inside

Determines whether a point <code>x</code> is inside a convex poltyope by checking whether
(1) all inequalities <code>A*x &lt;= b</code> are satisfied or
(2) the point <code>x</code> is in the convex hull of the vertices in <code>V</code>.

Implements Gibbs sampling and Bayes factors for multinomial models with
linear inequality constraints on the vector of probability parameters. As
special cases, the model class includes models that predict a linear order
of binomial probabilities (e.g., p[1] < p[2] < p[3] < .50) and mixture models
assuming that the parameter vector p must be inside the convex hull of a
finite number of predicted patterns (i.e., vertices). A formal definition of
inequality-constrained multinomial models and the implemented computational
methods is provided in: Heck, D.W., & Davis-Stober, C.P. (2019).
Multinomial models with linear inequality constraints: Overview and improvements
of computational methods for Bayesian inference. Journal of Mathematical
Psychology, 91, 70-87. <doi:10.1016/j.jmp.2019.03.004>.
Inequality-constrained multinomial models have applications in the area of
judgment and decision making to fit and test random utility models
(Regenwetter, M., Dana, J., & Davis-Stober, C.P. (2011). Transitivity of
preferences. Psychological Review, 118, 42–56, <doi:10.1037/a0021150>) or to
perform outcome-based strategy classification to select the decision strategy
that provides the best account for a vector of observed choice frequencies
(Heck, D.W., Hilbig, B.E., & Moshagen, M. (2017). From information
processing to decisions: Formalizing and comparing probabilistic choice models.
Cognitive Psychology, 96, 26–40. <doi:10.1016/j.cogpsych.2017.05.003>).

Daniel W. Heck

multinomineq

Bayesian Inference for Multinomial Models with Inequality
Constraints

inside function

<dl><dt>x</dt>
<dd>a vector of length equal to the number of columns of <code>A</code> or <code>V</code>
(i.e., a single point in D-dimensional space) or matrix of points/vertices (one per row).</dd>
<dt>A</dt>
<dd>a matrix with one row for each linear inequality constraint and one
column for each of the free parameters. The parameter space is defined
as all probabilities <code>x</code> that fulfill the order constraints <code>A*x &lt;= b</code>.</dd>
<dt>b</dt>
<dd>a vector of the same length as the number of rows of <code>A</code>.</dd>
<dt>V</dt>
<dd>a matrix of vertices (one per row) that define the polytope of
admissible parameters as the convex hull over these points
(if provided, <code>A</code> and <code>b</code> are ignored).
Similar as for <code>A</code>, columns of <code>V</code> omit the last value for each
multinomial condition (e.g., a1,a2,a3,b1,b2 becomes a1,a2,b1).
Note that this method is comparatively slow since it solves linear-programming problems
to test whether a point is inside a polytope (Fukuda, 2004) or to run the Gibbs sampler.</dd></dl>

Arguments

Check Whether Points are Inside a Convex Polytope — inside

<dl>

<dt>x</dt>
<dd>a vector of length equal to the number of columns of <code>A</code> or <code>V</code>
(i.e., a single point in D-dimensional space) or matrix of points/vertices (one per row).</dd>


<dt>A</dt>
<dd>a matrix with one row for each linear inequality constraint and one
column for each of the free parameters. The parameter space is defined
as all probabilities <code>x</code> that fulfill the order constraints <code>A*x &lt;= b</code>.</dd>


<dt>b</dt>
<dd>a vector of the same length as the number of rows of <code>A</code>.</dd>


<dt>V</dt>
<dd>a matrix of vertices (one per row) that define the polytope of
admissible parameters as the convex hull over these points
(if provided, <code>A</code> and <code>b</code> are ignored).
Similar as for <code>A</code>, columns of <code>V</code> omit the last value for each
multinomial condition (e.g., a1,a2,a3,b1,b2 becomes a1,a2,b1).
Note that this method is comparatively slow since it solves linear-programming problems
to test whether a point is inside a polytope (Fukuda, 2004) or to run the Gibbs sampler.</dd>

</dl>

inside: Check Whether Points are Inside a Convex Polytope

Description

Usage

Arguments

See Also

Examples