### Table 2: Distributions and their conjugate priors

1994

"... In PAGE 40: ...stablish properties of it. In Sections 5.3.2 and 5.3.3 it is shown how this property forms the basis of several fast Bayesian learning algorithms looking at multiple models, including decision trees [Bun91b], and Bayesian networks [SL90, SDLC93, Bun91d]. Table2... In PAGE 44: ...2.3 Recognizing and using the exponential family As a nal note, how can we apply these operations automatically to a graphical model? Which distributions are exponential family and which have conjugate distributions with normalizing con- stants in closed form? Table2 gives a selection of distributions, and their conjugate distribution. Further details can be found in most textbooks on probability distributions.... In PAGE 54: ...emma 5.8 Consider the context of Lemma 5.1. Then the model likelihood or evidence, given by evidence(M) = p(x1; : : :; xNjy1; : : :; yN; M), can be computed as: evidence(M) = p( j ) QN j=1 p(xjjyj; ) p( j 0) = Z ( 0) Z ( )ZN 2 : For the distributions in Table 1 with priors in Table2 , Table 4 gives their matching evidence derived using Lemma 5.8 and cancelling a few common terms.... In PAGE 55: ...Learning with Graphical Models 55 p(var2 = 1jvar1 = 0; 2) = 2;0j0 ; p(var2 = 1jvar1 = 1; 2) = 2;0j1 : If we use Dirichlet priors for these parameters, as shown in Table2 , then the priors are: ( 1; 1 ? 1) Dirichlet( 1;0; 1;1) ; ( 2;0jj; 1 ? 2;0jj) Dirichlet( 2;0jj; 2;1jj) for j = 0; 1 ; where 2;0j0 is aprior independent of 2;0j1. Arguments for choosing the values of these parameters are given later in Section 6.... In PAGE 55: ... Each get their own parameters, su cient statistics, and contribution to the evidence. Conjugate priors from Table2 (using xjy Gaussian) are indexed accordingly as: ijjj ijj Gaussian( 0;ijj; 0;ijj 2 ijj ) for i = 1; 2 and j = 0; 1 ; ?2 ijj Gamma( 0;ijj=2; 0;ijj) for i = 1; 2 and j = 0; 1 ; Notice that 0;ijj is one dimensional when i = 0 and two dimensional when i = 2. Suitable su cient statistics for this situation are read from Table 3 by looking at the data summaries used there.... ..."

Cited by 189

### Table 3: Distributions and their conjugate priors

1994

"... In PAGE 25: ... Once the posterior distribution is found, and assuming it is one of the standard distributions, the property can easily be established. Table3 in Appendix B gives some standard conjugate prior distri- butions for those in Table 2, and Table 4 gives their matching posteriors. More extensive summaries of this are given by DeGroot (1970) and Bernardo and Smith (1994).... In PAGE 41: ... The evidence for some common exponential family distributions is given in Appendix B in Table 5 For instance, consider the learning problem given in Figure 24. Assume that the variables var1 and var2 are both binary (0 or 1) and that the parameters 1 and 2 are interpreted as follows: p(var1 = 0j 1) = 1 ; p(var2 = 0jvar1 = 0; 2) = 2;0j0 ; p(var2 = 0jvar1 = 1; 2) = 2;0j1 : If we use Dirichlet priors for these parameters, as shown in Table3 , then the priors are: ( 1; 1 ? 1) Dirichlet( 1;0; 1;1) ; ( 2;0jj; 1 ? 2;0jj) Dirichlet( 2;0jj; 2;1jj) for j = 0; 1 ; where 2;0j0 is a priori independent of 2;0j1. The choice of priors for these distributions is discussed in (Box amp; Tiao, 1973; Bernardo amp; Smith, 1994).... In PAGE 42: ...statistics, and contribution to the evidence. Conjugate priors from Table3 in Appendix B (using yjx Gaussian) are indexed accordingly as: ijjj ijj Gaussian( 0;ijj; 0;ijj 2 ijj ) for i = 1; 2 and j = 0; 1 ; ?2 ijj Gamma( 0;ijj=2; 0;ijj) for i = 1; 2 and j = 0; 1 : Notice that 0;ijj is one-dimensional when i = 0 and two-dimensional when i = 2. Suitable su cient statistics for this situation are read from Table 4 by looking at the data summaries used there.... In PAGE 58: ... Further details and more extensive tables can be found in most Bayesian textbooks on probability distributions (DeGroot, 1970; Bernardo amp; Smith, 1994). Table3 gives some standard conjugate prior distributions for those in Table 2, and Table 4 gives their matching posteriors (DeGroot, 1970; Bernardo amp; Smith,... In PAGE 60: ...Distribution Evidence j C-dim multinomial Beta(n1 + 1; : : :; nC + C)=Beta( 1; : : :; C) yjx Gaussian det1=2 0 N=2 det1=2 ?(( 0+N)=2) ( 0+N)=2 ?( 0=2) 0=2 0 x Gamma 0 0 ?(N + 0) ?( 0)?PN i=1 xi+ 0 N + 0 for xed x d-dim Gaussian det 0=2 S0 ( )dN=2 det( 0+N)=2(S+S0) Nd 0 (N+N0)d Qd i=1 ?(( 0+N?1?i)=2) ?(( 0?1?i)=2) Table 5: Distributions and their evidence 1994). For the distributions in Table 2 with priors in Table3 , Table 5 gives their matching evidence derived using Lemma 6.4 and cancelling a few common terms.... ..."

Cited by 189

### Table 4: Distributions and matching conjugate posteriors

1994

"... In PAGE 25: ... Once the posterior distribution is found, and assuming it is one of the standard distributions, the property can easily be established. Table 3 in Appendix B gives some standard conjugate prior distri- butions for those in Table 2, and Table4 gives their matching posteriors. More extensive summaries of this are given by DeGroot (1970) and Bernardo and Smith (1994).... In PAGE 42: ... Conjugate priors from Table 3 in Appendix B (using yjx Gaussian) are indexed accordingly as: ijjj ijj Gaussian( 0;ijj; 0;ijj 2 ijj ) for i = 1; 2 and j = 0; 1 ; ?2 ijj Gamma( 0;ijj=2; 0;ijj) for i = 1; 2 and j = 0; 1 : Notice that 0;ijj is one-dimensional when i = 0 and two-dimensional when i = 2. Suitable su cient statistics for this situation are read from Table4 by looking at the data summaries used there. This can be simpli ed for x1 because d = 1 and y1 for the Gaussian is uniformly 1.... In PAGE 42: ... Denote x1j0 and x1j1 as the sample means of x1 when var1 = 0; 1, respectively, and s2 1j0 and s2 1j1 their corresponding sample variances. This cannot be done for the second case, so we use the notation from Table4 , where ; ; from Table 4 become, respectively, 2jj; 2jj; 2jj. Change the vector y to (1; x1) when making the calculations indicated here.... In PAGE 58: ... Further details and more extensive tables can be found in most Bayesian textbooks on probability distributions (DeGroot, 1970; Bernardo amp; Smith, 1994). Table 3 gives some standard conjugate prior distributions for those in Table 2, and Table4 gives their matching posteriors (DeGroot, 1970; Bernardo amp; Smith,... ..."

Cited by 189

### Table 3: Distributions and matching conjugate posteriors

1994

"... In PAGE 42: ...Learning with Graphical Models 42 prior distributions for those in Table 1, and Table3 gives their matching posteriors [DeG70]. The parameters for these priors can be set using standard reference priors [BT73], or elicited from a domain expert.... In PAGE 55: ... Conjugate priors from Table 2 (using xjy Gaussian) are indexed accordingly as: ijjj ijj Gaussian( 0;ijj; 0;ijj 2 ijj ) for i = 1; 2 and j = 0; 1 ; ?2 ijj Gamma( 0;ijj=2; 0;ijj) for i = 1; 2 and j = 0; 1 ; Notice that 0;ijj is one dimensional when i = 0 and two dimensional when i = 2. Suitable su cient statistics for this situation are read from Table3 by looking at the data summaries used there. This can be simpli ed for x1 because d = 1 and y1 for the Gaussian is uniformly 1.... In PAGE 55: ... Denote x1j0 and x1j1 as the sample means of x1 when var1 = 0; 1 respectively, and s2 1j0 and s2 1j1 their corresponding sample variances. For the second case, this cannot be done, so we use the notation from table 3, where ; ; from Table3 become, respectively, 2jj; 2jj; 2jj. Change the vector y to (1; x1) when making the calculations indicated here.... In PAGE 62: ... The su cient statistics in this case are all counts given by nj = N Xi=1 1classi=j = #class = j ; nv;kjj = N Xi=1 1classi=j1varv;i=k = #varv = k and class = j : The expected su cient statistics computed from the rules of probability for a given set of parameters and 1; 2; 3 are given by nj = N Xi=1 p(classi = jjvar1;i; var2;i; var3;i; phi; 1; 2; 3) ; nv;kjj = N Xi=1 p(classi = jjvar1;i; var2;i; var3;i; phi; 1; 2; 3) 1varv;i=k : Thanks to Lemma 5.2, these expected su cient statistics can be computed automatically for most exponential family distributions, and all those in Table3 . Once su cient statistics are computed for any of the distributions in Table 3, posterior means or modes of the model parameters, in this case and 1; 2; 3, can be found automatically.... In PAGE 62: ...2, these expected su cient statistics can be computed automatically for most exponential family distributions, and all those in Table 3. Once su cient statistics are computed for any of the distributions in Table3 , posterior means or modes of the model parameters, in this case and 1; 2; 3, can be found automatically. 1.... In PAGE 62: ... (b) Recompute and 1; 2; 3 to be equal to their mean (or mode in classic EM) conditioned on the su cient statistics. For the posterior distributions in Table3 , these can be looked up in standard tables, and in most cases found via Lemma 5.2.... ..."

Cited by 189

### Table 1 Summary of conjugate priors for the Gaussian distribution and their respective values used in our implementation.

2000

Cited by 3

### Table 1: Distributions for each parameter of a number of exponential family distributions if the model is to satisfy conjugacy constraints. Conjugacy also holds if the distributions are replaced by their multivariate counterparts e.g. the distribution conjugate to the precision matrix of a multivariate Gaussian is a Wishart distribution. Where None is specified, no standard distribution satisfies conjugacy.

2005

"... In PAGE 15: ... A variable with an Exponential or Poisson distribution can have a gamma prior over its scale or mean respectively, although, as these distributions do not lead to hierarchies, they may be of limited interest. These constraints are listed in Table1 . This table can be encoded in implementations of the variational message passing algorithm and used during initialisation to check the conjugacy of the supplied model.... ..."

Cited by 22

### Table 7. Simultaneous effect of multiple conjugate length functions

"... In PAGE 10: ... The number of different length functions was 10 or 100. Table7 displays the results. For each experiment we provide the success rate of the normal form length function, the minimum and maximum success rates attained by any conjugate length function, as well as the average and standard deviation of the distributions.... ..."

### Table 1: Distributions for each parameter of a number of exponential family distributions if the model is to satisfy conjugacy constraints. Conjugacy also holds if the dis- tributions are replaced by their multivariate counterparts e.g. the distribution conjugate to the precision matrix of a multivariate Gaussian is a Wishart distri- bution. Where None is specified, no standard distribution satisfies conjugacy.

2005

"... In PAGE 16: ... A variable with an Exponential or Poisson distribution can have a Gamma prior over its scale or mean respectively, although, as these distributions do not lead to hierarchies, they may be of limited interest. These constraints are listed in Table1 . This table can be encoded in implementations of the Variational Message Passing algorithm and used during initialisation to check the conjugacy of the supplied model.... ..."

Cited by 22

### Table 10: Timings and performance measurements for the conjugate gradient bench- mark (QCD2) distributed in one dimension on the SUPRENUM Total problem Processor Problem size Predicted Actual Performance size Con guration per processor Time (s) Time (s) (M op/s) 2 43

1991

Cited by 17

### Table 2 Results from various conjugate search methods

2003

"... In PAGE 5: ... This software has been reported to work relatively better than other commercial software packages [4]. Table2 shows the performance of DEM generation using each method. For each DEM produced, its accuracy was estimated with 3 arc-second Digital Terrain Elevation Model (DTED) distributed by NIMA.... ..."