1.1 Motivation

This is a study in the development of a  Bayesian cognitive engineering model and Bayesian psychometric decision procedure. It is accompanied by the reuse and integration of psychological meta-analysis data. All computations are supported by code written in the Turing-complete functional probabilistic programming language WebPPL. We feel being in the tradition of Westmeyer (1975), Bessiere, Laugier and Siegwart (2008), Pearl (2009), Lee and Wagenmakers (2013), Goodman, Tenenbaum and The ProbMods Contributors (2016), and Levy and Mislevy (2016, 2020). We pursue several goals:

First, we lean on the new and paradoxical metaphor of a cautious gunslinger. We think that a whole range of risky situations (Lefèvre, Vasquez, and Laugier, 2014) can be embedded into this metaphor. The agent has to answer himself three increasing complex counterfactual and metaphorical questions: 1) "Can I draw my revolver fast enough, if my opponent needs only \(\tau_*\) milliseconds to do so ?", 2) "Can I draw my revolver as fast as a randomly selected person of a (younger) reference population, if my opponent needs only \(\tau_*\) milliseconds to do so ?", 3) "Is the probability of drawing my revolver slower than a randomly selected person of a (younger) reference population at most \(p=0.05\) if my opponent needs only \(\tau_*\) milliseconds ?".

Second, the above described gunslinger metaphor can be mapped to the framework of Bayesian decision strategies. We want to show by way of example that within this framework the research question transfer the locus of longitudinal control in Partial Autonomous Driver Assistant Systems (PADAS) can be tackled.

Third, evidence-based priors for our generative Bayesian models are obtained by reuse of meta-analytical results. For demonstration purposes we reuse reaction-time interval estimates of Card, Moran, and Newell's (CMN's) meta-analysis, the Model Human Processor (MHP). According to the MHP total simple reaction times (SRTs) of an arbitrary computer user are composed from three latent time components related to perception, cognition, and motor processes.

Fourth, the modification of priors to posterior probability distributions is weighted by a likelihood function, which is used to consider the SRT data from a single subject as evidence and to measure how plausibly the alternative prior hypotheses generate these data. Posteriors are obtained by runs of the Metropolis-Hastings Markov-Chain-Monte-Carlo (MH-MCMC) algorithm provided in Turing-complete, functional WebPPL.

Fifth, we want to demonstrate the expressiveness and usefulness of WebPPL in computing posterior distributions and personal probabilities of risk. When SRT-specific values-at-risk (Linsmeier & Pearson, 2000; Cottin & Döhler, 2009, p.114ff; Petters & Dong, 2016, p. 178) are externally provided prior-risk-probabilities can be compared to posterior risk-probabilities. It can be checked whether there is a substantial or even striking increase, which we call risk-excess. This way it is possible to answer the above mentioned questions. So, hazardous scenarios (e.g. traffic scenarios) with only a few behavioral data of a single subject (e.g. a driver) can be mapped to the paradoxical and counterfactual cautious gunslinger scenario and to Bayesian psychometric decision procedures.

1.2 Generation of Simple Reaction Times (SRTs) under MHP Guidance

Card, Moran, and Newell's Model Human Processor (MHP)

In their seminal book The Psychology of Human-Computer Interaction (PHCI) Card, Moran, and Newell (CMN, 1983) present the solution of several Human-Computer Interaction (HCI) design problems. The proposed solutions are based on some basic and abstract human information-processing mechanisms and are summarized in the Model Human Processor (MHP). MHP is a (simplified) engineering model and a static calculation guide of the human perceptual-cognitive-motor system. "It can be divided into three interacting subsystems: (1) the perceptual system, (2) the motor system, and (3) the cognitive system, each with its own memories and processors" (CMN, 1983, p.24).

This way, MHP can be used as a simple static calculation guideline, e.g. to predict human response times. According to CMN MHP-guided design activities belong to applied information-processing psychology supporting engineering activities like task analysis, calculation, and approximation (CMN, 1983, p.9f; 1986).  A more recent meta-analysis can be found in Jastrzembski and Charness (2007).

In MHP the meta-analytic knowledge is quantified by time intervals with interval boundaries and 'typical' values. "We can define three versions of the model: one in which all the parameters listed are set to give the worst performance (Slowman), one in which they are set to give the best performance (Fastman), and one set for a nominal performance (Middleman)." (CMN, 1983, p.44)

Interval data from even more recent meta-analyses can easily be integrated into our Bayesian SRT model by substituting the CMN intervals by more recent ones. As an example we refer to Gratzer and Becke (2009, p.126). They present interval data similar to CMN (1983) but for reaction phases in braking events. They report intervals for basic reaction time ('Reaktionsgrundzeit'), conversion time ('Umsetzzeit'), response time ('Ansprechzeit') which add up to total reaction time.

MHP-Composition of Processor Cycle Times

The uncertainties in the parameters of the MHP are captured by three subversions of the MHP (CMN, 1983, p.44):

• Middleman is the version ... in which all the parameters ... are set to give the normal perfomance.
• Fastman is the version ... in which all the parameters ... are set to give the best perfomance.
• Slowman is the version ... in which all the parameters ... are set to give the worst perfomance.

Cycle times of hypothetical perceptual (P), cognitive (C), and motor (M) processor are reported according to the interval-template

$$ \tau_X := \tau_{X_{Middleman}} [\tau_{X_{Fastman}} \sim \tau_{X_{Slowman}}] \text{  ;    } X = P, C, M   \qquad(1) $$

and similarly for the total reaction time \( T \)

$$\tau_T := \tau_{T_{Middleman}} [\tau_{T_{Fastman}} \sim \tau_{T_{Slowman}}] \qquad(2)$$

According to CMN  \(\tau_T\) should be the sum of the specific component cycle times \(\tau_{X_{Zman}} \; ; X = P, C, M \; ; Z = Middle, Fast, Slow \) :

$$\tau_{T_{Zman}} = \sum_{X \in \{P,C,M\}} \tau_{X_{Zman}} \qquad(3)$$

The meaning of (1) is that \( \tau_X \) is ranging from \(\tau_{X_{Fastman}}\) to \(\tau_{X_{Slowman}}\) with a 'typical' value \(\tau_{X_{Middleman}}\) and (2) has a similar meaning for the total reaction time \(\tau_T\). CMN not only provide quantitative intervals for (1) but also for (2) which obey the constraints (3). But a given left side of (2) can be fulfilled by many more 3-tuples \((\tau_P, \tau_C, \tau_M) \) not considered in (3) on the right-hand side. This is why we introduce for modelling purposes a new less constrained variable \(\tau_\Sigma \) replacing \(\tau_T\).

$$ \tau_\Sigma := \tau_P + \tau_C + \tau_M = \sum_{X \in \{P,C,M\}} \tau_X . \qquad(4) $$

The semantics of a 'typical value' is not formally specified by CMN (CMN, 1983, p.44f). We attempt various formal interpretations of the ambiguous term 'typical value' through statistics such as mode, median, and mean. These interpretations lead to different weakly informed prior belief distributions.

MHP-\(\tau_X\)-intervals

Cycle-times \(\tau_X \) and related \(\tau_X \; ; \; X \in \{ P, C, M \}\)-intervals are displayed in Table 1 (CMN, Fig 2.1, p.26):

• \(\tau_P := 100 \; [  \; 50 \sim 200] \text{ msec } \) ; cycle time of perceptual processor (CMN, 1983, p.32f\)
• \(\tau_C :=  \; 70 \; [ \; 25 \sim 170] \text{ msec } \) ; cycle time of cognitive processor (CMN, 1983, p.42f)\)            (Table 1)
• \(\tau_M := \;  70 \; [ \; 30 \sim 100] \text{ msec } \) ; cycle time of motor processor (CMN, 1983, p.34f)
• \(\tau_T := 240 \; [105 \sim 470] \text{ msec } \) ; total reaction time (CMN, 1983, Fig 2.1, p.26, p.66, p.433f)

Generation of Simple Reaction Times (SRTs) in MHP's Example 9

One of the standard problems in The Psychology of Human-Computer Interaction (CMN, 1983) to motivate the use of MHP is example 9: A user sits before a computer display terminal. Whenever any symbol appears, he is to press the space bar. What is the time between signal and response? (CMN, 1983, p.66).

Solution. "Let us follow the course of processing through the Model Human Processor in Figure ... The user is in some state of attention to the display ...When some physical depiction of the letter A (we denote it \(\alpha\)) appears, it is processed by the Perceptual Processor, giving rise to a physically-coded representation of the symbol (we write it \(\alpha'\)) in the Visual Image Store and very shortly thereafter to a visually coded symbol (we write it \(\alpha''\)) in Working Memory... This process requires one Perceptual Process cycle \(\tau_P \). The occurrence of the stimulus is connected with a response...,requiring one Cognitive Processor cycle, \(\tau_C \). The motor system then carries out the actual physical movement to push the key..., requiring one Motor Processor cycle, \(\tau_M \). Total time required is \(\tau_P + \tau_C + \tau_M \). Using Middleman values, the total time required is 100 + 70 + 70 = 240 msec. Using Fastman and Slowman values gives a range 105 ~ 470 msec. " \( \blacksquare \) (CMN, Fig 2.1, p.26, p.66, p.433f)

These are the cycle times in msec of the hypothetical perceptual processor, the cognitive processor, and the motor processor, respectively.

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Because the MHP-intervals provide only scarce information in form of the 'typical' values, the lower, and the upper bounds we chose as weakly informative priors the triangular (Pt. 2.1-2.3) and the \(\Gamma\)-distribution (Pt. 2.4). When using triangular priors the 'inspired guesses' are the mode, the median, and the mean, and again the mean in the case of \(\Gamma\) priors.

Here in Pt 2.1 the "inspired guess" is the mode of the triangular prior pdf. In other words we force the MHP-value \( \tau_{X_{Middleman}} \) to be the mode of the prior pdf.

The PDF is

$$ f_{Triangle}(x | a, b, c) := \left\{ \begin{array}{clr} 0 & \text{    for  } x \lt a, \\ \frac{2(x-a)}{(b-a)(c-a)} & \text{    for  } a \leq x \lt c, \\ \frac{2}{(b-a)} & \text{    for  } x = c, & \qquad(8) \\ \frac{2(b-x)}{(b-a)(b-c)} & \text{    for  } c \lt x \leq b, \\ 0 & \text{    for  } b \lt x. \end{array} \right .$$

The generation of triangular-distributed random variates is done by the following mapping. Given a random variate \(U\) drawn from the uniform distribution in the interval \((0, 1)\), then the variate

$$ X_{Triangle} | a, b, c := \left\{ \begin{array}{clr} a + \sqrt{U(b-a)(c-a)} & \text{    for  } 0 \lt U \lt F(c), & \qquad(9.1) \\ b - \sqrt{(1-U)(b-a)(b-c)} & \text{    for  } F(c) \leq U \lt 1 & \qquad(9.2) \end{array} \right.$$

where $$F(c | a, b) := (c - a) / (b - a). $$.

The prior pdfs are displayed in Figs 2.1.01 - 05. In Fig 2.1.01 - 03 the MHP-values \( \tau_{X_{Middleman}} \) were defined to be the mode of the corresponding triangular priors. The \( \tau_\Sigma\)-distribution in Fig 2.1.04 is the convolution of the priors \(\tau_P, \tau_C, \tau_M\) in Figs 2.1.01 - 03. This was obtained by summing the samples of the latent components (4).

Fig 2.1.05 displays the prior pdf \(\sigma_{\tau_{\Sigma}} \sim \Gamma(k=4, \theta=20)\) for the standard deviation \(\sigma_{\tau_{\Sigma}}\) of the Gaussian likelihood function (10) for i.i.d. SRT-data

$$  L_N(\tau_\Sigma, \sigma_{\tau_{\Sigma}} |SRTs) := \prod_{i=1}^m N(SRT_i | \tau_\Sigma, \sigma_{\tau_{\Sigma}}) \text{  ; m = #SRTs = nr of SRT-data.} \qquad(10) $$

We chose \(\Gamma(4, 20)\) because we thought that a mean of \( \mu=80\) and a \(\sigma=40\) for the \(\Gamma\)-pdf of the \(\sigma_{\tau_{\Sigma}}\) would provide sufficient unexplained variation in the Gaussian likelihood independent of the true variation due to \(\tau_\Sigma\). The independence assumption by introducing the product in (10) could be criticized because all SRT-data (Fig 3.1) stem from a single subject. But for the moment we stick to this assumption.

Three parametrizations for the \(\Gamma\) distribution are known. We chose that one with shape \(k\) and scale \(\theta\) parameter (with \(k > 0,\;  \theta > 0)\). Under this selection expectation, variance, and standard deviation are functions of these hyperparameters \(k\) and \(\theta\):

$$E(\sigma_{\tau_{\Sigma}}|k,\theta) =  \mu(\sigma_{\tau_{\Sigma}}|k,\theta) = k\theta = 4 \cdot 20 = 80, $$
$$ $$
$$ Var(\sigma_{\tau_{\Sigma}}|k,\theta) = \sigma^2(\sigma_{\tau_{\Sigma}}|k,\theta) = k\theta^2 = 4 \cdot \theta^2 = 4 \cdot 20^2 = 1600 $$
and $$\sqrt {Var(\sigma_{\tau_{\Sigma}}|k,\theta)} = \sigma(\sigma_{\tau_{\Sigma}}|k,\theta) = \sqrt {k}\theta\ = \sqrt{4} \cdot 20 = 40.$$

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As a second alternative for a weakly informative prior we chose the triangular distribution with the median as an "inspired guess" for MHP's 'typical value'. In other words we force the MHP-value \(\tau_{X_{Middleman}}\) to be the median of the prior pdf.

The median of a triangular distribution is defined as:

$$ median_{Triangle} = \left\{ \begin{array}{ccc}  a + \sqrt{\frac{(b-a)(c-a)}{2}}, & \text{   for  } c \geq \frac{a+b}{2}, & \qquad(11.1) \\ b - \sqrt{\frac{(b-a)(b-c)}{2}}, & \text{   for  } c \lt \frac{a+b}{2} & \qquad(11.2) \end{array} \right. $$

If we interpret the 'typical' value \(\tau_{X_{Middleman}}\) as the median of a \(Triangle(a, b, c)\)-pdf with bounds \(a = \tau_{X_{Fastman}}\) and \(b = \tau_{X_{Slowman}}\) then we need to determine the unknown third parameter \(c = mode_{triangle}\) from the information given by \(median_{Triangle}, \tau_{X_{Fastman}}, \text{ and } \tau_{X_{Slowman}}\). In short, we apply the function \(h_{median_{Triangle}}\) to instantiated \(median_{Triangle}, \tau_{X_{Fastman}}, \text{ and } \tau_{X_{Slowman}}\) statistics. The result of the function application provides the desired \(mode_{Triangle} = c \).

$$c := mode_{Triangle} = h_{median_{Triangle}}( \tau_{X_{Fastman}}, \tau_{X_{Slowman}},\tau_{X_{Middleman}})  \qquad(12)$$

which is

$$ c := mode_{Triangle} = h_{median_{Triangle}}(a, b, median_{Triangle}) \qquad(13)$$

First, we derive \(c\) when \( c \geq \frac{a+b}{2}\; \) and \(md \equiv median :\)

$$(md - a) = \sqrt{\frac{(b-a)(c-a)}{2}}$$

$$(md - a)^2 = \frac{(b-a)(c-a)}{2}$$

$$2(md - a)^2 = (b-a)(c-a)$$

$$\frac{2(md - a)^2}{(b-a)} = (c-a)$$

$$c := h_{median_{Triangle}}(a, b, md_{Triangle}) = \frac{2(md - a)^2}{(b-a)} + a \; \qquad(14)$$

Second, we derive \(c\) when \( c \lt \frac{a+b}{2}\;:\)

$$(md - b) = - \sqrt{\frac{(b-a)(b-c)}{2}}$$

$$- (md - b) =  \sqrt{\frac{(b-a)(b-c)}{2}}$$

$$(md - b)^2 =\frac{(b-a)(b-c)}{2}$$

$$2(md - b)^2 = (b-a)(b-c)$$

$$\frac{2(md - b)^2}{(b-a)} = (b-c)$$

$$\frac{2(md - b)^2}{(b-a)} - b = - c$$

$$c := h_{median_{Triangle}}(a, b, md_{Triangle}) = - \frac{2(md - b)^2}{(b-a)} + b  \; \qquad(15)$$

The prior pdfs are displayed in Figs 2.2.01 - 05. We recognize that in Figs 2.2.01 - 03 the MHP-values \( \tau_{X_{Middleman}} ; X = P, C, M \) were defined to be the median of the corresponding triangular prior. The \( \tau_\Sigma\)-distribution in Fig 2.2.04 is the convolution of the priors \(\tau_P, \tau_C, \tau_M\) in Figs 2.2.01 - 03. This was obtained by summing the samples of the latent components (4).

Fig 2.2.05 displays the prior \(\sigma_{\tau_{\Sigma}} \sim \Gamma(k:=4, \theta:=20)\)-pdf for the standard deviation \(\sigma_{\tau_{\Sigma}}\) of the Gaussian likelihood function (10) for i.i.d. SRT-data.

$$  L(\tau_\Sigma, \sigma_{\tau_{\Sigma}}|SRTs) := \prod_{i=1}^m N(SRT_i; \tau_\Sigma, \sigma_{\tau_{\Sigma_{prior}}}) \text{  ; m = #SRTs}.  \qquad(10) $$

This distribution deviates from  Fig.2.1.05 only by sampling errors due to the MH-MCMC sampling process.

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As a third alternative for a weakly informative prior we chose the triangular distribution with the mean as an "inspired guess" for MHP's 'typical value'. In other words we force the MHP-value \( \tau_{X_{Middleman}} \) to be the mean of the prior pdf.

The mean of a triangular distribution is defined as:

$$ mean_{Triangle} = \frac{a+b+c}{3}\qquad(16)$$

If we interpret the 'typical' value \(\tau_{X_{Middleman}}\) as the mean of a \(Triangle(a, b, c)\) pdf with bounds \(a = \tau_{X_{Fastman}}\) and \(b = \tau_{X_{Slowman}}\) we need to obtain the unknown third parameter \(c = mode_{triangle}\) of \(Triangle(a, b, c)\) from the information given by \(mean_{Triangle}, \tau_{X_{Fastman}}, \text{ and } \tau_{X_{Slowman}}\). In short, we apply the function \(h_{mean_{Triangle}}\) to instantiated statistics \(mean_{Triangle}, \tau_{X_{Fastman}}, \text{ and } \tau_{X_{Slowman}}\) to generate \(c\):

$$c := mode_{Triangle} = h_{mean_{Triangle}}( \tau_{X_{Fastman}}, \tau_{X_{Slowman}},\tau_{X_{Middleman}}) \qquad(17)$$

which is

$$ c := mode_{Triangle} = h_{mean_{Triangle}}(a, b, mean_{Triangle}) \qquad(18)$$

where

$$c := h_{mean_{Triangle}}(a, b, mean_{Triangle}) = 3 \cdot mean_{Triangle} - (a + b). \qquad(19)$$

The prior pdfs are displayed in Fig 2.3.01 - 05. We recognize that in Fig 2.3.01 - 03 the MHP-values \( \tau_{X_{Middleman}} \) were defined to be the mean of the corresponding triangular prior. The \( \tau_\Sigma\)-distribution in Fig 2.3.04 is the convolution of the priors \(\tau_P, \tau_C, \tau_M\) in Fig 2.3.01 - 03. This was obtained by summing the samples of the latent components (4).

Fig.2.3.05 displays the prior \(\sigma_{\tau_{\Sigma}} \sim \Gamma(k:=4, \theta:=20)\)-pdf for the standard deviation of the Gaussian likelihood function (10). This distribution deviates from  Fig 2.1.05 and Fig 2.2.05 only by sampling errors due to the MH-MCMC sampling process.

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As a fourth alternative for a weakly informative prior we chose the Gamma pdf. We treat all process cycle times \(\tau_X \) as gamma distributed waiting times though other distributions are also popular by cognitive psychologists (Luce, 1986; Wickens, 1982; Van Zandt, 2000).

There are several reasons for this decision. First, because the 'typical value' \( \tau_{X_{Middleman}} \) does not lay in the middle of the CMN-interval it is not wise to use the mean of a symmetric distribution like a Gaussian as the 'typical value' of a prior. In contrast the Gamma pdf can be left-skewed. Second, if we assume that the processor cycle times \( \tau_X ; X = P, C, M \) are results of many independent subprocesses with exponential waiting times then their sum is gamma distributed. The convolution of exponential distributions is a Gamma distribution. In a special case when some parameters are equal a similar result is true for independent Gamma-distributed subprocesses. For the general case with nonidentical parameters no closed solution is known. This general case can be handled quite easily within the sampling approach of WebPPL.

Here the 'inspired guess' is the mean of the Gamma prior pdf. In other words we force the CMN-value \( \tau_{X_{Middleman}} \) to be the mean \( \mu_\Gamma(X) \) of the Gamma prior pdf.

Convolution of Waiting Time and Processor Cycle Time Distributions

According to CMN the total cycle time \(\tau_T\) should be the sum of the specific component cycle times \(\tau_{X_{Zman}} \; ; X = P, C, M \; ; Z = Middle, Fast, Slow \) :

$$\tau_{T_{Zman}} = \sum_{X \in \{P,C,M\}} \tau_{X_{Zman}} \qquad(3)$$

We decided that the prior pdfs of the subcomponent cycle times \( \tau_X \) should be Gamma-distributed.  The convolution of Gammas is known to be a Gamma only in the special case when one or both parameters of the to be convoluted Gammas are identical. It seems that in our general case where all parameters are different we have to compute the result of the convolution within a numerical simulation within WebPPL.

Extracting Gamma Distribution Parameters from MHP's Empirical Constraints

There exist three slightly different parametrizations of the gamma distribution. We chose that one with shape parameter \(k\) and scale parameter \(\theta\) (\(k>0, \theta > 0)\). Mean, variance, and standard deviation of a gamma-distributed random variable \(X\) are functions of these parameters:

$$E_\Gamma(X|k,\theta) = k\theta =  \mu_\Gamma(X), \qquad(2.4.20.1)$$

$$Var_\Gamma(X|k,\theta) = k\theta^2 =\sigma_\Gamma^2(X), \qquad(2.4.20.2) $$

and $$\sqrt {Var_\Gamma(X)} = \sqrt {k}\theta\ = \sigma_\Gamma(X). \qquad(2.4.20.3)$$

In the WebPPL scripts we use the substitutions \(k/a\) ("\(a\) for \(k\)"),  \(\theta/b\) ("\(b\) for \(\theta\)"), \(\mu_\Gamma/m\) ("\(m\) for \(\mu_\Gamma\)"), and \(\sigma_\Gamma/s\) ("\(s\) for \(\sigma_\Gamma\)").

To specify the Gamma parameters \(k, \theta \) we map the concepts of CMN's empirical intervals

$$ \tau_X := \tau_{X_{Middleman}} [\tau_{X_{Fastman}} \sim \tau_{X_{Slowman}}]  \;\;; \;\;   X = P, C, M. \qquad(1)$$

to the parameters \(\mu_\Gamma\) and \(\sigma_\Gamma\) of the Gamma pdf and the spread parameter \(r\) (the identifiers \(a, b, m, s, r \) are symbols in WebPPL scripts): 

                                                                        $$ mode_\Gamma(X|a,b) = (a-1)b \qquad(2.4.21.1)$$

$$\mu_\Gamma(X|a,b) = ab =: m \qquad(2.4.21.2)$$

$$\sigma_\Gamma(X|a,b) = \sqrt{a}b =: s \qquad(2.4.21.3)$$

$$\tau_{X_{Fastman}} \approx (ab - r\sqrt{a}b) =: (m-rs) \qquad(2.4.22.1)$$

and

$$\tau_{X_{Slowman}} \approx (ab + r\sqrt{a}b) =:  m+rs,\qquad(2.4.22.2)$$

where \(r\) is the number of standard deviations ('nSigma' in WebPPL) measuring the distance between interval boundaries \(\tau_{X_{Fastman}}\) or  \(\tau_{X_{Slowman}}\) and mean \(\mu_\Gamma(X) \).

If we assume that the distance between interval mean \(\mu_\Gamma(X) \) and interval bounds \(\tau_{X_{Fastman}} \) or \(\tau_{X_{Slowman}}\) can be approximated by (2.4.22) so that \( \tau_{X_{Fastman}} = (\mu_X - r\sigma_X)\) and \( \tau_{X_{Slowman}} = (\mu_X + r\sigma_X)\) we can use Chebyshev's inequality

$$P(|X-\mu| \ge r\sigma) \le \frac{1}{r^2}\qquad(2.4.23)  $$

to make an approximate determination of \(r\) according:

$$ r\sigma\text{-Intervals Derived from Chebyshev's Inequality}$$

$$\text{Probability of Staying Within and Risk of Leaving Interval} $$

$$\begin{array}{|c|c|c|} \hline \\
r & P(|X-\mu| \le r\sigma) \ge 1 - \frac{1}{r^2} & P(|X-\mu| \ge r\sigma) \le \frac{1}{r^2}   \\   \text{ } & P(\text{Within CMN-Interval}) & P(\text{Outside CMN-Interval}) \\ \hline \text{...} & \text{.........} & \text{.........} \\
3 & P(|X-\mu| \le r\sigma) \ge 0.8888 & P(|X-\mu| \ge r\sigma) \le \frac{1}{9} = 0.1111 \\ 4 & P(|X-\mu| \le r\sigma) \ge 0.9375 & P(|X-\mu| \ge r\sigma) \le \frac{1}{16} = 0.0625 \\    \text{...} & \text{.........} & \text{.........} \\  \hline \end{array} \qquad(2.4.24) $$

So the width of an \(r\sigma\)-interval  (\(r = 1,2,3, ... \)) is

$$[\tau_{X_{Slowman}} - \tau_{X_{Fastman}}] \approx 2r\sigma_X = 2r\sqrt{a}b =: 2rs.$$

We chose \(r = 3\) because the risk of observing a random value outside the \(3\sigma\)-inverval with probability \(p \le 0.1111\) is sufficiently small.

The semantics of the CMN-intervals can be now be approximated by the parameters of a gamma \( \Gamma(a, b) \)-distribution:

$$\hat \tau_X := \hat \tau_{X_{Middleman}} [\hat \tau_{X_{Fastman}} \sim \hat \tau_{X_{Slowman}}] \approx  ab[ab-r\sqrt{a}b \sim ab+r\sqrt{a}b] \qquad(2.4.25).$$

Of course this approximation includes an error, because this interval is symmetric about the mean, whereas this is not true for the CMN-interval and its 'typical value' \(\tau_{X_{Middleman}}\).  Later when have generated the \( \hat \tau_\Gamma(X) \)-distributions we can compare our generated \( \hat \tau_\Gamma(X) \)-intervals with those from CMN and compute the probabilities or risks of exceeding the CMN-interval limits. As we shall see the probability of exceeding the CMN-intervals by a sample from our simulated Gamma-distributions is negligible small.

Now, we have to identify the parameters or \(a\) and \(b\). Solving towards \(a\) and \(b\) we get:

$$a := \frac{ab}{b}=\frac{m}{b}$$

$$b := \frac{[\tau_{X_{Slowman}} - \tau_{X_{Fastman}}]}{2r \sqrt{a}}$$

 We substitute \(a = \frac{m}{b}\) into the formula for b:

$$b=\frac{[\tau_{X_{Slowman}} - \tau_{X_{Fastman}}]}{2r \sqrt{\frac{m}{b}}}$$

squaring both sides:

$$b^2=\frac{[\tau_{X_{Slowman}} -\tau_{X_{Fastman}}]^2}{2^2r^2 \left(\frac{m}{b}\right)}=\frac{[\tau_{X_{Slowman}} -\tau_{X_{Fastman}}]^2}{2^2r^2}\frac{b}{m}$$

Cancelling \(b\) on both sides:

$$b := \frac{[\tau_{X_{Slowman}} - \tau_{X_{Fastman}}]^2}{2^2r^2 m} \qquad(2.4.26)$$

Now, \(b\) is totally identified by the estimators or empirical constraints \(\tau_{X_{Slowman}}\), \(\tau_{X_{Fastman}}\), and \( \mu_\Gamma(X) = m \). This is not yet the case for \(a\). So we substitute \(b\) back into the equation for \(a\). This results in:

$$a := \frac{m}{b}=m*\frac{2^2r^2m}{[\tau_{X_{Slowman}} - \tau_{X_{Fastman}}]^2}=\frac{(2rm)^2}{[\tau_{X_{Slowman}} - \tau_{X_{Slowman}}]^2}\qquad(2.4.27)$$

We check the results:

$$ab=\frac{(2rm)^2}{[\tau_{X_{Slowman}} - \tau_{X_{Fastman}}]^2} * \frac{[\tau_{X_{Slowman}} - \tau_{X_{Fastman}}]^2}{(2r)^2 m}=\frac{2^2r^2m^2}{2^2r^2m}=m $$

The simulation to specify the shape of the prior Gamma-pdfs starts with two constants \(nTrials\) and \(nSigma\). \(nTrials\) is relevant for the WebPPL-\(Infer\) function which generates \( nTrials (= 50000) \) samples \( \hat \tau_X \sim \Gamma_X(a_X,b_X) \) with the \(forward\) method. The second constant is \(nSigma\)  which determines the interval width \(2 \cdot nSigma \cdot  \sigma = 6 \cdot \sigma \). We have chosen \(nSigma = 3\) because the probability that a \( \hat \tau_X \) is sampled outside the interval should be a rare event. According to Chebyshev's Inequality (s.above) the probability of staying within the \(3\sigma\)-interval is for any distribution is \(p < 0.1111\). We shall demonstrate that our generated \(tau_\Gamma(X)\)-priors stay within the CMN-intervals with a probability \(p < 0.98\) and leave the CMN-intervals with \(p< 0.02\).

In the first step of the simulation we generate a data structure containing the \(\tau_X \)-intervals as published by CMN (2.4.28). We added a column \(\tau_{X_{middle}}\). This designates the middle of the interval:

$$\tau_{X_{middle}} = \frac{\tau_{X_{Fastman}} + \tau_{X_{Slowman}}} {2}$$

$$\tau_X \text{-CMN-intervals}$$

$$\begin{array}{|c|ccc|} \hline
\tau_X & \tau_{X_{Middleman}} & \tau_{X_{Fastman}}  & \tau_{X_{middle}} & \tau_{X_{Slowman}} \\
\hline
\tau_P & 100 & 50 & 125 & 200 \\
\tau_C &   70 & 25 &   97.5 & 170 \\
\tau_M &  70 & 30 &   65 & 100 \\ \hline \tau_\Sigma = \sum_{X \in {P, C, M}} \tau_X & 240 & 105 & 287.5 & 470 \\ \hline
 \end{array}  \qquad(2.4.28)$$

The last line of the above table is a result of summing the \(\tau_X\)-interval values.

Now, the Gamma-parameters \(a_{\tau_X}\) and \(b_{\tau_X}\) can be obtained by (2.4.27) and (2.4.26). These values are documented in the second and third columns of the following table (2.4.29). From the two parameters \(a_{\tau_X}\) and \(b_{\tau_X}\) we can compute \(mode_{\tau_X}, \mu_{\tau_X}\) and \( \sigma_{\tau_X}\) of the Gamma pdf according to (2.4.21.1), (2.4.21.2), and (2.4.21.3). These parameters can be found in columns 4 - 6 of the following table (2.4.29).

$$\text{                                                                                           }$$

$$\Gamma_{\tau_X} \text{-parameters} $$

$$\begin{array}{|c|cc|ccc|} \hline
\hat \tau_X & a_{\tau_X} (2.4.27) & b_{\tau_X} (2.4.26) & mode_{\tau_X} (2.4.21.1) & \mu_{\tau_X} (2.4.21.2) & \sigma_{\tau_X} (2.4.21.3) \\ \hline
\hat \tau_P & 16.00 & 6.25 & 93.75 & 100 & 25 \\
\hat \tau_C &   8.39 & 8.34 & 61.65 &   70  & 24.1 \\ \hat \tau_M & 36.00 & 1.94 & 68.05 &  70 & 11.66 \\ \hat \tau_T & 15.56 & 15.42 & 224.58 & 240 & 60.6 \\\hline
 \end{array} \qquad(2.4.29) $$

$$\text{                                                                                           }$$

Having identified the basic \(\Gamma\)-parameters \(a_X (2.4.27)\) and \(b_X (2.4.26) \) we are able to reconstruct the \( \hat \tau_X\)-intervals according (2.4.25). These are found in columns 4 - 6 in the table (2.4.30) below. As expected the \( \hat \tau_{X_{Middleman}} \) are nearly identical to the CMN's \( \tau_{X_{Middleman}} \) though the lower and upper \( \hat \tau_X\)-interval bounds deviate slightly from their CMN-\( \tau_X\)-counterparts. The best fit is for the motor-processor cycle-time \(\tau_M\).

As can be seen from (2.4.31) and (2.4.32; second to last column in table (2.4.30)), sampling is done for \( \hat\tau_X \) from the the gamma distributions parametrized with the \(a_{\tau_X} (2.4.27) \) and \(b_{\tau_X} (2.4.26) \). \(a_{\tau_X} \) and \(b_{\tau_X} \) were directly obtained from CMN's interval values. (2.4.32.5; table 2.4.30) is computed in a different way than (2.4.32.4). Whereas (2.4.32.4) is directly computed from CMN's intervals by (2.4.26) and (2.4.27), the case for (2.4.32.5) is different. (2.4.32.5) is the sum of the sampled values (2.4.32.1) - (2.4.32.3). So (2.4.32.5) is in some way a numeric equivalent to a convolution of distributions (2.4.32.1) - (2.4.32.3). The range of this 'convoluted' interval is a bit smaller than the values (2.4.32.4) obtained directly from CMN's interval. Because this distribution is only existent as an array of nTrials samples it has no parameters \( a_{\tau_X} \) and \( b_{\tau_X} \). So we wanted to condense this array of numeric particles into a parametrized Gamma pdf with \( a_{\tau_X} \) and \( b_{\tau_X} \). These parameter estimates are presented in columns 2-3 in the last line of the next table (2.4.33). Only with these two parameters at hand we can derive the interval values (columns 4-6 in last line of table (2.4.30). But, interval values of (2.4.32.5) and (2.4.32.6) are nearly identical.

$$ \hat \tau_X \sim \Gamma_{\tau_X}(a_{\tau_X}, b_{\tau_X}) ; X \in {P, C, M, T} \qquad(2.4.31) $$

$$\text{Model-generated } \hat \tau_X \text{-intervals (2.4.25)  and Sampled Distributions } \hat \tau_X \sim \Gamma_{\tau_X}(a_{\tau_X},b_{\tau_X})$$

$$\begin{array}{|c|cc|ccc|lc|c|} \hline\hat \tau_X & a_{\tau_X} & b_{\tau_X} & \hat \tau_{X_{Middle}} & \hat \tau_{X_{Fast}} & \hat \tau_{X_{Slow}} & \hat \tau_X \sim \Gamma_{\tau_X}(a_{\tau_X}, b_{\tau_X}) & (2.4.32.k) & \text{Fig} \\ \hline \hat \tau_P & 16.0 & 6.3 & 100.0 & 24.7 & 175.2 & \hat \tau_P \sim \Gamma_{\tau_P}(a_P,b_P) & (2.4.32.1) & \text{2.4.1} \\ \hat \tau_C & 8.4 & 8.3 &  69.8 & -2.5 & 142.2 & \hat \tau_C \sim \Gamma_{\tau_C}(a_C,b_C) & (2.4.32.2) & \text{2.4.2} \\ \hat \tau_M & 36.0 & 1.9 & 70.1 & 35.2 & 104.9 & \hat \tau_M \sim \Gamma_{\tau_M}(a_M,b_M) & (2.4.32.3) & \text{2.4.3} \\ \hat \tau_T & 15.6 & 15.4 & 240.0 & 56.9 & 423.1 & \hat \tau_T \sim \Gamma_{\tau_T}(a_T,b_T) & (2.4.32.4) & \text{2.4.4} \\ \hline \hat \tau_\sum & - & - & 239.9 & 129.7 & 350.5 & \hat \tau_\sum = \sum_{X \in {P, C, M}} \hat \tau_X & (2.4.32.5) & \text{2.4.5} \\ \hat \tau_\sum & 42.4 & 5.7 & 240.1 & 129.8 & 350.5 & \hat \tau_\sum \sim \Gamma_\sum(42.4, 5.7) & (2.4.32.6) & \text{2.4.6} \\ \hline \end{array}\\ \qquad(2.4.30)$$

Now, we study the risk of generating \(\Gamma_{\hat\tau_X}\)-random values falling outside the CMN-interval boundaries. These risk-probabilities are compiled in the following table (2.4.33):

$$\text{Probability of a \(\Gamma_{\hat\tau_X}\)-generated Samples of Falling Outside the CMN-Interval Boundaries}$$

$$\begin{array}{|c|cc|cc|cc|} \hline
\hat \tau_X & a_{\tau_X} & b_{\tau_X} & \hat \tau_{X_{Fastman}} & \hat \tau_{X_{Slowman}} & P(\hat\tau_X < \tau_{X_{Fastman}}) & P(\hat\tau_X > \tau_{X_{Slowman}}) \\
\hline
\hat \tau_P        & 16.0 & 6.3   &   50 & 200  & 0.00878 & 0.00054 \\
\hat \tau_C        &   8.4 & 8.3   &   25 & 170  & 0.00828 & 0.00074 \\
\hat \tau_M       & 36.0 & 1.9   &   30 & 100  & 0.00000 & 0.00926 \\ \hline                                                               \hat \tau_T        & 15.6 & 15.4 & 105 & 470  & 0.00294 & 0.00138 \\ \hat \tau_\sum & 42.4 & 5.7   & 105 & 470  & 0.00000 & 0.00000 \\ \hline 
 \end{array} \qquad(2.4.33)$$

We see that the risk of a Gamma-generated \(\hat\tau_X\)-sample falling outside a CMN-interval is as small as \(p < 0.009\). This means that our Gamma distributions are not generating values untypical for the CMN-intervals. From this aspect they are good priors derived from the CMN-intervals. Maybe our Gamma-generated \(3\sigma\)-intervals are too narrow. We have to study this question, when visualizing the Gamma distributions.

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Sampling according to our generative model is done with four sampling subprocesses:

$$ \text{1. Unconditional sampling triangular priors for interval modes as 'typical' values: }$$

$$ \tau_P \sim Triangle_{mode}(a=50, b=200, c=100)  $$

$$ \tau_C \sim Triangle_{mode}(a=25, b=170, c=70)  $$

$$ \tau_M \sim Triangle_{mode}(a=30, b=100, c=70) $$

$$ \text{2. Unconditional sampling priors for the standard deviation in the Gaussian likelihood: } $$

$$ \sigma_{\tau_{\Sigma}} \sim \Gamma(k=4, \theta=20) $$

$$ \text{3. Computing the deterministic function sum of priors: } $$

$$ \tau_\Sigma = \sum_{X \in \{P,C,M\}} \tau_X$$

$$ \text{4. Conditional sampling from Gaussian likelihood with variable priors and fixed SRTs:} $$

$$ (\tau_P, \tau_C, \tau_M, \tau_\Sigma, \sigma_{\tau_{\Sigma}} | SRTs) \sim L_N( \tau_\Sigma, \sigma_{\tau_\Sigma} | SRTs ) = \prod_{i=1}^m N(SRT_i | \tau_\Sigma,  \sigma_{\tau_{\Sigma}}) $$

The sampled values of the priors are summed up to \(\tau_\Sigma \). Then the likelihood of the SRT-data is evaluated conditional on the prior latent \(\tau_\Sigma \) and the prior \(   \sigma_{\tau_{\Sigma}} \).

Besides the code for function oneSampleOfModel the most central code line in the WebPPL-script is:

var posterior = Infer({model:oneSampleOfModel, method:'MCMC', samples: nTrials, burn:myBurnPeriod, lag:myLag})

The function Infer generates the conditional 4-tuple-samples of the joint posterior distribution:

$$ (\tau_P, \tau_C, \tau_M, \tau_\Sigma, \sigma_{\tau_{\Sigma}} | SRTs) \qquad{(23.12)} $$

with the help of the Markov-Chain Monte-Carlo (MCMC)-method (Robert & Casella, 2004, ch.7; Murphy, 2012, ch.23; Lunn et al, 2014, ch.4) with the parameters \(nTrials = 6E4 = 60 000\), \(myBurnPeriod = nTrials \cdot 0.10\), and \(myLag=10 \).

\(nTrials \) is the number of random trials or samples, \(myBurnPeriod \) is the length of the 'burn-in'-period at the beginning of the MCMC-process, and \(myLag\) is the sampling jump distance of the algorithm to avoid autocorrelation between the samples. A lag of 10 means that only each 10th sample is kept; all others are discarded. Also all samples of the 'burn-in' period are discarded.

All generated samples (23) constitute the support of the joint posterior pdf

$$P ( \tau_P, \tau_C, \tau_M,\tau_\Sigma, \sigma_{\tau_{\Sigma}} | SRT),  \qquad(24) $$

which posterior marginal densities \(Triangle_{mode}( \tau_X | SRT)\) and \(f( \tau_\Sigma | SRT)\) (Fig 3.1.01-04) are reconstructed from the posterior samples (23) by kernel density estimation methods (Hastie, Tibshirani & Friedman, 2001, ch.6.6.1; Murphy, 2012, ch.14.7.2). In addition we display the posterior pdf of the standard deviation of the Gaussian likelihood in Fig 4.1.05. Our SRT-data are distributed with approximately \( \sigma_{\tau_\Sigma} = 85 \) around the marginal posterior \(\tau_\Sigma | SRT \).

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Sampling according to our generative model is done with four sampling subprocesses:

$$ \text{1. Unconditional sampling triangular priors for interval medians as 'typical' values: } $$

$$ \tau_P \sim Triangle_{median}(a=50, b=200, c=66.7) $$

$$ \tau_C \sim Triangle_{median}(a=25, b=170, c=32.1) $$

$$ \tau_M \sim Triangle_{median}(a=30, b=100, c=75.7) $$

$$ \text{2. Unconditional sampling priors for standard deviation of Gaussian likelihood: } $$

$$ \sigma_{\tau_\Sigma} \sim \Gamma(k=4, \theta=20) $$

$$ \text{3. Computing the deterministic function sum of priors: } $$

$$\tau_\Sigma = \sum_{X \in \{P,C,M\}} \tau_X$$

$$ \text{4. Conditional sampling from Gaussian likelihood with variable priors and fixed SRTs:} $$

$$ (\tau_P, \tau_C, \tau_M, \tau_\Sigma, \sigma_{\tau_\Sigma} | SRTs) \sim L_N( \tau_\Sigma, \sigma_{\tau_\Sigma} ; SRTs ) = \prod_{i=1}^m Gaussian(SRT_i | \tau_\Sigma, \sigma_{\tau_\Sigma}). $$

The function Infer generates the conditional 4-tuple-samples of the joint posterior distribution (23)

$$( \tau_P, \tau_C, \tau_M, \tau_\Sigma, \sigma_{\tau_\Sigma}) | SRT  $$

by the Markov-Chain Monte-Carlo (MCMC)-method (Robert & Casella, 2004, ch.7; Murphy, 2012, ch.23; Lunn et al, 2014, ch.4) with the parameters \(nTrials = 6E4 = 60 000\), \(myBurnPeriod = nTrials \cdot 0.10\), and \(myLag=10 \).

All generated samples (23) constitute the support of the joint posterior pdf

$$P ( \tau_P, \tau_C, \tau_M,\tau_\Sigma, \sigma_{\tau_\Sigma} | SRT),  \qquad(24) $$

which marginal densities \(Triangle_{median}( \tau_X | SRT)\) and \(f( \tau_\Sigma | SRT)\) (Fig 3.2.01-04) are reconstructed from the posterior samples (23) by kernel density estimation methods (Hastie, Tibshirani & Friedman, 2001, ch.6.6.1; Murphy, 2012, ch.14.7.2). In addition we display the posterior pdf of the standard deviation of the Gaussian likelihood in Fig 4.2.05. Our SRT-data are distributed with approximately \( \sigma_{\tau_\Sigma} = 86 \) around the marginal posterior \(\tau_\Sigma | SRT \).

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Sampling according to our generative model is done with four sampling subprocesses:

$$ \text{1. Unconditional sampling triangular priors for interval means as 'typical' values: } $$

$$ \tau_P \sim Triangle_{mean}(a=50, b=200, c=50.0) $$

$$ \tau_C \sim Triangle_{mean}(a=25, b=170, c=15.0) $$

$$ \tau_M \sim Triangle_{mean}(a=30, b=100, c=80.0) $$

$$ \text{2. Unconditional sampling priors for standard deviation of Gaussian likelihood: } $$

$$ \sigma_{\tau_\Sigma} \sim \Gamma(k=4, \theta=20) $$

$$ \text{3. Computing the deterministic function sum of priors: }$$

$$\tau_\Sigma = \sum_{X \in \{P,C,M\}} \tau_X
$$

$$ \text{4. Conditional sampling from Gaussian likelihood with variable priors and fixed SRTs:} $$

$$ (\tau_P, \tau_C, \tau_M, \tau_\Sigma, \sigma_{\tau_\Sigma} | SRTs) \sim L_N( \tau_\Sigma, \sigma_{\tau_\Sigma} ; SRTs ) = \prod_{i=1}^m N(SRT_i | \tau_\Sigma, \sigma_{\tau_\Sigma}). $$

The function Infer generates the conditional 4-tuple-samples of the joint posterior distribution (23):

$$( \tau_P, \tau_C, \tau_M, \tau_\Sigma, \sigma_{\tau_\Sigma}) | SRT $$

with the help of the Markov-Chain Monte-Carlo (MCMC)-method (Robert & Casella, 2004, ch.7; Murphy, 2012, ch.23; Lunn et al, 2014, ch.4) with the parameters \(nTrials = 6E4 = 60 000\), \(myBurnPeriod = nTrials \cdot 0.10\), and \(myLag=10 \).

All generated samples (23) constitute the support of the joint posterior pdf

$$P ( \tau_P, \tau_C, \tau_M,\tau_\Sigma, \sigma_{\tau_\Sigma} | SRT),  \qquad({24)} $$

whose marginal densities \(Triangle_{mean}( \tau_X | SRT)\) and \(f( \tau_\Sigma | SRT)\) (Fig 4.3.01-04) are reconstructed from the posterior samples (23) by kernel density estimation methods (Hastie, Tibshirani & Friedman, 2001, ch.6.6.1; Murphy, 2012, ch.14.7.2). In addition we display the posterior pdf of the standard deviation of the Gaussian likelihood in Fig 4.3.05. Our SRT-data are distributed with approximately \( \sigma_{\tau_\Sigma} = 86 \) around the marginal posterior \(\tau_\Sigma | SRT \).

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Sampling according to our generative model is done with four sampling subprocesses (44.1 - 44.4):

$$ \text{1. Unconditional sampling Gamma priors for interval means as 'typical' values: } $$

$$ \tau_P \sim \Gamma(a=16, b=6.3)  \qquad{(44.1)} $$

$$ \tau_C \sim \Gamma(a=8.4, b=8.3)  \qquad{(44.2)} $$

$$ \tau_M \sim \Gamma(a=36, b=1.9)  \qquad{(44.3)} $$

$$ \text{2. Unconditional sampling priors for standard deviation of Gaussian likelihood: } $$

$$ \sigma_{\tau_\Sigma} \sim \Gamma(k=4, \theta=20) \qquad{(35.10)=(44.4)} $$

$$ \text{3. Computing the function value of priors: } \tau_\Sigma = \sum_{X \in \{P,C,M\}} \qquad{(35.11)=(44.5)} $$

$$ \text{4. Conditional sampling from Gaussian likelihood with priors and fixed SRTs:} $$

$$ (\tau_P, \tau_C, \tau_M, \tau_\Sigma | SRTs) \sim L( \tau_\Sigma, \sigma_{\tau_\Sigma} ; SRTs ) = \prod_{i=1}^m Gaussian(SRT_i | \tau_\Sigma, \sigma_{\tau_\Sigma}). \qquad{(35.12)=(44.6)} $$

The function Infer generates the conditional 4-tuple-samples of the joint posterior distribution:

$$( \tau_P, \tau_C, \tau_M, \tau_\Sigma) | SRT \qquad(35.12)=(45) $$

with the help of the Markov-Chain Monte-Carlo (MCMC)-method (Robert & Casella, 2004, ch.7; Murphy, 2012, ch.23; Lunn et al, 2014, ch.4) with the parameters \(nTrials = 5E4 = 50 000\), \(myBurnPeriod = nTrials \cdot 0.10\), and \(myLag=10 \).

All generated samples (45) constitute the support of the joint posterior pdf

$$P ( \tau_P, \tau_C, \tau_M,\tau_\Sigma| SRT),  \qquad(46) $$

whose marginal densities \(f( \tau_P | SRT)\), \(f( \tau_C | SRT)\), \(f( \tau_M | SRT)\), and \(f( \tau_\Sigma | SRT)\) (Fig 4.4.01-04) are reconstructed from the posterior samples (35.12) by kernel density estimation methods (Hastie, Tibshirani & Friedman, 2001, ch.6.6.1; Murphy, 2012, ch.14.7.2). In addition we display the posterior pdf of the standard deviation of the Gaussian likelihood in Fig 4.4.05. We see that our SRT-data are distributed with approximately \( \sigma_{\tau_\Sigma} = 89 \) around the marginal posterior \(\tau_\Sigma | SRT \).

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Psychometric assessment is always problematic when there are only a few repeated measurements of the same person (Levy and Mislevy, 2016). The same is true for measuring simple reaction times (SRTs) when measuring the fitness of an individual vehicle driver (UbiCar, 2022). Test scores suffer from measurement errors and state fluctuations of the subject due to fatigue, alcohol, drugs, fever, depressive mood, or even heavy food (CogniFit, 2022),

5.1 Questions concerning risks from the ego-perspective

In this situation, we suggest exploiting the diagnostic potential of our generative Bayesian SRT model. For demonstration purposes we present the ego-perspective of the new, paradoxical, and methaporical scenario of a cautious gunslinger. The agent has to answer himself three increasing complex counterfactual and metaphoric questions:

• 1) "Can I draw my revolver fast enough, if my opponent needs only \(\tau_*\) milliseconds to do so ?"
• 2) "Can I draw my revolver as fast as a randomly selected person of a (younger) reference population, if my opponent needs only \(\tau_*\) milliseconds to do so ?"
• 3) "Is the probability of drawing my revolver slower than a randomly selected person of a (younger) reference population at most \(p=0.05\) if my opponent needs only \(\tau_*\) milliseconds ?"

In a first step, we assume that there is only one fixed critical SRT threshold (SRT-at-risk) \( \tau_*\) that separates safe from risky SRT behavior (Linsmeier & Pearson, 1996). A behavior is safe for the person under study if the subject's SRT is below the SRT-at-risk \( \tau_*\). Consequently, a behavior is unsafe or dangerous if the subject's SRT is above the SRT-at-risk \( \tau_*\). In this case, harmful consequences occur for the subject, brought about by an opponent ('gunslinger').

5.2 Answer to question 1

The external defined SRT-at-risk \( \tau_{\Sigma_*} \) defines a threshold on the support of the prior \( \tau_{\Sigma} \) so that

$$P(\tau_{\Sigma} > \tau_{\Sigma_*}) = \alpha_{*_{prior}}(\tau_{\Sigma_*}). \qquad{(25)}$$

After testing the subject we have a set of personal SRTs and we can compute the personal risk probability by marginalizing and filtering the posterior \( f_{posterior}( \tau_P, \tau_C, \tau_M,\tau_\Sigma, \sigma_{\tau_{\Sigma}} | SRTs)\)

$$P(\tau_{\Sigma} > \tau_{\Sigma_*} | SRTs) = \alpha_{*_{posterior}}(\tau_{\Sigma_*}) \qquad{(26)} $$

$$ \alpha_{*_{posterior}}(\tau_{\Sigma_*}) = \int_{\tau_{\Sigma} > \tau_{\Sigma_*}} \int_{\tau_P} ...  \int_{\sigma_{\tau_{\Sigma}}} \, f_{posterior}( \tau_P, \tau_C, \tau_M,\tau_\Sigma, \sigma_{\tau_{\Sigma}} | SRTs) \; \text{d} \sigma_{\tau_{\Sigma}} ...  \text{d} \tau_P \,  \text{d} \tau_\Sigma. $$

Now with (26) it is possible to answer question 1 of our gunslinger scenario. The only problem left is to define the vague concept 'fast enough' by a tolerable probability \(\alpha_{*_{posterior}}(\tau_{\Sigma_*})\).

In WebPPL (26) will be solved numerically by Monte-Carlo methods. First we marginalize the posterior pdf with the Infer-function. Then the marginal posterior will be filtered to obtain the particles satisfying \(\tau_{\Sigma} > \tau_{\Sigma_*} | SRTs \). In the last step the probability \(\alpha_{*_{posterior}}(\tau_{\Sigma_*})\) can be estimated by the ratio of the length of this filtered array to the length of the unfiltered marginal posterior array.

5.3 Answer to question 2

Next the psychometric personal risk-excess calculation is done by comparing the posterior personal risk probability \(\alpha_{*_{posterior}}\) with that from the prior risk \(\alpha_{*_{prior}}\)

$$ P(\tau_{\Sigma} > \tau_{\Sigma_*} | SRTs) - P(\tau_{\Sigma} > \tau_{\Sigma_*})   = \alpha_{*_{posterior}}(\tau_{\Sigma_*}) - \alpha_{*_{prior}}(\tau_{\Sigma_*}) \qquad{(27)} $$

$$  \alpha_{*_{diff}}(\tau_{\Sigma_*}) = \alpha_{*_{posterior}}(\tau_{\Sigma_*}) - \alpha_{*_{prior}}(\tau_{\Sigma_*}) \qquad{(28)} $$

Risk-excess (27, 28) is a monotonic increasing function of the standard deviation of priors and the \(\tau_{\Sigma_*}-\)thresholds. The posterior \(3 \sigma_{\tau_\Sigma} \)-intervals from pt 4.1 - 4.4 are summarized in Table 2.

$$
\begin{array}{|c|c|c|c|} \hline
\tau_{P,C,M}\text{-priors}  & \text{posterior }\sigma_{\tau_{\Sigma}} & \text{posterior } 3\sigma_{\tau_{\Sigma}} \text{- interval} & \text{interval range} \\ \hline
Triangle_{mode} & 24.2 & 310.0\; [237.5 \sim 382.5] & 145.0\; msec \\
Triangle_{median} & 25.1 & 307.2\;  [232.1 \sim 382.8] & 150.7\; msec \\                                          Triangle_{mean} & 25.6 & 306.6\;  [229.7 \sim 383.4] & 153.7\; msec \\
\Gamma & 25.5 & 294.9 \; [218.5 \sim 371.3] & 152.8\;  msec\\ \hline
\end{array} \qquad(\text{Table } 2)
$$

The widest posterior interval is obtained from \(Triangle_{mean} \)-priors of the \(\tau_\Sigma\)-components. This prior was most strongly influenced by the subjects' data. Thus the use of this posterior for risk excess calculations is most unfavorable for the person under study, when the subject is suspected to be slower than a randomly chosen person from the reference population.

Answering question 2 we compute (28) for the range \(\{\tau_{\Sigma_*} | \tau_{Middleman} \le \tau_{\Sigma_*} \le \tau_{Slowman} \} \)
$$\alpha_{*_{posterior}}(\tau_{\Sigma_*}) - \alpha_{*_{prior}}(\tau_{\Sigma_*}). \; ; \; \tau_{Middleman} \le \tau_{\Sigma_*} \le \tau_{Slowman} \qquad{(29)}$$

Results of (29) are displayed in Fig 5.11 - Fig 5.22. The most important results can be seen for \(\tau_{\Sigma_*} \)  in Fig 5.14, Fig 5.18, and Fig 5.22. The maximal risk-excess is nearby the threshold \(\tau_{\Sigma_*} = 290 \;msec \). This is the most unfavorable threshold for the subject. Depending on the prior the person-specific risk-excess ranges from p=0.25 to p=0.65. Risk-excess reduces with growing threshold \(\tau_{\Sigma_*} \) more and more till it reduces surprisingly to zero. This happens for threshold values \(\tau_{\Sigma_*} \gt 340 \; msec \).

Now we have provided a formal answer to question 2 of the counterfactual and metaphorical gunslinger scenario. This is even true for varying thresholds \((\tau_{\Sigma_*})\).

5.4 Answer to question 3

Only if the risk-excess (27, 28) is substantial greater than e.g. \(p =.05\) then the subject's SRT should be considered as more risky than that of any randomly chosen subject from the reference population. We think that p = 0.05 can be accepted by convention. To answer question 3 we formalize it as

$$\tau_{X_{crit}} = \min_{\rm \tau_{\Sigma_*} \in \{\tau_{Middleman} \le \tau_{\Sigma_*} \le \tau_{Slowman}\}}
\alpha_{*_{diff}}(\tau_{\Sigma_*}) \le 0.05 \qquad{(30)} $$

(30) has two meanings. First, it is the greatest upper bound of risky challenges (= the slowest, least demanding/dangerous challenge with risky consequences). Second, it is the least lower bound of nonrisky challenges (= fastest or most demanding/dangerous challenge with nonrisky consequences).

The critical thresholds \(\tau_{X_{crit}}\; msec \; ; \; X \in \{P, C, M, \Sigma \} \) are collected in Table 3. They partition single subject's risk-excessive from non-risk-excessive SRT-regions:

$$ \begin{array}{|c|c|c|c|c|} \hline
prior & \tau_{P_{crit}} msec & \tau_{C_{crit}} msec & \tau_{M_{crit}}  msec & \tau_{\Sigma_{crit}} msec\\     \hline
Triangle_{mode}    & 172  & 144 & 83.8 & 336.6 \\
Triangle_{median} & 178  & 148 & 86.8 & 341.2 \\
Triangle_{mean}    & 178  & 150 & 88.0 & 341.2 \\                                                                                                         \hline
\end{array} \qquad(\text{Table } 3)
$$

The entries of Table 3 can be identified quite easily in Fig 5.11 - Fig 5.22. They have the following meaning. If the opponent's SRT \(\tau_{X_*}\) (in process \(X \in \{P, C, M, \Sigma \} \)) is \(\tau_{X_*} > \tau_{X_{crit}} \) then the probability that the subject is slower than a randomly selected subject from the reference population is below p=0.05. In other words we can be quite certain that the single subject's SRT is no more risky than that of any subject from the reference population when the opponent's SRT \(\tau_{X_*}\) is greater than \(\tau_{X_{crit}}\).

Following the entries from Table 3 we can answer question 3 when we know the opponent's SRT-value \(\tau_*\). Furthermore we can see that the choice of prior is not important for certain risk-avoiding decisions. Let's concentrate on the \(Triangle_{mode}\)-prior. E.g. is the SRT-value-at-risk \(\tau_\Sigma < 336.6 \; msec\) the increase in single subject's prior-to-posterior risk probability (risk-excess (27)) is greater than 0.05 (Fig 5.14). Otherway round we can say that if the SRT-value-at-risk \(\tau_\Sigma \ge 336.6 \; msec\) the increase in single subject's prior-to-posterior risk probability (risk-excess (27)) is smaller than 0.05. This means that for all SRT-values-at-risk greater than 336.6 msec the reaction times of our single subject in the age of 72 years do not include a significant higher risk than a typical (younger) person of the MHP-reference population. This result is also true for the two other priors and a slightly greater \(\tau_{\Sigma_{crit}}=341.2\)  (Fig 5.18, and Fig 5.22).

The results of this kind of individual  Bayesian risk calculation are much more precise than general statements such as "... that the chronological age of a driver cannot be a clear indicator of his sensory-motor performance." (Cohen, 2009, p.231) Instead we think that our Bayesian model combines in a near ideal way results of meta-analyses and with evidence-based single-case diagnostics.

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this is a DRAFT, please send comments to: claus.moebus@uol.de

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