Hessian eigenvalue warnings with Student-t model

[Gregor Gorjanc]

Hi,

I have developed a model assuming Gaussian likelihood. I wanted to test this assumption by using Student-t model. However, I am receiving warnings about negative Hessian eigenvalues (printout shown below). I have centred and scaled the modelled variable as well as covariates and the Gaussian model fits ok (I believe) Is there any guidance what I should do/test/check etc.?

Many thanks!

Regards, Gregor

> source(file="Data.R")

> Fit = inla(data=Data,

+            formula=TimeS ~ Gender +

+                            DistanceS +

+                            ClimbPerDistanceS +

+                            f(RaceF, model="iid") +

+                            f(DateYearMonthI, model="ar1", values=DateYearMonthsI, replicate=RunnerI))

> summary(Fit)

Call:

"inla(formula = ..1, data = data, control.compute = control.compute)"

  

Time used:

 Pre-processing    Running inla Post-processing           Total

         1.8929         15.4059          1.1940         18.4928

  

Fixed effects:

                     mean     sd 0.025quant 0.5quant 0.975quant    mode kld

(Intercept)        0.2279 0.0432     0.1431   0.2279     0.3130  0.2277   0

GenderM           -0.1463 0.0384    -0.2215  -0.1464    -0.0706 -0.1466   0

DistanceS          1.0083 0.0288     0.9517   1.0083     1.0649  1.0083   0

ClimbPerDistanceS  0.2070 0.0278     0.1520   0.2070     0.2616  0.2071   0

  

Random effects:

Name      Model

 RaceF   IID model

DateYearMonthI   AR1 model

  

Model hyperparameters:

                                           mean    sd 0.025quant 0.5quant 0.975quant   mode

Precision for the Gaussian observations 35.4040 1.426     32.627  35.4011    38.2218 35.432

Precision for RaceF                     12.3002 2.019      8.892  12.1009    16.8185 11.686

Precision for DateYearMonthI            45.7837 8.247     31.611  45.0933    63.9827 43.772

Rho for DateYearMonthI                   0.9975 0.001      0.995   0.9977     0.9989  0.998

  

Expected number of effective parameters(std dev): 189.46(10.82)

Number of equivalent replicates : 7.922

  

Deviance Information Criterion (DIC) ...............: -897.57

Deviance Information Criterion (DIC, saturated) ....: 1701.54

Effective number of parameters .....................: 192.02

  

Watanabe-Akaike information criterion (WAIC) ...: -859.73

Effective number of parameters .................: 197.96

  

Marginal log-Likelihood:  247.99

Posterior marginals for linear predictor and fitted values computed

> Fit = inla(data=Data,

+            formula=TimeS ~ Gender +

+                            DistanceS +

+                            ClimbPerDistanceS +

+                            f(RaceF, model="iid") +

+                            f(DateYearMonthI, model="ar1", values=DateYearMonthsI, replicate=RunnerI),

+            family="T")

summary(Fit)

  

  

        *** WARNING *** Eigenvalue 1 of the Hessian is -794.583 < 0

        *** WARNING *** Set this eigenvalue to 11.957

        *** WARNING *** This have consequence for the accurancy of

        *** WARNING *** the approximations; please check!!!

        *** WARNING *** R-inla: Use option inla(..., control.inla = list(h = h.value), ...)

        *** WARNING *** R-inla: to chose a different  `h.value'.

  

  

        *** WARNING *** Eigenvalue 4 of the Hessian is -37.1948 < 0

        *** WARNING *** Set this eigenvalue to 11.957

        *** WARNING *** This have consequence for the accurancy of

        *** WARNING *** the approximations; please check!!!

        *** WARNING *** R-inla: Use option inla(..., control.inla = list(h = h.value), ...)

        *** WARNING *** R-inla: to chose a different  `h.value'.

  

> summary(Fit)

  

Call:

"inla(formula = ..1, family = \"T\", data = data, control.compute = control.compute)"

Time used:

 Pre-processing    Running inla Post-processing           Total

         1.0208        283.0098          1.7501        285.7807

  

Fixed effects:

                     mean     sd 0.025quant 0.5quant 0.975quant    mode kld

(Intercept)        0.1812 0.0337     0.1150   0.1812     0.2473  0.1812   0

GenderM           -0.1447 0.0369    -0.2171  -0.1447    -0.0723 -0.1447   0

DistanceS          0.9791 0.0172     0.9452   0.9791     1.0128  0.9792   0

ClimbPerDistanceS  0.2011 0.0164     0.1690   0.2011     0.2333  0.2011   0

  

Random effects:

Name      Model

 RaceF   IID model

DateYearMonthI   AR1 model

  

Model hyperparameters:

                                            mean      sd 0.025quant 0.5quant 0.975quant    mode

precision for the student-t observations 16.4962  1.6281    14.4113  16.1411    20.5230 14.9767

degrees of freedom for student-t         14.4602  1.8518    10.4502  14.6746    17.3813 15.7504

Precision for RaceF                      58.0571 16.7488    31.9985  55.7801    97.3127 51.5082

Precision for DateYearMonthI             54.4213  5.6391    44.1914  54.1278    66.3475 53.5454

Rho for DateYearMonthI                    0.9984  0.0005     0.9973   0.9984     0.9991  0.9986

  

Expected number of effective parameters(std dev): 140.21(0.00)

Number of equivalent replicates : 10.71

  

Deviance Information Criterion (DIC) ...............: -1183.49

Deviance Information Criterion (DIC, saturated) ....: 252.53

Effective number of parameters .....................: -229.21

  

Watanabe-Akaike information criterion (WAIC) ...: -862.19

Effective number of parameters .................: 85.55

  

Marginal log-Likelihood:  88.35

Posterior marginals for linear predictor and fitted values computed

[Helpdesk]

Hi

yes, the problem is deep. the issue is that student-t is not log-
concave, and the question is what to do if the local hessian for the
log student-t (like for one observation) has the wrong sign. its like
getting variance < 0, so the contribution for the student-t observation
will locally inflate the variance, not reducing it.

its possible to truncate this at 0, meaning that if the hessian has the
wrong sign, set it to zero. you can do this using

control.inla = list(cmin = 0)

where the default value is -Inf which means do nothing. its unclear
what is the better thing to do, beyond the obvious, which comes with a
higher cost.

to be honest, I do not like the student-t anymore ;-)

H

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[Haakon Bakka]

You can add informative priors to all the model components. This often helps. Check out

>> inla.doc("pc.prec")

and

>> inla.doc("pccor1")

PS. You can of course do a scenario analysis of the T parameter: Fix it, run the model, compare results across fixed T parameters.

Kind regards,

Haakon Bakka

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> -group.
> For more options, visit https://groups.google.com/d/optout.

--
Håvard Rue
Helpdesk
he...@r-inla.org

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