Rational Foundations of Fast and Frugal Heuristics: The Ecological Rationality of Strategy Selection via Improper Linear Models

Abstract

Research on “improper” linear models has shown that predetermined weighting schemes for the linear model, such as equally weighting all predictors, can be surprisingly accurate on cross-validation. We review recent advances that can characterize the optimal choice of an improper linear model. We extend this research to the understanding of fast and frugal heuristics, particularly to the ecologically rational goal of understanding in which task environments given heuristics are optimal. We demonstrate how to test this model using the Recognition Heuristic and Take the Best heuristic, show how the model reconciles with the ecological rationality program, and discuss how our prescriptive, computational approach could be approximated by simpler mental rules that might be more descriptive. Echoing the arguments of van Rooij et al. (Synthese 187:471–487, 2012), we stress the virtue of having a computationally tractable model of strategy selection, even if one proposes that cognizers use a simpler heuristic process to approximate it.

Appendices

Appendix 1

Prior work has shown that simple heuristics, such as TTB, can outperform standard statistical methods, such as regression, in many task environments (Katsikopoulos et al. 2010). There is a parallel literature on comparing improper linear models to standard regression techniques (Dana and Dawes 2004; Davis-Stober 2011; Dawes 1979; Wainer 1976), demonstrating that improper linear models often perform quite favorably for task environments where information is limited, i.e., sample and effect sizes are small to modest in size. There is converging evidence that simple decision heuristics and improper linear models work well for similar reasons. For example, the loss function we have used, mean squared error, can be decomposed into the sum of an estimator’s squared bias and its variance. Said simply, one can improve overall accuracy, mean squared error, by systematically accepting a small amount of bias while reducing the variability of the estimates. As described by Dana (2008) and Davis-Stober et al. (2010a), this is one interpretation for the strong performance of improper linear models when compared to standard estimation methods. The improper models are biased as the weighting policies are pre-determined, yet variance is greatly reduced for the same reason; on the other hand, standard methods, such as regression, are unbiased in their estimates but are more variable from sample to sample.

In this appendix, we summarize the main results from Davis-Stober et al. (2010a) that underlie our approach. We refer readers to the original papers for detailed proofs. Throughout, we consider the standard regression model where $\textit{Y} = \varvec{X\beta } + \epsilon$, where $\varvec{X}$ is a known $n \times p$ design matrix, $\epsilon \sim (0, \; \sigma ^{2}\varvec{I}_{n \times n})$, $\varvec{\beta }$ is a set of unknown population weights, and the matrix $\varvec{X'X}$ is full rank and positive-definite. We now formerly define our estimator of $\varvec{\beta }$ that conforms to the weighting policy of a suitably chosen improper linear model. We term this estimator the constrained estimator.

Definition

(Davis-Stober et al. 2010a). Define $\varvec{a}$ to be an exogenously chosen $p \times 1$ weighting vector, with $\Vert \varvec{a}\Vert ^{2} > 0$. Assume $\textit{Y} \sim (\varvec{X\beta }, \; \sigma ^{2}I_{n \times n})$ with i.i.d. sampling. The constrained estimator ${\hat{\varvec{\beta }}}_{\varvec{a}}$ is an estimator of $\varvec{\beta }$ and is defined as follows:

$$\begin{aligned} {\hat{\varvec{\beta }}}_{\varvec{a}} = \varvec{a}k, \end{aligned}$$

(5)

where $k=(\varvec{a'X'Xa})^{-1}\varvec{a'X'y}$.

This formulation allows us to compare the mean squared error (a.k.a. loss) of the constrained estimator ${\hat{\varvec{\beta}}}_{\varvec{a}}$ to more traditional estimators such as the standard regression estimator, Ordinary Least Squares (OLS). We can also calculate the loss of ${\hat{\varvec{\beta}}}_{\varvec{a}}$ under various choices of exogenously chosen weights, $\varvec{a}$. In other words, given different predictor cue structures, $\varvec{X}$, we can determine which choices of $\varvec{a}$ result in a constrained estimator that incurs less loss than others.

A potential stumbling block is that the true value of the population weights, $\varvec{\beta }$, is unknown. Because the parameter k is a scalar value and cannot change the pre-determined relationships among the estimates in ${\hat{\varvec{\beta}}}_{\varvec{a}}$, which are determined by $\varvec{a}$, the constrained estimator is both biased and inconsistent. Further, we don’t know, a priori, how biased the constrained estimator will be for any particular choice of $\varvec{a}$. In other words, if our exogenously chosen weights are, perhaps by luck or ecological rationality principles, very similar to the population weights then the resulting constrained estimator would not be very biased, resulting in a relatively small value of mean squared error. On the other hand, a choice of exogenously chosen weights could be quite different than $\varvec{\beta }$, resulting in a larger bias, hence larger mean squared error, all things equal.

We solve this problem by solving for the maximum possible mean squared error. The following theorem provides a tight upper bound for the mean squared error of ${\hat{\varvec{\beta}}}_{\varvec{a}}$, and this value can be expressed as a function of the exogenously chosen weights, $\varvec{a}$, and the matrix $\varvec{X}$.

Theorem 1

(Davis-Stober et al. 2010a). Assume $\varvec{a}$ is chosen exogenously and without loss of generality let $\Vert \varvec{a}\Vert = 1, \Vert \varvec{\beta }\Vert ^{2} < \infty , and\,\textit{Y} \sim (\varvec{X\beta }, \sigma ^{2}\varvec{I}_{n \times n})$ with i.i.d. sampling. The mean squared error (MSE) of ${\hat{\varvec{\beta}}}_{\varvec{a}}$ is bounded above by the following,

$$\begin{aligned} MSE_{\hat{\varvec{\beta }}_{\varvec{a}}} \le \Vert \varvec{\beta }\Vert ^{2} \left( \frac{\varvec{a}'(\varvec{X}'\varvec{X})^{2} \varvec{a}}{(\varvec{a}'\varvec{X}'\varvec{Xa})^{2}} \right) + \frac{\sigma ^{2}}{\varvec{a'X'Xa}}. \end{aligned}$$

(6)

This maximal MSE is attained when the population weights, $\varvec{\beta }$, is a scale multiple of the vector

$$\begin{aligned} \varvec{\beta }^{\varvec{*}} = \varvec{X'Xa} - \varvec{a(a'X'Xa)}. \end{aligned}$$

(7)

Theorem 1 provides two important pieces of information. First, for any choice of weights $\varvec{a}$ and matrix $\varvec{X'X}$, it explicitly describes the maximum loss that the constrained estimator can incur. Second, it provides the population weights, $\varvec{\beta }^{\varvec{*}}$, under which this maximal loss occurs. Not surprisingly, this worst case set of population weights is geometrically orthogonal to $\varvec{a}$ (Davis-Stober et al. 2010a). Under this “worst case” condition, the improper linear model being deployed, $\varvec{a}$, is maximally unfit given the true weighting relationship among the dependent variable and predictors, $\varvec{\beta }^{\varvec{*}}$.

Theorem 1 also gives an explicit description of maximal loss in terms of the well-known bias-variance trade-off. As is well-known, mean squared error can be written as the sum of an estimator’s squared bias and its variance. More formally, for any estimator, ${\hat{\varvec{\beta}}}$,

$$\begin{aligned} MSE_{\hat{\varvec{\beta}}} = \Vert E({\hat{\varvec{\beta}}})-\varvec{\beta }\Vert ^{2} + tr(\varvec{\Psi }_{\hat{\varvec{\beta }}}), \end{aligned}$$

(8)

where $\varvec{\Psi }_{\hat{\varvec{\beta }}}$ is the covariance matrix of $\hat{\varvec{\beta }}$, and “tr” denotes the trace operator. Returning to Theorem 1, the maximal value of mean squared error for the constrained estimator is written as a sum of its squared bias, $\Vert \varvec{\beta }\Vert ^{2} \left( \frac{\varvec{a}'(\varvec{X}'\varvec{X})^{2} \varvec{a}}{(\varvec{a}'\varvec{X}'\varvec{Xa})^{2}} \right)$, and the sum of its variance, $\frac{\sigma ^{2}}{\varvec{a'X'Xa}}$, see Davis-Stober et al. (2010a) for the complete derivation and proof. The value, $\Vert \varvec{\beta }\Vert ^{2}$, in the bias term can be equated with overall effect size and does not depend on the direction of the population weights, only the sum of their squared values. The remaining values in the bias and variance terms are only functions of $\varvec{a}$ and $\varvec{X'X}$. From this interpretation, the relationship between a choice of $\varvec{a}$ and the predictor variables is clear. Minimizing the maximal loss that one could incur depends upon optimizing the relationship between $\varvec{a}$ and $\varvec{X'X}$, as the product $\varvec{X'Xa}$ features prominently in both the maximal bias and variance terms in Eq. (8). This is quite intuitive, as for mean-centered predictor variables, $\varvec{X'X} = (n-1)\varvec{C}$, where $\varvec{C}$ is the covariance matrix among the predictor cues. In other words, to minimize maximum loss, the mini–max criterion, we need only find the best choice of $\varvec{a}$ for a particular predictor cue covariance structure. The following theorem gives precisely this result.

Theorem 2

(Davis-Stober et al. 2010a). Assume $\varvec{X}$ is given, and without loss of generality let $\Vert \varvec{a}\Vert ^{2} = 1$. Define $\lambda _{\max }$ as the largest eigenvalue of the matrix $\varvec{X'X}$. Assume $\Vert \varvec{\beta }\Vert ^{2} < \infty$. The weight vector $\varvec{a}$ that is mini–max with respect to all exogenously chosen $\varvec{a} \in \mathbb {R}^{p}$, is the eigenvector corresponding to the largest eigenvalue of $\varvec{X'X}$. The maximal value of $MSE_{\hat{\varvec{\beta }}_{\varvec{a}}}$ is bounded by the following inequality:

$$\begin{aligned} MSE_{\hat{\varvec{\beta }}_{\varvec{a}}} \le \Vert \varvec{\beta }\Vert ^{2} + \frac{\sigma ^{2}}{\lambda _{\max }}. \end{aligned}$$

(9)

Putting everything together, Theorems 1 and 2 provide necessary and sufficient conditions for finding an optimal choice of improper weighting policy, according to the mini–max criterion, given a predictor cue covariance structure $\varvec{C}$. By Theorem 2, this optimal weighting policy will be mini–max with respect to all other exogenously chosen weights that could be considered.

Appendix 2

Proof of Result 1

The proof follows by direct calculation.

$$\begin{aligned} \varvec{Cov}_{\varvec{cascade}}\varvec{a}_{1} = \left( \begin{array}{cccccc} 1+c2^{2p-2} &{} c2^{2p-3} &{} c2^{2p-4} &{} \cdots &{} \cdots &{} c2^{p-1}\\ c2^{2p-3} &{} 1+c2^{2p-4} &{} c2^{2p-5} &{} \cdots &{} \cdots &{} c2^{p-2}\\ c2^{2p-4} &{} c2^{2p-5} &{} 1+c2^{2p-6} &{} \cdots &{} \cdots &{} c2^{p-3}\\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \cdots &{} \vdots \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ c2^{p-1} &{} c2^{p-2} &{} c2^{p-3} &{} \cdots &{} c2^{1} &{} 1+c\\ \end{array}\right) \left( \begin{array}{c} \frac{1}{2^{0}}\\ \frac{1}{2^{1}}\\ \frac{1}{2^{2}}\\ \vdots \\ \vdots \\ \frac{1}{2^{p-1}}\\ \end{array}\right) , \end{aligned}$$

which is equal to

$$\begin{aligned} \left( \begin{array}{c} \frac{1+c2^{2p-2}}{2^{0}} + \frac{c2^{2p-3}}{2^{1}}+ \frac{c2^{2p-4}}{2^{2}} + \cdots + \frac{c2^{p-1}}{2^{p-1}}\\ \frac{c2^{2p-3}}{2^{0}} + \frac{1+c2^{2p-4}}{2^{1}}+ \frac{c2^{2p-5}}{2^{2}} + \cdots + \frac{c2^{p-2}}{2^{p-1}}\\ \frac{c2^{2p-4}}{2^{0}} + \frac{c2^{2p-5}}{2^{1}}+ \frac{1+c2^{2p-6}}{2^{2}} + \cdots + \frac{c2^{p-3}}{2^{p-1}}\\ \vdots \\ \vdots \\ \frac{c2^{p-1}}{2^{0}} + \frac{c2^{p-2}}{2^{1}}+ \frac{c2^{p-3}}{2^{2}} + \cdots + \frac{1+c}{2^{p-1}}\\ \end{array}\right) , \end{aligned}$$

which simplifies to

$$\begin{aligned} \left( \begin{array}{c} \frac{1+c2^{2p-2}}{2^{0}} + \frac{c2^{2p-4}}{2^{0}}+ \frac{c2^{2p-6}}{2^{0}} + \cdots + \frac{c2^{0}}{2^{0}}\\ \frac{c2^{2p-2}}{2^{1}} + \frac{1+c2^{2p-4}}{2^{1}}+ \frac{c2^{2p-6}}{2^{1}} + \cdots + \frac{c2^{0}}{2^{1}}\\ \frac{c2^{2p-2}}{2^{2}} + \frac{c2^{2p-4}}{2^{2}}+ \frac{1+c2^{2p-6}}{2^{2}} + \cdots + \frac{c2^{0}}{2^{2}}\\ \vdots \\ \vdots \\ \frac{c2^{2p-2}}{2^{p-1}} + \frac{c2^{2p-4}}{2^{p-1}}+ \frac{c2^{2p-6}}{2^{p-1}} + \cdots + \frac{1+c2^{0}}{2^{p-1}}\\ \end{array}\right) = \left( \begin{array}{c} \frac{1 + c\sum _{i=0}^{p-1}2^{2i}}{2^{0}}\\ \frac{1 + c\sum _{i=0}^{p-1}2^{2i}}{2^{1}}\\ \frac{1 + c\sum _{i=0}^{p-1}2^{2i}}{2^{2}}\\ \vdots \\ \vdots \\ \frac{1 + c\sum _{i=0}^{p-1}2^{2i}}{2^{p-1}}\\ \end{array}\right) = \left(1 + c\sum _{i=0}^{p-1}2^{2i}\right)\left( \begin{array}{c} \frac{1}{2^{0}}\\ \frac{1}{2^{1}}\\ \frac{1}{2^{2}}\\ \vdots \\ \vdots \\ \frac{1}{2^{p-1}}\\ \end{array}\right) . \end{aligned}$$

Thus, $\varvec{a}_{1}$ is an eigenvector of $\varvec{Cov}_{\varvec{cascade}}$ with an eigenvalue equal to $\lambda _{max}=1 + c\sum _{i=0}^{p-1}2^{2i}$. It is routine to show that all other eigenvectors of $\varvec{Cov}_{\varvec{cascade}}$ have an eigenvalue equal to 1, thus $\lambda _{max}$ is maximal for any positive value of c. $\square$