DPQ (version 0.5-3)

lbeta: (Log) Beta Approximations

Description

Compute log(beta(a,b)) in a simple (fast) or asymptotic way.

Usage

lbetaM   (a, b, k.max = 5, give.all = FALSE)
lbeta_asy(a, b, k.max = 5, give.all = FALSE)
lbetaMM  (a, b, cutAsy = 1e-2, verbose = FALSE)
 betaI(a, n)
lbetaI(a, n)
logQab_asy(a, b, k.max = 5, give.all = FALSE)
Qab_terms(a, k)

Value

a fast or simple (approximate) computation of lbeta(a,b).

Arguments

a, b, n

the Beta parameters, see beta; n must be a positive integer and “small”.

k.max

..

give.all

logical ..

cutAsy

cutoff value from where to switch to asymptotic formula.

verbose

logical (or integer) indicating if and how much monitoring information should be printed to the console.

k

the number of terms in the series expansion of Qab_terms(), currently must be in \({0, 1, .., 5}\).

Author

Martin Maechler

Details

All lbeta*() functions compute log(beta(a,b)).

We use \(Qab = Qab(a,b)\) for $$Q_{a,b} := \frac{\Gamma(a + b)}{\Gamma(b)},$$ which is numerically challenging when \(b\) becomes large compared to a, or \(a \ll b\).

With the beta function $$B(a,b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)} = \frac{\Gamma(a)}{Qab},$$ and hence $$\log B(a,b) = \log\Gamma(a) + \log\Gamma(b) - \log\Gamma(a+b) = \log\Gamma(a) - \log Qab,$$ or in R, lBeta(a,b) := lgamma(a) - logQab(a,b).

Indeed, typically everything has to be computed in log scale, as both \(\Gamma(b)\) and \(\Gamma(a+b)\) would overflow numerically for large \(b\). Consequently, we use logQab*(), and for the large \(b\) case logQab_asy() specifically, $$\code{logQab(a,b)} := \log( Qab(a,b) ).$$

Note this is related to trying to get asymptotic formula for \(\Gamma\) ratios, notably formula (6.1.47) in Abramowitz and Stegun.

Note how this is related to computing qbeta() in boundary cases, and see algdiv() ‘Details’ about this.

We also have a vignette about this, but really the problem has been adressed pragmatically by the authors of TOMS 708, see the ‘References’ in pbeta, by their routine algdiv() which also is available in our package DPQ.

References

Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. New York: Dover. https://en.wikipedia.org/wiki/Abramowitz_and_Stegun provides links to the full text which is in public domain.
Formula (6.1.47), p.257

See Also

R's beta function; algdiv().

Examples

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