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The precisions of both forms of noise are modelled by Gamma distributions: f(αi) ∼ Ga(ai, bi) whereaiis the shape parameter andbiis the rate parameter. Correspondingly,

the probability density function is as follows:

f(αi) = bai i Γ(ai) αai−1 i e −biαi _(2.90)

An illustrative graph is shown in Fig. 2.10 with different values ofai and bi whereai/bi

is the mean of the distribution, and ai/b2i is the variance.

Figure 2.10: Probability density distribution of the precision of the noise

Let us consider how the hyperpriors ai and bi in both types of noise (system noise and

measurement noise) influence the posterior distribution of the parameters. As explained in Section2.5, setting all the hyperparameters (a1,a2,b1,b2) to 0.001 can provide weakly

informative hyperpriors for the precision of both forms of noise. Different combinations of the hyperprior setting have been used to study the influence of the hyperpriors over the estimation error for the noise intensity and the value of the free energy. The estimation error for both forms of noise intensity can be calculated as follows:

RMSEα = s (aˆ1 ˆ b1 −100)2_{+ (}ˆa2 ˆ_b2 −100)2 2 (2.91)

First, fix the hyperpriors for the system noise, a1 and b1, to 0.001, and consider the

influence of the hyperpriors of the measurement noise over the free energy value and the posterior distributions of both forms of noise intensity. Varying the values ofa2 and

b2 results in different values of free energy as shown in Fig. 2.11 (a). It is clear from

the figure that the free energy values along the diagonal lines (parallel to the indicated line) are close. The free energy values are maximised along the indicated diagonal line log(a2) = log(b2) + 2, where the prior mean is a2/b2 = 100. This means that the prior

mean of the measurement noise intensity has a non-negligible influence over the free energy values. The RMSE between the estimated mean of the noise precision and the true noise precision was calculated according to (2.91). The logarithm of the RMSEα

Figure 2.11: (a)The value of free energy and (b) RMSEα for different combinations of the shapea2and rateb2hyperpriors for the measurement noise. Note that the figure is shown in log-scale (with a base of 10). All the free energy valuesF in this graph are

negative, so the logarithm of the free energy is calculated by−lg(|F |).

of the measurement noise is shown in Fig. 2.11 (b). These results agree with the free energy results shown in Fig. 2.11 (a): when the prior mean of the measurement noise precision a2/b2 is set to 100, F is maximised and RMSEα is minimised. Along the

indicated diagonal line in both Fig.2.11 (a) and Fig.2.11 (b), the variance of the prior distribution for the measurement noise precision,a2/b22, decreases when log(b2) increases.

enough, the variance of the prior distribution of the noise intensity should be at least 102, which means thatb2 should be no larger than 1. When b2 is set to 1, anda2 is 102, the

posterior estimations for the hyperparameters are: ˆa1 = 100.001, ˆb1 = 5.31, ˆa2 = 200,

ˆ_{b2 = 2.22, and the free energy is−21, which is the largest value. It is worth noticing that} the posterior mean of the system noise precision ˆa1/ˆb1 = 19, and the posterior mean of

the measurement noise precision is ˆa2/ˆb2 = 90. Both estimated precisions are smaller

than the real precisions of 100, with the measurement noise precision closer to the real value. The underestimated noise precision indicates that the VB method tends to be conservative in terms of noise precision inference.

Second, fixing the hyperpriors for the measurement noise, a2 and b2, to 0.001, and

consider the influence of the hyperpriors of the system noise over the free energy value and the posterior distribution of both forms of noise intensity. Varying the value of a1

and b1 from 10−4 to 105 (order 10 incremental) results in different values of the free

energyF as shown in Fig.2.12(a) and the RMSEαas shown in Fig.2.12(b). Note that

varying the value of a1 and b1 from 10−4 to 105 has covered a huge range of possible

noise intensities from 10−9 to 109, which is considered sufficiently wide. Similar to the mean of the prior distribution of the measurement noise intensity, the mean of the prior distribution of the system noise,a1/b1, also influences the value ofFand the RMSEα. It

is worth noticing the high free energy values on the top left corner of Fig.2.12(a). Take a1 = 10 and b1 = 10−4 for example, the free energy value, F = 473, and is large. This

indicates that the probability of the model being true with the inferred parameters is high. However, from Fig.2.12(b), the RMSEαwith hyperpriorsa1= 10−4 andb1 = 101

is as high as 7.8×105, indicating an inaccurate inference. The contradictory result of a high free energy and a high RMSEαis caused by the bad choice of the hyperpriors of the

system noise with a mean value of a1/b1 = 10−5 and a variance of a1/b2i = 10−6. With

such a strong informative hyperprior, the posterior hyperparameters ˆa1 = 110, ˆb1= 10−4

are completely dominated by the prior distribution. The artificial high confidence level of the small system noise precision boosts up and dominates the value of the free energy. Therefore, the mean of the priora1/b1 is held at 100 indicated by the straight line shown

in Fig.2.12(a). Finally,a1= 100 andb1 = 1 are selected based on the free energy value

of 2. It is the largest among all of the combinations of the hyperparameters except for the top left corner where the free energy values do not reflect the goodness of fit. The posterior hyperparameter values are ˆa1 = 200, ˆb1 = 1.54, ˆa2 = 100.001, ˆb2 = 10.30.

Figure 2.12: (a)The value of free energy and (b) RMSEα for different combinations of the shape a2 and rateb2 hyperpriors for the system noise. Note that the figure is

shown in log-scale (with a base of 10).

The posterior mean of the system noise is ˆa1/ˆb1 = 130, which is overestimated and the

posterior mean of the measurement noise is ˆa1/ˆb1 = 9.71, which is underestimated.

The optimal free energies of the second step (fixing the hyperpriors for the measurement noise) are higher than for the first step (fixing the hyperpriors for the system noise). Therefore, the hyperpriors are finally chosen as: a1 = 100,b1 = 1,a2=b2= 0.001. The

mean of the posterior distribution for the parameters θ and the covariance matrix Σ_θˆ are as follows: ˆ θ =      0.099 0.020 −5.03      Σ_θˆ=      1.53×10−6 1.13×10−7 −3.11×10−5 1.13×10−7 2.60×10−8 −6.58×10−6 −3.11×10−5 −6.58×10−6 0.0017     