Stochastic Frames (2020)

Abstract: In the frame hypothesis (CITATION), human concepts are equated with frames, which extend feature lists by a functional structure consisting of attributes and values. For example, a bachelor is represented by the attributes gender and marital status and their values ‘male’ and ‘unwed’. This paper makes the point that for many applications of concepts in cognition, including for concepts to be associated with lexemes in natural languages, the right structures to assume are not merely frames but stochastic frames in which attributes are associated with probability distributions over values. The paper introduces the idea of stochastic frames and suggests three applications: vagueness, ambiguity, and typicality.

## stochastic frames stochastic frames are the stochastic version of the frames defined in @xcite . this incorporates the probabilistic semantics of sutton (2015) (also see @xcite in which the basic idea is that the nodes are associated probability distributions over possible values. formally, a (minimally) stochastic frame has a recursive attribute-value structure with a central node with a value x and where each node in the frame has a type from a type hierarchy, just as classical frames do. where (minimal) stochastic frames differ from classical frames is that the values of attributes need not be categorical (given as particular entities). instead, they may be probability distributions over entities/values of the relevant type. for example, for a stochastic frame f that contains an attribute color, the range of this attribute is a probability distribution over entities of the type color (points in the color space). if f contains an attribute height, the range of this attribute is a probability distribution over entities of the type height (values on a measurement scale). a simple example is given in figure 3 for a cat, felix, where the agent does not know what color fur felix has. this contrasts with the non-stochastic, classical frame in figure 2 in which a categorical value for felix's fur color is recorded. some values in the stochastic frame can be categorical (technically, an assignment of probability 1 to a single value). for example, the value in the stochastic felix frame for the attribute fur is assumed to be categorical in this way if the agent knows that felix has fur. however, in stochastic frames, values of attributes may be distributions over multiple values each @xmath0. for example, a distribution over colors for the value of attribute fur (such that this distribution may be generated by the agent's experiences of the typical fur colors of cats, see section 3.3). this minimal conception of a stochastic frame is extended by the addition of constraints. constraints are encoded as conditional probabilities that capture stochastic relations between the values that different nodes of the frame can obtain (see §2.2). as a further extension, probabilities can also be embedded at the level of attributes, not just values. for example, the probability that cats have fur (that the cat frame has a fur attribute), given that an agent may consider fur-less cats to be a possibility. the minimal and extended characterisations of stochastic frames given above extend the notion of classical frames. stochastic values (and attributes) allow us to model the uncertainty an agent has about a particular entity or class of entities. combined with stochastic constraints on relations be-tween nodes, as we will show, we can furthermore model the knowledge that agents have regarding the distributions of properties of entities of some particular class. stochastic frames and classical frames more or less converge when it comes to ground frames, i.e., frames for specific individuals in which every attribute has a categorical value. for example, the stochastic frame of a particular instance of 'bachelor' (say john) resembles the corresponding classical frame. (the difference between them is that, technically, the values of attributes in a stochastic ground frame are probability distributions over a single value in a classical frame.) where classical and stochastic frames diverge is with respect to uninstantiated frames and frames which are only partially grounded (where some attribute values are not given categorical values). a set of ground frames can be taken as the list of observations on which an uninstantiated stochastic frame or a partially grounded stochastic frame is based, where the list of observations corresponds to the probabilities the frame assigns. this is the case for the partially grounded stochastic frame in figure 3 . the frame assigns categorical values to the referent of felix and to the stuff that makes up felix's fur, but the distribution over colors of fur is based on observed instances of cats and the colors of fur they have. thinking in terms of ground instances gives a simple transition from thinking in terms of belonging to classes with a given probability to thinking in terms of distributions over values. ## final remarks the paper presents a notion of a stochastic frame that represents concepts and the linguistic knowledge of agents in terms of attribute-value structures in which values may only occur with some probability. we outlined how probabilistic constraints on stochastic frames facilitate reasoning about probable features (attribute values) in conditions of uncertainty. what comes out of this can be interpreted a formalisation of the prototype theory of concepts in which all other theories of concepts can be understood as special cases. by trivialising the distributions, one obtains the classical view. (products of) regions in conceptual spaces are obtained by deriving such regions from actual distributions (it becomes hard to see such an account of concepts as properly different from the prototype view). taking this common core, we also outlined three areas in which stochastic frames have obvious applications: vague predication, lexical ambiguity, and the typicality of kinds. a shared property of these phenomena is arguably that, in all cases, individuals must reason with complex, multifaceted concepts in conditions of uncertainty, be this uncertainty about the extension of a term (vagueness), uncertainty about the meaning of a term in use (lexical ambiguity), or uncertainty about properties of the instances (the typicality of kinds). for an explanation of all of these cases, we seem to need not only something along the lines of a probabilistic component to drive the reasoning process and model graded or fuzzy phenomena, but also a means of applying this reasoning tool to different aspects or properties of the entities being reasoned about. representational structures such as frames give us the structure we need in this respect. stochastic frames, therefore, give us the right combination of conceptual structure and a formal theory of reasoning.