Bayesian Analysis of the Behrens-Fisher Problem under a Gamma Prior ()
1. Introduction
Consider a hypothesis testing problem about the difference of two means as follows: Let
and
and no assumption is made about
and
. Then testing the hypothesis stated as
(1)
based on random samples of size
and
respectively, and the assumption that
and
are independent is known as the Behrens-Fisher problem.
Lindley [3] showed that using the same data, the conclusion of a hypothesis test from a frequentist perspective could differ from that of a Bayesian perspective. It was shown that as
, the posterior probability under H0 tends to 1. This result holds irrespective of the prior probability assigned to H0. For discussions and arguments concerning Lindley’s paradox, see Spanos [4] and Robert [5] . As an extension of the methodology of Yin [1] , Yin and Li [2] made an attempt to solve the Behrens-Fisher Problem as well as Lindley’s Paradox using a noninformative prior. In this work, we propose to examine the performance of the methodology of Yin [1] in solving simultaneously, the Behrens-Fisher problem and Lindley’s paradox when Gamma priors are assigned to the unknown variances.
2. Literature Review
Scheffe [6] showed that for the Behrens-Fisher problem, there does not exist convenient tests and confidence intervals by constructing a test statistic based on a linear and a quadratic form. He established this result by showing that there exists no symmetric solution to the Behrens-Fisher problem using this approach. Fraser and Streit [7] derived a valid solution for the Behrens-Fisher problem using arbitrary absolutely continuous error distributions. They used a structural approach, where the random fluctuations apparent in the experiment were generated by a random variable with known distribution. Robinson [8] investigated the discrepancy between the coverage probabilities for the Behrens-Fisher intervals and the intervals of the nominal significance level. He advocates the use of the Behrens-Fisher test unless a proper Bayesian test is considered appropriate. Tsui and Weerahandi [9] proposed the use of generalized pivotal quantities and generalized p-values in the case of hypothesis testing in the presence of nuisance parameters given by
(2)
where
,
,
is the variance of sample i calculated from a sample of size
and
. Zheng et al. [10] proposed an approach to solving the Behrens-Fisher problem in such a way that the
Type-II error and the length of the confidence interval is controlled conditioned on a specified Type-I error by using Stein’s two-stage sampling scheme. Ozkip et al. [11] compared the different methods of solving the Behrens-Fisher problem to see which test outperforms the rest.
Degroot [12] commented on the reaction of some Bayesians on the use of informative priors. He disagrees with the notion that diffuseness of a prior distribution reflects ignorance about the distribution of such a parameter. Berger and Sellke [13] investigated the relationship between the p-value and the Bayesian measure of evidence against the null hypothesis for a two-sided hypothesis testing problem and concluded that the two measures of evidence were irreconcilable. Casella and Berger [14] investigated the discrepancy between the Bayesian measure of evidence, that is, the posterior probability that H0 is true, and the p-value, in a one-sided hypothesis testing problem under the same class of priors as in Berger and Sellke [13] but concluded that the two measures of evidence were reconcilable. Berger and Delampady [15] also investigated the discrepancy between the p-value and the Bayesian measure of evidence and concluded that using a noninformative prior does not necessarily solve Lindley’s paradox.
Meng [16] proposed a Bayesian counterpart of the generalized p-value to allow the “Test Statistic” depend on both the data and unknown (nuisance) parameters and thus permit a direct measure of the discrepancy between sample and population quantiles. Unlike the generalized p-value of Tsui and Weerahandi [9] that requires the use of a pivotal quantity as a test variable whose tail area probabilities are free of nuisance parameters, only the specification of prior distributions are required for the posterior predictive p-value which is given as
(3)
where
,
and
.
Ghosh and Kim [17] proposed an approach of constructing a prior different from Jeffreys’ independent prior that leads to a credible interval whose asymptotic coverage probability matches the frequentist coverage probability more accurately than Jeffreys’ interval. Yin [1] developed a Bayesian testing procedure that solves Lindley’s Paradox in testing a precise null hypothesis in the one sample case. Instead of the conventional Bayesian approach, this new procedure avoids the dichotomy of the parameter space. Let
be a random sample from a distribution with density
, where
is unknown and belongs to the parameter space
, and let
have a prior density
, then the new Bayesian measure of evidence is
(4)
where
is the posterior expectation of
under the prior
. A smaller value of
means a bigger distance between
and the true
and therefore, suggests stronger evidence against the null hypothesis H0.
Yin and Li [2] extended the methodology of Yin [1] to solve simultaneously, the Behrens-Fisher problem and Lindley’s Paradox using Jeffreys’ objective prior given as
(5)
They showed that the posterior distribution of
, the difference of the two means
and
is given as
(6)
and the Bayesian measure of evidence under Jeffreys’ independent prior was given by
(7)
where
is a t variable with
degrees of freedom. This approach was shown to solve the two problems and also, to yield credible intervals that actually possess
coverage probability. This was however only demonstrated where a non-informative prior was used and the use of an informative prior was recommended in order to see how the methodology performs under such conditions.
3. Main Results
Let there exist samples of sizes
and
from
and
respectively. Under the assumption of independence, and letting x = (x1, x2), the likelihood function is given by
(8)
where
Now, let
,
, and
. The marginal posterior distribution of
is given by
(9)
So, we have that
(10)
Let
Then we have that
(11)
Now, let
And then (11) simplifies to
(12)
where clearly, (12) is the kernel of the joint distribution of two independent t random variables
and
with
and
degrees of freedom respectively.
Also, we have that
And this implies that the posterior distribution of
, the difference of the two means
and
is given as
(13)
and
where
is a random variable that follows a t distribution with
degrees of freedom, where
. The Bayesian measure of evidence under a Gamma Prior is then given by
(14)
To establish that the Bayesian measure of evidence of Yin (2012) solves the paradox in Lindley (1957) when a Gamma prior is assigned to the nuisance parameters, we need to show that
(15)
Recall that
where Z is a standard normal random variable. Now, under the Gamma prior, we have that
(16)
Then, it can easily be shown that
(17)
which implies that (15) holds. To show that (17) holds, we now need to show that
(18)
Let
,
,
and
then we have that
(19)
Secondly, let
, then we have that
(20)
Since from (19) and (20) we have that
, it has been shown that the Bayesian measure of evidence of Yin (2012) under the Gamma prior solves the paradox in Lindley [3] .
Consequently, since it can be easily seen from (13) that the posterior distribution of
is symmetric about its expected value,
, then Theorem 2 of Yin and Li [2] applies here. This implies that under the Gamma Prior, the Bayesian measure of evidence of Yin [1] yields the
credible intervals for
centered at
.
Lemma 1. Let Gamma Priors be assigned to the precisions
and
. Then, for values of
and
, the Posterior distribution of
, denoted by
is the same as the Posterior distribution under Jeffreys’ independent prior given by
Proof. By considering the values of
and
that satisfy
and
, we can safely assume that
especially where
, and
is sufficiently large,
. Then by setting
,we have from (10) that
(21)
which is the kernel of the joint distribution of two independent t random variables with
having
degrees of freedom and
having
degrees of freedom respectively. The nit can be easily seen that
and consequently,
(22)
where
is a t random variable with
degrees of freedom.+
Lemma 1 shows that the posterior distribution of the difference in means under Jeffreys’ independent prior is a special case of the posterior distribution of the difference in means under the Gamma prior.
4. Simulation Results and Discussion
For the purpose of this discussion, we shall refer to the methodology of Yin [1] as the New Bayesian measure of evidence. The Metropolis-Hastings algorithm was used for the simulation with a thinning length of 12. The values in Table 1 were obtained by fixing the following values:
. These results reveal that for the different sample sizes, whether large or small, equal or unequal, the conclusions of a hypothesis test based on either the Generalized p-value, the Posterior Predictive p-value, the New Bayesian measure of evidence under the objective prior, or the New Bayesian measure of evidence under the Gamma prior are in the same direction. However, the New Bayesian measure of evidence under the Gamma prior gives consistently smaller evidence against the null hypothesis, whether the sample sizes are equal or unequal except for large sample sizes where the new Bayesian measure of evidence gives stronger evidence against the null compared to the Posterior Predictive P-value.
On the other hand, the values in Table 2 were obtained by fixing the following values:
. The values in this table reflect the accuracy of the approximation of the new Bayesian measure of evidence under Jeffreys’ independent prior to the new Bayesian measure of evidence under the Gamma prior. Results here show that when the sample sizes are at least 30, the approximation seems to be good and the values of the
do not need to be far less than 1. The approximation is good only contingent on the fact that the values of the
are less than 1.
Thirdly, the values in Table 3 were obtained by fixing the following values:
. The values in this table also reflect for smaller variances, the accuracy of the approximation of the new Bayesian measure of evidence under Jeffreys’ independent prior by the new Bayesian measure of evidence under the Gamma prior. In a similar manner, results here show that the approximation is equally good for smaller variances. In fact, the approximation is good where samples sizes can be at least as large as 10 so long as the values of the
are considerably less than 1. Note that the parameter values are fixed to demonstrate the behaviour of the conclusion from the New Bayesian measure of evidence under different circumstances like when the sample variances are small or moderate or large. Also, in Table 2, the values were fixed to see how well the New Bayesian measure of evidence under Jeffreys’ prior can be approximated by the New Bayesian measure of evidence under the Gamma prior.
Table 1. The four different probability values for different values of n1 and n2.
Table 2. The four different probability values for different values of
and
.
Table 3. The four different probability values for different values of
and
.
Finally, Lehmann’s data on measures of driving times from following two different routes and Sahu’s data on scores of surgical and non-surgical treatments both displayed as Table 1 and Table 2 respectively in Yin and Li [2] were used as real examples to demonstrate the performance of the four measures of evidence (results not shown). All conclusions were in the same direction for all four measures of evidence against the null hypothesis.
5. Conclusions
In this paper, we looked at the Bayesian analysis of the Behrens-Fisher problem using the methodology of Yin [1] by assigning Gamma Priors to the two unknown variances. We were able to show analytically, that the Bayesian measure of evidence of Yin [1] solves simultaneously, the Behrens-Fisher problem and Lindley’s paradox when Gamma Priors are assigned to the unknown variances. In fact, we were able to show that the solution obtained by Yin and Li [2] is a special case of ours, where Gamma Prior is used instead of Jeffreys’ independent prior.
Simulation results further confirm the fact that extending the methodology of Yin [1] while assigning Gamma Prior to each of the nuisance parameters also solves Lindley’s paradox. This implies that the prowess of the methodology of Yin [1] does not only lie in the use of noninformative priors. In fact, simulation results reveal that for large sample sizes, the measure of evidence against the null hypothesis is stronger when the nuisance parameters are assigned Gamma Priors with carefully selected parameter values.