Rate Tracking

Introduction

Rates and Events

Rates are a fundamental way to characterize the recurrence of punctate events, like finding food, having a lightbulb burn-out, or completing a task. A rate \(r(t)\) is an abstraction in the sense that it is not directly observable. However, it is a useful abstraction. To define a rate, we need to be able to characterize an event, in terms of a description stringent enough to detect it’s occurrence and separate it from other phenomena, but general enough that event categories naturally recur. In this sense events are naturally coupled with concept and category formation, and a hierarchical similarity structure induced by categories will immediately extend to rates. More formally, events are tuples involving at least three types of data (event type, event time, event context), which provide the ability to detect the event, record it’s occurrence in a timebase that it relative to a context which provides the conditional factors needed to predict future events from their previous occurrence. We usually neglect the specification of the event and presume that the conditions, context and event definition are a solved problem or common knowledge. Please keep in mind that event specification is critical, but we will focus on measurement and tracking of event rates under the assumption that the specification is a “solved problem”.

The virtue of working with rates rather than events is twofold. First rates are an appropriate representation for forecasting, imbuing the agent with the capacity to predict event recurrence, and to condition this forecast on previous events. Second and more subtly, correlations between event occurrence can happen at the level of rates per se, despite individual events being statistically independent. Rates can be represented as instantaneous probabilities, such that integrating a rate over a finite interval of time gives a probability of occurrence in that interval \(P_i(event type i occurs \in \[t_1, t_2\])= \int_{t_1}^{t_2} r(t) dt\). Independent sampling for different events types \(i\) and \(j\) yields independent events. In other words, if we take these events and compute a joint occurrence table, they will be independent conditional on the rates. However, the rates themselves may be correlated across time, such that the rate of one event type drives the future rate of another type. The importance of this fact for animal conditioning was emphasized by Charles Gallistel[] who recognized that even simple animals are able to detect coincidences between the rates of events like sensory cues and reward delivery even when the events are statistically independent.

Here we will look at the way an organism might estimate rates of occurrence, and more importantly, to track them. We have two goals in mind. First, to build a representation for animal cognition critical for many tasks, including foraging, reinforcement learning, task monitoring and scheduling, dynamic decision making (including reaction times), social interactions, etc. Secondly, the rate tracking problem introduces the representation of second order uncertainty about rates that corresponds to a critical measure of confidence that we hypothesize is monitored for curiosity-driven exploration.

Rate Representation

Many different approaches have been developed that are related the problem of mathematically describing event generation processes. These constitute the generative models for event formation.

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References

  1. CMS/CERN. A New Boson with a Mass of 125 GeV Observed with the CMS Experiment at the Large Hadron Collider. Science 338, 1569-1575 (2012). Link

  2. Barry R Holstein. The mysterious disappearance of Ettore Majorana. J. Phys.: Conf. Ser. 173, 012019 IOP Publishing, 2009. Link