Table 7:

Fitting results for quadratic variance model (Model #4 in Table 1) for each of our straight, narrowing, and circular tunnel tasks. For each type of distribution, we use Steering law 1 to predict mean (or location) parameter, and quadratic variance model(Model #4 in Table 1) to predict the variance (or scale) parameter, as explained in Table 2. a and b are parameters of Steering law, and c and d are parameters of variance models. Parameter estimations are shown in mean and 95% credible interval of posterior distributions. Fitness results are reported in AIC and WAIC metrics. As shown in the three tasks, the Gamma, Lognormal, Extreme value and exGaussian distributions outperformed Gaussian and truncated Gaussian distributions.

Model		Model Parameters (Mean and 95% Credible Interval)					Information Criteria
Variance Model	Distribution Type	a	b	c	d	k	AIC	WAIC
σ2 = c + d · ID2 (Straight tunnel)	Gaussian	−0.1 [−0.2, −0.03]	0.08 [0.08, 0.09]	−0.019 [−0.05, 0.016]	0.003 [0.003, 0.004]	N/A	23371.9	23368.4
	Truncated Gaussian	−0.448 [−1.059, 0.031]	0.047 [0.008, 0.081]	0.121 [−0.106, 0.371]	0.007 [0.005, 0.009]	N/A	22896.4	22891.7
	Lognormal	−0.14 [−0.23, −0.028]	0.088 [0.08, 0.097]	−0.053 [−0.165, 0.118]	0.005 [0.004, 0.006]	N/A	22713.5	22709.0
	Gamma	−0.112 [−0.183, −0.034]	0.084 [0.078, 0.091]	−0.022 [−0.061, 0.026]	0.003 [0.003, 0.003]	N/A	22732.1	22727.4
	Extreme value	−0.108 [−0.175, −0.04]	0.082 [0.076, 0.088]	−0.016 [−0.045, 0.018]	0.002 [0.002, 0.003]	N/A	22909.5	22905.1
	exGaussian	−0.056 [−0.092, −0.022]	0.08 [0.076, 0.084]	0.0004 [0.00001, 0.001]	0.004 [0.0003, 0.004]	0.02 [0.02, 0.002]	22692.1	22685.9
σ2 = c + d · ID2 (Narrowing tunnel)	Gaussian	0.365 [0.282, 0.45]	0.078 [0.07, 0.086]	0.158 [0.109, 0.209]	0.003 [0.003, 0.004]	N/A	23179.9	23176.1
	Truncated Gaussian	0.257 [0.08, 0.422]	0.066 [0.048, 0.082]	0.26 [0.156, 0.395]	0.005 [0.003, 0.006]	N/A	22955.6	22951.2
	Lognormal	0.453 [0.338, 0.575]	0.074 [0.064, 0.085]	0.565 [0.354, 0.789]	0.003 [0.002, 0.005]	N/A	22956.1	22951.7
	Gamma	0.378 [0.289, 0.47]	0.077 [0.068, 0.086]	0.226 [0.157, 0.302]	0.003 [0.002, 0.004]	N/A	22866.0	22861.2
	Extreme value	0.372 [0.281, 0.457]	0.077 [0.068, 0.085]	0.19 [0.134, 0.249]	0.003 [0.002, 0.003]	N/A	22948.3	22943.6
	exGaussian	0.223 [0.153, 0.298]	0.092 [0.084, 0.1]	0.055 [0.037, 0.075]	0.005 [0.005, 0.006]	0.01 [0.01, 0.01]	22943.5	22937.3
σ2 = c + d · ID2 (Circular tunnel)	Gaussian	−0.005 [−0.11, 0.099]	0.277 [0.267, 0.286]	0.032 [−0.026, 0.102]	0.007 [0.006, 0.007]	N/A	24253.4	24249.7
	Truncated Gaussian	−0.01 [−0.122, 0.097]	0.277 [0.267, 0.287]	0.035 [−0.021, 0.103]	0.007 [0.006, 0.007]	N/A	24252.2	24248.3
	Lognormal	−0.008 [−0.104, 0.095]	0.278 [0.268, 0.287]	0.002 [−0.065, 0.077]	0.007 [0.007, 0.008]	N/A	24177.5	24172.9
	Gamma	−0.006 [−0.11, 0.097]	0.277 [0.267, 0.287]	0.012 [−0.043, 0.075]	0.007 [0.006, 0.008]	N/A	24166.3	24161.8
	Extreme value	−0.009 [−0.115, 0.098]	0.278 [0.269, 0.289]	−0.006 [−0.074, 0.067]	0.008 [0.007, 0.009]	N/A	24197.0	24192.4
	exGaussian	−0.012 [−0.112, 0.09]	0.277 [0.268, 0.287]	0.023 [0.0008, 0.067]	0.007 [0.006, 0.008]	0.02 [0.02, 0.02]	24210.8	24204.7