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In this paper, we present a novel Bayesian adaptive dual controller (ADC) for autonomously programming deep brain stimulation devices. We evaluated the Bayesian ADC’s performance in the context of reducing beta power in a computational model of Parkinson’s disease, in which it was tasked with finding the set of stimulation parameters which optimally reduced beta power as fast as possible. Here, the Bayesian ADC has dual goals: (a) to minimize beta power by exploiting the best parameters found so far, and (b) to explore the space to find better parameters, thus allowing for better control in the future. The Bayesian ADC is composed of two parts: an inner parameterized feedback stimulator and an outer parameter adjustment loop. The inner loop operates on a short time scale, delivering stimulus based upon the phase and power of the beta oscillation. The outer loop operates on a long time scale, observing the effects of the stimulation parameters and using Bayesian optimization to intelligently select new parameters to minimize the beta power. We show that the Bayesian ADC can efficiently optimize stimulation parameters, and is superior to other optimization algorithms. The Bayesian ADC provides a robust and general framework for tuning stimulation parameters, can be adapted to use any feedback signal, and is applicable across diseases and stimulator designs. [ABSTRACT FROM AUTHOR]

Noise is a prevalent and sometimes even dominant aspect of many biological processes. While many natural systems have adapted to attenuate or even usefully integrate noise, the variability it introduces often still delimits the achievable precision across biological functions. This is particularly so for visual phototransduction, the process responsible for converting photons of light into usable electrical signals (quantum bumps). Here, randomness of both the photon inputs (regarded as extrinsic noise) and the conversion process (intrinsic noise) are seen as two distinct, independent and significant limitations on visual reliability. Past research has attempted to quantify the relative effects of these noise sources by using approximate methods that do not fully account for the discrete, point process and time ordered nature of the problem. As a result the conclusions drawn from these different approaches have led to inconsistent expositions of phototransduction noise performance. This paper provides a fresh and complete analysis of the relative impact of intrinsic and extrinsic noise in invertebrate phototransduction using minimum mean squared error reconstruction techniques based on Bayesian point process (Snyder) filters. An integrate-fire based algorithm is developed to reliably estimate photon times from quantum bumps and Snyder filters are then used to causally estimate random light intensities both at the front and back end of the phototransduction cascade. Comparison of these estimates reveals that the dominant noise source transitions from extrinsic to intrinsic as light intensity increases. By extending the filtering techniques to account for delays, it is further found that among the intrinsic noise components, which include bump latency (mean delay and jitter) and shape (amplitude and width) variance, it is the mean delay that is critical to noise performance. As the timeliness of visual information is important for real-time action, this delay could potentially limit the speed at which invertebrates can respond to stimuli. Consequently, if one wants to increase visual fidelity, reducing the photoconversion lag is much more important than improving the regularity of the electrical signal. [ABSTRACT FROM AUTHOR]

Mathematical methods of information theory appear to provide a useful language to describe how stimuli are encoded in activities of signaling effectors. Exploring the information-theoretic perspective, however, remains conceptually, experimentally and computationally challenging. Specifically, existing computational tools enable efficient analysis of relatively simple systems, usually with one input and output only. Moreover, their robust and readily applicable implementations are missing. Here, we propose a novel algorithm, SLEMI—statistical learning based estimation of mutual information, to analyze signaling systems with high-dimensional outputs and a large number of input values. Our approach is efficient in terms of computational time as well as sample size needed for accurate estimation. Analysis of the NF-κB single—cell signaling responses to TNF-α reveals that NF-κB signaling dynamics improves discrimination of high concentrations of TNF-α with a relatively modest impact on discrimination of low concentrations. Provided R-package allows the approach to be used by computational biologists with only elementary knowledge of information theory. [ABSTRACT FROM AUTHOR]

The article offers information on the challenging scientific problems faced by humanity. Topics discussed include information on the efforts to empower Africans to join the global neuroscience community; discussions on the BRAIN initiative in the U.S. and the Human Brain Project in Europe; and the information on the limiting factor for computational neuroscience development in Africa.

Emerging RNA-based approaches to disease detection and gene therapy require RNA sequences that fold into specific base-pairing patterns, but computational algorithms generally remain inadequate for these secondary structure design tasks. The Eterna project has crowdsourced RNA design to human video game players in the form of puzzles that reach extraordinary difficulty. Here, we demonstrate that Eterna participants’ moves and strategies can be leveraged to improve automated computational RNA design. We present an eternamoves-large repository consisting of 1.8 million of player moves on 12 of the most-played Eterna puzzles as well as an eternamoves-select repository of 30,477 moves from the top 72 players on a select set of more advanced puzzles. On eternamoves-select, we present a multilayer convolutional neural network (CNN) EternaBrain that achieves test accuracies of 51% and 34% in base prediction and location prediction, respectively, suggesting that top players’ moves are partially stereotyped. Pipelining this CNN’s move predictions with single-action-playout (SAP) of six strategies compiled by human players solves 61 out of 100 independent puzzles in the Eterna100 benchmark. EternaBrain-SAP outperforms previously published RNA design algorithms and achieves similar or better performance than a newer generation of deep learning methods, while being largely orthogonal to these other methods. Our study provides useful lessons for future efforts to achieve human-competitive performance with automated RNA design algorithms. [ABSTRACT FROM AUTHOR]

Computational modeling plays an important role in modern neuroscience research. Much previous research has relied on statistical methods, separately, to address two problems that are actually interdependent. First, given a particular computational model, Bayesian hierarchical techniques have been used to estimate individual variation in parameters over a population of subjects, leveraging their population-level distributions. Second, candidate models are themselves compared, and individual variation in the expressed model estimated, according to the fits of the models to each subject. The interdependence between these two problems arises because the relevant population for estimating parameters of a model depends on which other subjects express the model. Here, we propose a hierarchical Bayesian inference (HBI) framework for concurrent model comparison, parameter estimation and inference at the population level, combining previous approaches. We show that this framework has important advantages for both parameter estimation and model comparison theoretically and experimentally. The parameters estimated by the HBI show smaller errors compared to other methods. Model comparison by HBI is robust against outliers and is not biased towards overly simplistic models. Furthermore, the fully Bayesian approach of our theory enables researchers to make inference on group-level parameters by performing HBI t-test. [ABSTRACT FROM AUTHOR]

Evaluating the future consequences of actions is achievable by simulating a mental search tree into the future. Expanding deep trees, however, is computationally taxing. Therefore, machines and humans use a plan-until-habit scheme that simulates the environment up to a limited depth and then exploits habitual values as proxies for consequences that may arise in the future. Two outstanding questions in this scheme are “in which directions the search tree should be expanded?”, and “when should the expansion stop?”. Here we propose a principled solution to these questions based on a speed/accuracy tradeoff: deeper expansion in the appropriate directions leads to more accurate planning, but at the cost of slower decision-making. Our simulation results show how this algorithm expands the search tree effectively and efficiently in a grid-world environment. We further show that our algorithm can explain several behavioral patterns in animals and humans, namely the effect of time-pressure on the depth of planning, the effect of reward magnitudes on the direction of planning, and the gradual shift from goal-directed to habitual behavior over the course of training. The algorithm also provides several predictions testable in animal/human experiments. [ABSTRACT FROM AUTHOR]

A well-established notion in cognitive neuroscience proposes that multiple brain systems contribute to choice behaviour. These include: (1) a model-free system that uses values cached from the outcome history of alternative actions, and (2) a model-based system that considers action outcomes and the transition structure of the environment. The widespread use of this distinction, across a range of applications, renders it important to index their distinct influences with high reliability. Here we consider the two-stage task, widely considered as a gold standard measure for the contribution of model-based and model-free systems to human choice. We tested the internal/temporal stability of measures from this task, including those estimated via an established computational model, as well as an extended model using drift-diffusion. Drift-diffusion modeling suggested that both choice in the first stage, and RTs in the second stage, are directly affected by a model-based/free trade-off parameter. Both parameter recovery and the stability of model-based estimates were poor but improved substantially when both choice and RT were used (compared to choice only), and when more trials (than conventionally used in research practice) were included in our analysis. The findings have implications for interpretation of past and future studies based on the use of the two-stage task, as well as for characterising the contribution of model-based processes to choice behaviour. [ABSTRACT FROM AUTHOR]

Grid cells have attracted broad attention because of their highly symmetric hexagonal firing patterns. Recently, research has shifted its focus from the global symmetry of grid cell activity to local distortions both in space and time, such as drifts in orientation, local defects of the hexagonal symmetry, and the decay and reappearance of grid patterns after changes in lighting condition. Here, we introduce a method that allows to visualize and quantify such local distortions, by assigning both a local grid score and a local orientation to each individual spike of a neuronal recording. The score is inspired by a standard measure from crystallography, which has been introduced to quantify local order in crystals. By averaging over spikes recorded within arbitrary regions or time periods, we can quantify local variations in symmetry and orientation of firing patterns in both space and time. [ABSTRACT FROM AUTHOR]

Classical decision theory postulates that choices proceed from subjective values assigned to the probable outcomes of alternative actions. Some authors have argued that opposite causality should also be envisaged, with choices influencing subsequent values expressed in desirability ratings. The idea is that agents may increase their ratings of items that they have chosen in the first place, which has been typically explained by the need to reduce cognitive dissonance. However, evidence in favor of this reverse causality has been the topic of intense debates that have not reached consensus so far. Here, we take a novel approach using Bayesian techniques to compare models in which choices arise from stable (but noisy) underlying values (one-way causality) versus models in which values are in turn influenced by choices (two-way causality). Moreover, we examined whether in addition to choices, other components of previous actions, such as the effort invested and the eventual action outcome (success or failure), could also impact subsequent values. Finally, we assessed whether the putative changes in values were only expressed in explicit ratings, or whether they would also affect other value-related behaviors such as subsequent choices. Behavioral data were obtained from healthy participants in a rating-choice-rating-choice-rating paradigm, where the choice task involves deciding whether or not to exert a given physical effort to obtain a particular food item. Bayesian selection favored two-way causality models, where changes in value due to previous actions affected subsequent ratings, choices and action outcomes. Altogether, these findings may help explain how values and actions drift when several decisions are made successively, hence highlighting some shortcomings of classical decision theory. [ABSTRACT FROM AUTHOR]

Although a standard reinforcement learning model can capture many aspects of reward-seeking behaviors, it may not be practical for modeling human natural behaviors because of the richness of dynamic environments and limitations in cognitive resources. We propose a modular reinforcement learning model that addresses these factors. Based on this model, a modular inverse reinforcement learning algorithm is developed to estimate both the rewards and discount factors from human behavioral data, which allows predictions of human navigation behaviors in virtual reality with high accuracy across different subjects and with different tasks. Complex human navigation trajectories in novel environments can be reproduced by an artificial agent that is based on the modular model. This model provides a strategy for estimating the subjective value of actions and how they influence sensory-motor decisions in natural behavior. [ABSTRACT FROM AUTHOR]

The precision of multisensory perception improves when cues arising from the same cause are integrated, such as visual and vestibular heading cues for an observer moving through a stationary environment. In order to determine how the cues should be processed, the brain must infer the causal relationship underlying the multisensory cues. In heading perception, however, it is unclear whether observers follow the Bayesian strategy, a simpler non-Bayesian heuristic, or even perform causal inference at all. We developed an efficient and robust computational framework to perform Bayesian model comparison of causal inference strategies, which incorporates a number of alternative assumptions about the observers. With this framework, we investigated whether human observers’ performance in an explicit cause attribution and an implicit heading discrimination task can be modeled as a causal inference process. In the explicit causal inference task, all subjects accounted for cue disparity when reporting judgments of common cause, although not necessarily all in a Bayesian fashion. By contrast, but in agreement with previous findings, data from the heading discrimination task only could not rule out that several of the same observers were adopting a forced-fusion strategy, whereby cues are integrated regardless of disparity. Only when we combined evidence from both tasks we were able to rule out forced-fusion in the heading discrimination task. Crucially, findings were robust across a number of variants of models and analyses. Our results demonstrate that our proposed computational framework allows researchers to ask complex questions within a rigorous Bayesian framework that accounts for parameter and model uncertainty. [ABSTRACT FROM AUTHOR]

Humans are remarkably adept at generalizing knowledge between experiences in a way that can be difficult for computers. Often, this entails generalizing constituent pieces of experiences that do not fully overlap, but nonetheless share useful similarities with, previously acquired knowledge. However, it is often unclear how knowledge gained in one context should generalize to another. Previous computational models and data suggest that rather than learning about each individual context, humans build latent abstract structures and learn to link these structures to arbitrary contexts, facilitating generalization. In these models, task structures that are more popular across contexts are more likely to be revisited in new contexts. However, these models can only re-use policies as a whole and are unable to transfer knowledge about the transition structure of the environment even if only the goal has changed (or vice-versa). This contrasts with ecological settings, where some aspects of task structure, such as the transition function, will be shared between context separately from other aspects, such as the reward function. Here, we develop a novel non-parametric Bayesian agent that forms independent latent clusters for transition and reward functions, affording separable transfer of their constituent parts across contexts. We show that the relative performance of this agent compared to an agent that jointly clusters reward and transition functions depends environmental task statistics: the mutual information between transition and reward functions and the stochasticity of the observations. We formalize our analysis through an information theoretic account of the priors, and propose a meta learning agent that dynamically arbitrates between strategies across task domains to optimize a statistical tradeoff. [ABSTRACT FROM AUTHOR]

Single-trial analyses have the potential to uncover meaningful brain dynamics that are obscured when averaging across trials. However, low signal-to-noise ratio (SNR) can impede the use of single-trial analyses and decoding methods. In this study, we investigate the applicability of a single-trial approach to decode stimulus modality from magnetoencephalographic (MEG) high frequency activity. In order to classify the auditory versus visual presentation of words, we combine beamformer source reconstruction with the random forest classification method. To enable group level inference, the classification is embedded in an across-subjects framework. We show that single-trial gamma SNR allows for good classification performance (accuracy across subjects: 66.44%). This implies that the characteristics of high frequency activity have a high consistency across trials and subjects. The random forest classifier assigned informational value to activity in both auditory and visual cortex with high spatial specificity. Across time, gamma power was most informative during stimulus presentation. Among all frequency bands, the 75 Hz 95 Hz band was the most informative frequency band in visual as well as in auditory areas. Especially in visual areas, a broad range of gamma frequencies (55 Hz 125 Hz) contributed to the successful classification. Thus, we demonstrate the feasibility of single-trial approaches for decoding the stimulus modality across subjects from high frequency activity and describe the discriminative gamma activity in time, frequency, and space. [ABSTRACT FROM AUTHOR]

Cooperation and competition between human players in repeated microeconomic games offer a window onto social phenomena such as the establishment, breakdown and repair of trust. However, although a suitable starting point for the quantitative analysis of such games exists, namely the Interactive Partially Observable Markov Decision Process (I-POMDP), computational considerations and structural limitations have limited its application, and left unmodelled critical features of behavior in a canonical trust task. Here, we provide the first analysis of two central phenomena: a form of social risk-aversion exhibited by the player who is in control of the interaction in the game; and irritation or anger, potentially exhibited by both players. Irritation arises when partners apparently defect, and it potentially causes a precipitate breakdown in cooperation. Failing to model one’s partner’s propensity for it leads to substantial economic inefficiency. We illustrate these behaviours using evidence drawn from the play of large cohorts of healthy volunteers and patients. We show that for both cohorts, a particular subtype of player is largely responsible for the breakdown of trust, a finding which sheds new light on borderline personality disorder. [ABSTRACT FROM AUTHOR]

Empirical evidence suggests the incentive value of an option is affected by other options available during choice and by options presented in the past. These contextual effects are hard to reconcile with classical theories and have inspired accounts where contextual influences play a crucial role. However, each account only addresses one or the other of the empirical findings and a unifying perspective has been elusive. Here, we offer a unifying theory of context effects on incentive value attribution and choice based on normative Bayesian principles. This formulation assumes that incentive value corresponds to a precision-weighted prediction error, where predictions are based upon expectations about reward. We show that this scheme explains a wide range of contextual effects, such as those elicited by other options available during choice (or within-choice context effects). These include both conditions in which choice requires an integration of multiple attributes and conditions where a multi-attribute integration is not necessary. Moreover, the same scheme explains context effects elicited by options presented in the past or between-choice context effects. Our formulation encompasses a wide range of contextual influences (comprising both within- and between-choice effects) by calling on Bayesian principles, without invoking ad-hoc assumptions. This helps clarify the contextual nature of incentive value and choice behaviour and may offer insights into psychopathologies characterized by dysfunctional decision-making, such as addiction and pathological gambling. [ABSTRACT FROM AUTHOR]

Decision-making in the real world presents the challenge of requiring flexible yet prompt behavior, a balance that has been characterized in terms of a trade-off between a slower, prospective goal-directed model-based (MB) strategy and a fast, retrospective habitual model-free (MF) strategy. Theory predicts that flexibility to changes in both reward values and transition contingencies can determine the relative influence of the two systems in reinforcement learning, but few studies have manipulated the latter. Therefore, we developed a novel two-level contingency change task in which transition contingencies between states change every few trials; MB and MF control predict different responses following these contingency changes, allowing their relative influence to be inferred. Additionally, we manipulated the rate of contingency changes in order to determine whether contingency change volatility would play a role in shifting subjects between a MB and MF strategy. We found that human subjects employed a hybrid MB/MF strategy on the task, corroborating the parallel contribution of MB and MF systems in reinforcement learning. Further, subjects did not remain at one level of MB/MF behaviour but rather displayed a shift towards more MB behavior over the first two blocks that was not attributable to the rate of contingency changes but rather to the extent of training. We demonstrate that flexibility to contingency changes can distinguish MB and MF strategies, with human subjects utilizing a hybrid strategy that shifts towards more MB behavior over blocks, consequently corresponding to a higher payoff. [ABSTRACT FROM AUTHOR]

Humans and animals are capable of evaluating actions by considering their long-run future rewards through a process described using model-based reinforcement learning (RL) algorithms. The mechanisms by which neural circuits perform the computations prescribed by model-based RL remain largely unknown; however, multiple lines of evidence suggest that neural circuits supporting model-based behavior are structurally homologous to and overlapping with those thought to carry out model-free temporal difference (TD) learning. Here, we lay out a family of approaches by which model-based computation may be built upon a core of TD learning. The foundation of this framework is the successor representation, a predictive state representation that, when combined with TD learning of value predictions, can produce a subset of the behaviors associated with model-based learning, while requiring less decision-time computation than dynamic programming. Using simulations, we delineate the precise behavioral capabilities enabled by evaluating actions using this approach, and compare them to those demonstrated by biological organisms. We then introduce two new algorithms that build upon the successor representation while progressively mitigating its limitations. Because this framework can account for the full range of observed putatively model-based behaviors while still utilizing a core TD framework, we suggest that it represents a neurally plausible family of mechanisms for model-based evaluation. [ABSTRACT FROM AUTHOR]

Direct reciprocity, or repeated interaction, is a main mechanism to sustain cooperation under social dilemmas involving two individuals. For larger groups and networks, which are probably more relevant to understanding and engineering our society, experiments employing repeated multiplayer social dilemma games have suggested that humans often show conditional cooperation behavior and its moody variant. Mechanisms underlying these behaviors largely remain unclear. Here we provide a proximate account for this behavior by showing that individuals adopting a type of reinforcement learning, called aspiration learning, phenomenologically behave as conditional cooperator. By definition, individuals are satisfied if and only if the obtained payoff is larger than a fixed aspiration level. They reinforce actions that have resulted in satisfactory outcomes and anti-reinforce those yielding unsatisfactory outcomes. The results obtained in the present study are general in that they explain extant experimental results obtained for both so-called moody and non-moody conditional cooperation, prisoner’s dilemma and public goods games, and well-mixed groups and networks. Different from the previous theory, individuals are assumed to have no access to information about what other individuals are doing such that they cannot explicitly use conditional cooperation rules. In this sense, myopic aspiration learning in which the unconditional propensity of cooperation is modulated in every discrete time step explains conditional behavior of humans. Aspiration learners showing (moody) conditional cooperation obeyed a noisy GRIM-like strategy. This is different from the Pavlov, a reinforcement learning strategy promoting mutual cooperation in two-player situations. [ABSTRACT FROM AUTHOR]

In a dynamic world, an accurate model of the environment is vital for survival, and agents ought regularly to seek out new information with which to update their world models. This aspect of behaviour is not captured well by classical theories of decision making, and the cognitive mechanisms of information seeking are poorly understood. In particular, it is not known whether information is valued only for its instrumental use, or whether humans also assign it a non-instrumental intrinsic value. To address this question, the present study assessed preference for non-instrumental information among 80 healthy participants in two experiments. Participants performed a novel information preference task in which they could choose to pay a monetary cost to receive advance information about the outcome of a monetary lottery. Importantly, acquiring information did not alter lottery outcome probabilities. We found that participants were willing to incur considerable monetary costs to acquire payoff-irrelevant information about the lottery outcome. This behaviour was well explained by a computational cognitive model in which information preference resulted from aversion to temporally prolonged uncertainty. These results strongly suggest that humans assign an intrinsic value to information in a manner inconsistent with normative accounts of decision making under uncertainty. This intrinsic value may be associated with adaptive behaviour in real-world environments by producing a bias towards exploratory and information-seeking behaviour. [ABSTRACT FROM AUTHOR]

Sound waveforms convey information largely via amplitude modulations (AM). A large body of experimental evidence has provided support for a modulation (bandpass) filterbank. Details of this model have varied over time partly reflecting different experimental conditions and diverse datasets from distinct task strategies, contributing uncertainty to the bandwidth measurements and leaving important issues unresolved. We adopt here a solely data-driven measurement approach in which we first demonstrate how different models can be subsumed within a common ‘cascade’ framework, and then proceed to characterize the cascade via system identification analysis using a single stimulus/task specification and hence stable task rules largely unconstrained by any model or parameters. Observers were required to detect a brief change in level superimposed onto random level changes that served as AM noise; the relationship between trial-by-trial noisy fluctuations and corresponding human responses enables targeted identification of distinct cascade elements. The resulting measurements exhibit a dynamic complex picture in which human perception of auditory modulations appears adaptive in nature, evolving from an initial lowpass to bandpass modes (with broad tuning, Q∼1) following repeated stimulus exposure. [ABSTRACT FROM AUTHOR]

Accurate estimation of neuronal receptive fields is essential for understanding sensory processing in the early visual system. Yet a full characterization of receptive fields is still incomplete, especially with regard to natural visual stimuli and in complete populations of cortical neurons. While previous work has incorporated known structural properties of the early visual system, such as lateral connectivity, or imposing simple-cell-like receptive field structure, no study has exploited the fact that nearby V1 neurons share common feed-forward input from thalamus and other upstream cortical neurons. We introduce a new method for estimating receptive fields simultaneously for a population of V1 neurons, using a model-based analysis incorporating knowledge of the feed-forward visual hierarchy. We assume that a population of V1 neurons shares a common pool of thalamic inputs, and consists of two layers of simple and complex-like V1 neurons. When fit to recordings of a local population of mouse layer 2/3 V1 neurons, our model offers an accurate description of their response to natural images and significant improvement of prediction power over the current state-of-the-art methods. We show that the responses of a large local population of V1 neurons with locally diverse receptive fields can be described with surprisingly limited number of thalamic inputs, consistent with recent experimental findings. Our structural model not only offers an improved functional characterization of V1 neurons, but also provides a framework for studying the relationship between connectivity and function in visual cortical areas. [ABSTRACT FROM AUTHOR]

Neuroprosthetic brain-computer interfaces function via an algorithm which decodes neural activity of the user into movements of an end effector, such as a cursor or robotic arm. In practice, the decoder is often learned by updating its parameters while the user performs a task. When the user’s intention is not directly observable, recent methods have demonstrated value in training the decoder against a surrogate for the user’s intended movement. Here we show that training a decoder in this way is a novel variant of an imitation learning problem, where an oracle or expert is employed for supervised training in lieu of direct observations, which are not available. Specifically, we describe how a generic imitation learning meta-algorithm, dataset aggregation (DA), can be adapted to train a generic brain-computer interface. By deriving existing learning algorithms for brain-computer interfaces in this framework, we provide a novel analysis of regret (an important metric of learning efficacy) for brain-computer interfaces. This analysis allows us to characterize the space of algorithmic variants and bounds on their regret rates. Existing approaches for decoder learning have been performed in the cursor control setting, but the available design principles for these decoders are such that it has been impossible to scale them to naturalistic settings. Leveraging our findings, we then offer an algorithm that combines imitation learning with optimal control, which should allow for training of arbitrary effectors for which optimal control can generate goal-oriented control. We demonstrate this novel and general BCI algorithm with simulated neuroprosthetic control of a 26 degree-of-freedom model of an arm, a sophisticated and realistic end effector. [ABSTRACT FROM AUTHOR]

Many everyday estimation tasks have an inherently discrete nature, whether the task is counting objects (e.g., a number of paint buckets) or estimating discretized continuous variables (e.g., the number of paint buckets needed to paint a room). While Bayesian inference is often used for modeling estimates made along continuous scales, discrete numerical estimates have not received as much attention, despite their common everyday occurrence. Using two tasks, a numerosity task and an area estimation task, we invoke Bayesian decision theory to characterize how people learn discrete numerical distributions and make numerical estimates. Across three experiments with novel stimulus distributions we found that participants fell between two common decision functions for converting their uncertain representation into a response: drawing a sample from their posterior distribution and taking the maximum of their posterior distribution. While this was consistent with the decision function found in previous work using continuous estimation tasks, surprisingly the prior distributions learned by participants in our experiments were much more adaptive: When making continuous estimates, participants have required thousands of trials to learn bimodal priors, but in our tasks participants learned discrete bimodal and even discrete quadrimodal priors within a few hundred trials. This makes discrete numerical estimation tasks good testbeds for investigating how people learn and make estimates. [ABSTRACT FROM AUTHOR]