Análisis de los espaciadores intergénicos ribosomales (RISA)

Bayes estimators can be evaluated also in terms of frequentist properties, in order to com- pare their performances to frequentist estimators (Carlin and Louis, 2008). Therefore, it is possible to propose procedures that allow to choose hyperparameters that minimize the frequentist MSE. The deduced prior settings can be particularly useful when a small sample is available and a point estimation is required.

−15 −10 −5 0 5 0 25 50 75 100 x log[f(x)] Prior GIG(0,0.001,γ0) GIG(0.5,0.001,γ0) IG(1,0.001) IG(0.001,0.001)

Figure 3.1: Log density of the weakly informative GIG distributions proposed and of the most common vague inverse gamma (IG) priors.

Since the proposed estimators have particularly complicated mathematical expressions, due to the presence of an innite sum of Bessel K functions, closed form relations to nd optimal parameters cannot be found, even if approximations are applied. Therefore, a numerical solution is rstly proposed. Afterwards, connections with results included in the paper by Fabrizi and Trivisano (2012) for the functional θa,b(1.4) are shown in order to have a strategy for the hyperparameters specication that does not involve a numerical software.

Numerical optimization

A ve parameter optimization problem appears over-dimensioned for the inferential purpose, especially in a small sample framework.

Some suggestions to reduce the dimensionality of the minimizing function can be argued by specifying the MSE expression.

Recalling the (3.13), the MSE of the Bayes estimator under quadratic loss (3.24) can be decomposed in the following way:

E   ˆ_θQB p − θp 2 = Ehe2(ψ ¯w+(1−ψ)ξ0)_g(V2₎2_{− 2θ} peψ ¯w+(1−ψ)ξ0g(V2) + θ2p i = θ2p " e2(1−ψ)(ξ0−ξ)+2ψ2 σ2_n −2φσ E "  g(V2)_{− e}(1−ψ)(ξ0−ξ)−3ψ2 σ2_2n +φσ 2# + +1_{− e}−ψ2 σ2n  , (3.46)

where g(V2_{)is a function of the sample variance V}2 _{and it is the only part of the expression} that includes the hyperparameters of the GIG distribution.

A rst way to simplify the research of the minimum is to specify n0 in a weakly informative but proper way, e.g. to obtain ψ = 0.98. Through this decision, the estimator performance would be robust to misspecication of ξ0, but the contribution of the normal prior instead of the at prior might be useful in containing the estimator variance.

However, in this prior setting, the target functional to minimize with respect to the remaining parameters (λ, δ, γ) is approximately equal to:

E "  g(V2_{)− exp}  φσ₋3σ 2 2n 2# , (3.47)

considering the (3.46) and letting ψ → 1. This approximation coincides with the result that would be obtained in case of improper prior on ξ.

Therefore, the functional to minimize is an expectation taken with respect the random variable V2_{, and from the log-normality assumption it holds that:}

nV2 _{∼ Ga} n− 1 2 , 1 2σ2  . (3.48)

By exploring the trend of function (3.47), the indication is that the searching problem is still over-dimensioned: it is not possible to nd a unique minimum point unless two GIG parameters are kept constant.

Because of the necessity of putting constraints on the parameters range, numerical algo- rithms that allow to satisfy this requirement are considered. In particular, a bounds con- strained quasi-Newton method that is implemented in the R package optimx (Nash et al., 2014) is employed.

The rst parameter to x is λ: it is a shape parameter and it appears in the order of the Bessel's K functions and an eventual numerical optimization algorithm for it might be too unstable. Following the idea of the previous section, the shape parameter λ is xed equal to 0. By observing the trends of the (3.47) in this case, it appears that a nite minimum cannot still be found optimizing with respect two parameters. Besides, a distinction in the procedure for quantiles above the median and below the median (case that will be dealt with later) seems to be necessary. Moreover, the dierent behaviour of the functional over the quantiles can provide some suggestions in order to x another parameter and to have the possibility of nding the third value through minimization (gure 3.2).

To capture the shape of the function the following strategies could be adopted:

ˆ p < 0.5: x γ equal to the minimum value that allows the existence of second posterior moment according to the rule in (3.45). Then minimize with respect to the parameter δ;

ˆ p > 0.5: x δ and minimize with respect to γ. Recalling the (3.44), it is possible to specify an informative value of δ, considering it as a contribution in terms of variance

4 8 12 0 5 10 15 δ γ 0.0025 0.0050 0.0075 Functional p=0.1, n=21, λ=0 4 8 12 0 5 10 15 δ γ 0.005 0.010 0.015 Functional p=0.25, n=21, λ=0 4 8 12 0 5 10 15 δ γ 0.025 0.050 0.075 Functional p=0.75, n=21, λ=0 4 8 12 0 5 10 15 δ γ 0.1 0.2 0.3 0.4 Functional p=0.9, n=21, λ=0

Figure 3.2: Behaviour of the functional in (3.47) for ˆθQB

p with respect to the parameters γ and δ,

keeping constant λ = 0 and considering p = (0.10, 0.25, 0.75, 0.90). The case n = 21 and σ2 _{= 1is}

considered.

of a prior sample. A general proposal could be δ = 1: in most applied problems, values of the variance in the log-scale σ2 _{are seldom greater than 2, so 1 can be read} as a reasonable guess for the size of an hypothetical deviation from the mean when n = 1. Of course, if the scale of the problem is totally dierent, the user can specify alternative values for δ.

Heuristically, searching for optimal γ for quantiles above the median, and optimal δ for those below, is in line with the specialization of these parameters in the GIG distribution: γ rules the right tail of the distribution that is not relevant when p < 0.5, while δ is more involved with the general spread of the distribution and is therefore more relevant to the shape of the lower tail.

In the practical context, it must be considered that σ2 _{is unknown and it appears in the} functional (3.47). This issue might be overcome by plugging into the expression the sample variance v2 _{or a guess s}2

0 if it is available. The latter procedure is advisable since it allows to remove from the MSE the part of variability caused by the use of v2_{. Moreover, the} procedure could be considered more rigorous from a Bayesian viewpoint, since data would not been used twice in the inferential procedure. However, it is not always possible to have a safe value, even if the procedure is quite robust to the misspecication of s2

0, as will be in the simulation study presented in the following chapter.

A considerable particular case is the median and its neighbourhood. In these cases, the Bayes estimator under relative quadratic loss seems to perform better than the posterior mean for several reasons. Considering the gure 3.3, it could be deduced that the MSE is minimized when λ → −∞ is chosen. Using the limiting form (A.8) for the Bessel K function, it is possible to prove that the Bayes estimator under quadratic loss equates the naive estimator exp{ ¯w_{} with that degenerate prior. As a consequence, the so called naive} estimator for the median represents the best case for the Bayes estimator under quadratic loss.

On the contrary, the Bayes estimator under relative quadratic loss ˆθRQB

p presents a non- monotone decreasing trend of the MSE only around the median. For this reason it appears to be the preferable estimator in this situation, but the characteristics of ˆθRQB

p deteriorate in a fast way departing from the median. This particular behaviour is related to the absence of σ in θ0.5: it is a case similar to the conditional estimator of θp dealt with in section 3.2.1. In this framework, the Bayes estimator under relative quadric loss resulted the minimum MSE choice. Moreover, because of the absence of σ in the estimand, the hyperparameters specication does not inuence the estimation step. Therefore the weakly informative prior could be used in that case.

4 8 12 0 5 10 15 δ γ 0.006 0.007 0.008 0.009 Functional p=0.5, n=21, λ=0, QB estimator 4 8 12 0 5 10 15 δ γ 0.00025 0.00050 0.00075 Functional p=0.5, n=21, λ=0, RQB estimator

Figure 3.3: Behaviour of the functional in (3.47) for ˆθQB 0.5 and ˆθ

RQB

0.5 with respect to the parameters

γ and δ, keeping constant λ = 0. The case n = 21 and σ2_{= 1is considered.}

Connection with the Bayes estimator of θa,b

In the paper by Fabrizi and Trivisano (2012) the problem of the Bayesian inference of the functional θa,b = exp{aξ + bσ2}, assuming a log-normal distribution for data, was studied. In that context, the prior (3.37) was considered, and the found posterior distribution of θa,b was a GH. Furthermore, a strategy to nd a minimum MSE Bayes estimator was proposed. Through an analytic approximation of the target functional to minimize, the optimal value of λ, for a xed δ, was found to be:

λopt = n_{− 3} 2 − (n_{− 1)(a}2_{+ 2nb)} 4nc − (a2_{+ 2nb)δ} 4ncσ2 , (3.49)

where c = b₋3a2

2n. Moreover, the following condition must hold: b /∈ − a2 2n,

3a2

2n. It is useful to set δ → 0 in order to remove the eect of σ2_{, avoiding the substitution with the} sample quantity v2 _{or a guess s}2

0. The value of γ was selected in order to be sure of the second posterior moment existence. According to the theory of the GH distribution it is:

γ0 = max  0, 4 a 2 n + b  + ε. (3.50)

Note that if a = 1 and b = 0, the median of the log-normal distribution is considered, then: θ1,0= θ0.5. This case violates the condition on b required for the validity of relation (3.49). This is in agreement with the MSE trend showed in gure 3.3, where a nite minimum cannot be found. As a consequence, the indications included in this section cannot be applied in the median estimation context.

In order to apply the relation (3.49) in the log-normal quantile estimation problem, the proposal is based on the idea that each functional specied by a couple (a, b) corresponds to a quantile θp too. Utilizing the similarity among the quantiles that are above or below the median, the relation (3.49) could be used to obtain a unique set of optimal values for all the quantiles that belongs to the same half cumulative distribution (i.e. p < 0.5 or p > 0.5). Even if the (3.49) is a function of n, the sample size does not signicantly change the nal result. An empirical general choice for λ could be −2 for the quantiles above the median, and 0.5 for the quantiles below the median, always keeping δ = ε and gamma equal to (3.45). For example, considering that in this case b = Φ−1(p)

σ , they correspond, with the intermediate value σ2 _{= 1 to the optimal value for p = 0.85 and p = 0.30, respectively.} This method produces surprisingly good performances in terms of frequentist MSE, as will be stressed in the following chapter about the simulation study, and it possesses an appealing property if the statistical analysis goal is the joint estimation of dierent quantiles: since a unique prior is specied, the user is sure that the estimation procedure maintains the logical order of the quantiles and counter-intuitive results are avoided.

Prior proposals: brief outline and software implementation

In the previous sections dierent prior specications were listed. Using gure 3.4 it is possible to sum up the proposals and x some notation. Starting from the weakly informative prior, the triplet (λ = 0, δ = ε, γ = γmin) produces the posterior mean of the p-th log-normal quantile ˆθQBw

p , but is also used in the median estimations issue with the Bayes estimator under relative quadratic loss ˆθRQBw

p . The value λ = 0.5 might be used too.

Then, a frequentist optimal (i.e. minimum MSE) Bayes estimator is considered, and an optimization algorithm is used for the hyperparameters choice. In this case the Bayes estimator under quadratic loss associated to the prior setting is ˆθQBn

p and in gure 3.4 priors examples are reported considering: p = 0.10 and p = 0.25 below the median (optimization with respect to δ) and p = 0.75 and p = 0.90 above the median (optimization with respect to γ).

−20 −15 −10 −5 0 5 0.0 2.5 5.0 7.5 10.0 x log[f(x)]

Prior BQa, p>0.5 _{BQa, p<0.5} Weakly Inf._{BQn, p=0.10} BQn, p=0.25_{BQn, p=0.75} BQn, p=0.90

Figure 3.4: Comparison of log-densities of dierent GIG priors. A random sample of size n = 21 generated from a log-normal having σ2_{= 0.5is used.}

Finally, an approximately optimal Bayes estimator is proposed using the similarities be- tween θp and θa,b. Two generic triplets are proposed for the lower and upper parts of the distribution. In this case the Bayes estimator is specied with ˆθQBa

p .

In the package BayesLN the function LN_Quant is provided in order to allow the user to carry out a Bayesian inferential procedure on the log-normal quantiles. To simplify the usage of the function, two prior specication settings are only proposed: the use of λ = 0 for the weakly informative prior, and the optimal prior obtained thorough numerical minimization of the MSE. Concerning the prior on ξ, a at improper prior is considered.