-shaped distributions, but the apparent M shape depends on binning; distributions can also shift from bell to M to U as more individuals decide.Source: Symmetric set-up · pp. 5–7
- With s = 2.2, the model fit stickleback choices across 19 symmetric experimental settings, with robust fits for s = 2–4 [42].The experimental settings varied group size and replica counts.Source: Figure 2 · p. 6, Symmetric set-up · p. 6
Symmetric set-up with modified replicas of animals
- For modified-replica choices, the model fit best with real-fish reliability sf = 2.9 and remained robust for sf = 2–4.The model depends on the ratio of the two replica reliability parameters, not their separate values.Source: Symmetric set-up with modified… · pp. 8–9
Asymmetric set-up
- For predator-present choices, the simpler asymmetric model fit best at s = 2.6 and remained robust for s = 2–4; undecided-fish reliability was rejected.The more complex model fit similarly, while differences in how fish relied on others by chosen side had little impact.Source: Asymmetric set-up · p. 9
- The simpler models show only very small fit improvements from distinguishing replicas from real fish, suggesting fish follow replicas as much as real fish.The Bayes Information Criterion rejects only parameter sfu.Source: Asymmetric set-up · p. 10
Model including dependencies
- With dependencies, choice probabilities vary among states sharing the same Δn, while the simpler model remains a good approximation to the dependency-aware model’s mean and most likely values.The comparison uses a′ X = 5 for the dependency-aware model and s = 2.5 for the simpler model.Source: Figure 8 · p. 10, Model including dependencies · p. 12
- The dependency-aware model matches experimental data robustly across cases using the same a′ X = 5, including choices involving two different replicas.The comparison covers symmetric, predator, and two-different-replica cases.Source: Figure 9 · p. 12, Model including dependencies · p. 12
- The model predicts that a fish may favor the less-chosen option because choices in a crowded group are less informative, while a fish opposing the group is highly informative.That opposing choice can outweigh the accumulated social information and make following the last fish more likely than following those who chose earlier.Source: Model including dependencies · pp. 12–13
Discussion
The model links probabilistic estimation to collective decisions, fitting stickleback choices across three experimental settings with a social reliability parameter generally between 2 and 4. It also yields predictions connecting individual estimation to collective behavior and movement.
Discussion
- Across three experimental settings, the model fits the data well with social reliability s between 2 and 4, and all data fit very well at s = 2.5.At s = 2.5, fish are interpreted as assuming conspecifics choose correctly 2.5 times more often than incorrectly for these experiments.Source: Discussion · p. 13
- The model uses fewer parameters than previous empirical fits, scales from simpler to more complex set-ups, and scores better by BIC than models in and.The paper presents this as a simpler, more interpretable complement to empirical fitting.Source: Discussion · p. 13
- Probabilistic estimation offers a proposed substrate for simple collective decision rules and a connection between collective behavior and brain-level decision processes.The discussion notes similar mathematical rules reported for Pharaoh’s ants, mosquito fish, and meerkats.Source: Discussion · p. 13
- The model can be combined with Bayesian foraging theory and assigns social information a weight that varies with reliability rather than switching it off when personal information is strong.For directly observed food, social and non-social information may have similar mathematical forms.Source: Discussion · p. 13
- The framework derives choice probabilities for M options and predicts that its symmetric-case expression should also fit data with M > 2 options.The model can use predicted locations or densities at a later time instead of current animal locations.Source: Discussion · p. 14
- The model predicts that manipulating reliabilities a and s_k can test whether collective behavior changes as predicted, including whether a fish’s food expectations alter its tendency to follow conspecifics.An alternative test is whether copying increases with how reliably a behavior informs about the environment.Source: Discussion · p. 14
- Estimating future rather than current locations predicts attraction and alignment toward future positions, and may also yield deviations from simple empirical rules.For a compact moving group, the example predicts movement toward the group’s future position.Source: Discussion · p. 14
Methods
The methods simulate group choices by applying an individual decision rule sequentially, fit model parameters by log likelihood, and compare fitted models using BIC.
Obtaining group behavior from the model of an individual
- Group behavior is simulated by having individuals decide between x and y in turn, recomputing model parameters after each decision; alternative simultaneous or undecided implementations made no relevant difference.For dependency models, initial values of S are averaged across possible paths to the observed initial state when path information is unavailable.Source: Obtaining group behavior from the… · p. 14
Fits
- Model parameters are optimized by iteratively adjusting each parameter to increase log likelihood until convergence; repeated random starts reached the same values.The authors report smooth, single-maximum log-likelihood curves for cases with one or two parameters.Source: Fits · p. 15
Bayesian Information Criterion
- BIC compares optimized models using fit and parameter count, with weights interpreted approximately as each model’s probability of being most correct under equal prior probabilities.For comparisons with, their parameters were re-optimized; selected parameters and single-individual data were excluded from BIC as described in Methods.Source: Bayesian Information Criterion · p. 15
Supporting Figures
The supporting figures compare model predictions with observed stickleback choice distributions across symmetric, replica, and asymmetric set-ups. BIC comparisons favor different model variants across those settings.
Supporting Figures
- In the symmetric set-up, the dependency model ranks first by BIC (w = 0.996), ahead of the enhanced independent model (w = 0.004).The simplified independent model has w = 3 · 10−11; the re-optimized model from has w = 5 · 10−13.Source: Supporting Figures · p. 20, Figure S1 · p. 20
- With two different replicas, the model neglecting dependencies ranks first by BIC (w = 0.9985), versus w = 0.001 for the dependency model and w = 0.0005 for.The reported log probabilities exclude single-individual data.Source: Supporting Figures · p. 21, Figure S2 · p. 21
- In the asymmetric set-up, the enhanced independent model (w = 0.52) and dependency model (w = 0.38) rank ahead of the simplified model (w = 0.098) by BIC.The re-optimized model from has w = 2.5 · 10−6; simulations adding data to two empty graphs still favored the first two models.Source: Supporting Figures · p. 22, Figure S3 · p. 22
- Figure S6 compares fit trends across three cases: RMSE minima and log-probability maxima indicate preferred a′_X values, while absolute curve heights are not comparable across cases.The summed global-probability curve is shifted by 1000 to fit the plotted scale.Source: Figure S6 · p. 25
Supporting text: Model for more than 2 options
The supplement extends the Bayesian decision model to M options and derives simpler cases under additional reliability assumptions.
Model for M options
- For M options, individuals estimate which option is best using non-social information and the other individuals’ behavior, yielding a general model applicable across experiments.Each term preserves a multiplicative relation between social and non-social information.Source: Model for M options · p. 26
- With independent behaviors grouped into L classes, the general model can have up to L(M −1) independent reliability parameters, though usually fewer are independent.Here, n_k counts individuals performing behavior β_k.Source: Model for M options · p. 26
One basic reliability parameter
- When behaviors mean choosing an option and correct-choice probability is the same across options, one reliability parameter s captures the correct-choice to mistake probability ratio.Non-informative behaviors have reliability parameter 1 and do not affect the model.Source: One basic reliability parameter · pp. 26–27
Symmetric case
- For three symmetric options, Fig. S7 plots P(X|C, B) against the differences n_x − n_y and n_x − n_z.In the symmetric case, the non-social parameters equal 1.Source: Symmetric case · p. 27, Figure S7 · p. 27