by Peter M Lee
X has a z distribution on and degrees of freedom, denoted
if , or equivalently if it has the density
Another definition is that if , then
The mean and variance are easily deduced from those of the log chi-squared distribution; they are
where is (as above) the digamma function, or approximately
The mode is zero.
Unless and are very small, the distribution of z is approximately normal. The z distribution was introduced by Fisher (1924).
A.21 Cauchy distribution
X has a Cauchy distribution with location parameter μ and scale parameter , denoted
if
Because the relevant integral is not absolutely convergent, this distribution does not have a finite mean, nor a fortiori, a finite variance. However, it is symmetrical about μ, and hence
The distribution function is
so that when then F(x)=1/4 and when then F(x)=3/4. Thus, and are, respectively, the lower and upper quartiles, and hence σ may be thought of as the semi-interquartile range. Note that for a normal distribution the semi-interquartile range is rather than σ.
It may be noted that the C(0, 1) distribution is also the Student’s t distribution on 1 degree of freedom
A.22 The probability that one beta variable is greater than another
Suppose π and ρ have independent beta distributions
Then
[see Altham (1969)]. For an expression (albeit a complicated one) for
see Weisberg (1972).
When α, β, γ and are large we can approximate the beta variables by normal variates of the same means and variances and hence approximate the distribution of by a normal distribution.
A.23 Bivariate normal distribution
The ordered pair (X, Y)T of observations has a bivariate normal distribution, denoted
if it has joint density
where
The means and variances are
and X and Y have correlation coefficient and covariance
Most properties follow from those of the ordinary, univariate and normal distribution. One point worth noting (which is clear from the form of the joint density function) is that if X and Y have a bivariate normal distribution, then they are independent if they are uncorrelated (a result which is not true in general).
A.24 Multivariate normal distribution
An n-dimensional random vector has a multivariate normal distribution with mean vector and variance–covariance matrix , denoted
if it has joint density function
It can be checked by finding the determinant of the variance–covariance matrix
and inverting it that the bivariate normal distribution is a special case.
A.25 Distribution of the correlation coefficient
If the prior density of the correlation coefficient is ρ, then its posterior density, given n pairs of observations (Xi, Yi) with sample correlation coefficient r, is given by
on writing . It can also be shown that
When r=0 the density simplifies to
and so if the prior is of the form
it can be shown that
has a Student’s t distribution on degrees of freedom.
Going back to the general case, it can be shown that
Expanding the term in square brackets as a power series in u, we can express the last integral as a sum of beta functions. Taking only the first term, we have as an approximation
On writing
it can be shown that
and hence that for large n
approximately [whatever is]. A better approximation is
Appendix B: Tables
Table B.1 Percentage points of the Behrens–Fisher distribution.
Table B.2 Highest density regions for the chi-squared distribution.
Table B.3 HDRs for the inverse chi-squared distribution.
Table B.4 Chi-squared corresponding to HDRs for log chi-squared.
Table B.5 Values of F corresponding to HDRs for log F.
Appendix C: R programs
Appendix D: Further reading
D.1 Robustness
Although the importance of robustness (or sensitivity analysis) was mentioned at the end of Section 2.3 on several normal means with a normal prior, not much attention has been devoted to this topic in the rest of the book. Some useful references are Berger (1985, Section 4.7), Box and Tiao (1992, Section 3.2 and passim.), Hartigan (1983, Chapter ), Kadane (1984) and O’Hagan and Forster (2004, Chapter ).
D.2 Nonparametric methods
Throughout this book, it is assumed that the data we are analyzing comes from some parametric family, so that the density p(x|θ) of any observation x depends on one or more parameters θ (e.g. x is normal of mean θ and known variance). In classical statistics, much attention has been devoted to developing methods which do not make any such assumption, so that you can, for example, say something about the median of a set of observations without assuming that they come from a normal distribution. Some attempts have been made to develop a Bayesian form of nonparametric theory, though this is not easy as it involves setting up a prior distribution over a very large class of densities for the observations. Useful references are Ferguson (1973), Florens et al. (1983), Dalal (1980), Hill (1988), Lenk (1991), Ghosh and Ramamoorthi (2003) and Hjort et al. (2010). A brief account is given by Müller and Quintana (2004).
D.3 Multivariate estimation
In order to provide a reasonably simple introduction to Bayesian statistics, avoiding matrix theory as far as possible, the coverage of this book has been restricted largely to cases where only one measurement is taken at a time. Useful references for multivariate Bayesian statistics are Box and Tiao (1992, Chapter ), Zellner (1971, Chapter ), Press (2009) and O’Hagan and Forster (2004, Sections 10.28–10.41).
D.4 Time series and forecasting
Methods of dealing with time series, that is, random functions of time, constitute an important area of statistics. Important books on this area from the Bayesian standpoint are West and Harrison (1989) and Pole et al. (1994). Information about software updates can be found at
A briefer discussion of some of the ideas can be found in Leonard and Hsu (2001, Section 5.3).
D.5 Sequential methods
Some idea as to how to apply Bayesian methods in cases where observations are collected sequentially through time can be got from Berger (1985, Chapter ) or O’Hagan and Forster (2004, Sections 3.55–3.57).
D.6 Numerical methods
This is the area in which most progress in Bayesian statistics has been made in recent years. Although Chapter is devoted to numerical methods, a mere sketch of the basic ideas has been given. Very useful texts with a wealth of examples and full programs in WinBUGS available on an associated website are Congdon (2002, 2005, 2006 and 2010). Those seriously interested in the application of Bayesian methods in the real world should consult Tanner (1993), Gelman et al. (2004), Carlin and Louis (2008), Gilks et al. (1996), French and Smith (1997) and Brooks (1998).
More recently books giving a useful and comprehensible treatment of Bayesian numerical methods using and WinBUGS include Albert (2009) (a particularly attractive treatment), Marin and Robert (2007), Robert and Casella (2010) and Ntzoufras (2009).
At a much lower level, Albert (1994) is a useful treatment of elementary Bayesian ideas using Minitab.
D.7 Bayesian networks
References on this topic (on which I am not an expert) include Jensen (1996), Jensen and Nielson (2010) and Neapolitan (2004).
D.8 General reading
Apart from Jeffreys (1939, 1948 and 1961), Berger (1985) and Box and Tiao (1992), which have frequently been referred to, some useful references are Lindley (1971a), DeGroot (1970) and Raiffa and Schlaifer (1961). The more recent texts by Bernardo and Smith (1994) and O’Hagan and Forster (2004) are very important and give a good coverage of the Bayesian theory. Some useful coverage of Bayesian methods for the linear model can be found in Broemling (1985). Linear Bayes methods are cove
red in Goldstein and Wooff (2007). Anyone interested in Bayesian statistics will gain a great deal by reading de Finetti (1972 and 1974–1975) and Savage (1972 and 1981). A useful collection of essays on the foundations of Bayesian statistics is Kyburg and Smokler (1964 and 1980), and a collection of recent influential papers can be found in Polson and Tiao (1995). The Valencia symposia edited by Bernardo et al. (1980–2011) and the `case studies’ edited by Gatsonis et al. (1993–2002) contain a wealth of material. A comparison of Bayesian and other approaches to statistical inference is provided by Barnett (1982). Nice recent textbook treatments at a lower level than this book can be found in Berry (1996) and Bolstad (2007).
A very nice book giving a treatment of Bayesian methods of great interest both to the layman and to the specialist is McGrayne (2011).
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