Fast computation of the Bayes Factor (same source v. different sources) for the Normal - Inverted Wishart model.

Implemented in C.

samesource_C(
  quest,
  ref,
  n.iter,
  B.inv,
  W.inv.1,
  W.inv.2,
  U,
  nw,
  mu,
  burn.in,
  verbose = FALSE,
  marginals = FALSE
)

Arguments

quest	the questioned dataset (a $n_q \times p$ matrix)
ref	the reference dataset (a $n_r \times p$ matrix)
n.iter	number of MCMC iterations excluding burn-in
B.inv	prior inverse of between-source covariance matrix
W.inv.1	prior inverse of within-source covariance matrix (questioned items)
W.inv.2	prior inverse of within-source covariance matrix (reference items)
nw	degrees of freedom
mu	prior mean ($p \times 1$)
burn.in	number of MCMC burn-in iterations
verbose	if TRUE, be verbose
marginals	if TRUE, also return the marginal likelihoods in the LR formula (default: FALSE)

Value

the log-BF value (base e), or a list with the log-BF and the computed marginal likelihoods:

value: the log-BF value (base e)
log_ml_Hp: log-BF numerator (from reference = questioned source)
log_ml_Hd_ref: log-BF denominator from reference source
log_ml_Hd_quest: log-BF denominator from questioned (!= reference) source

Details

The hypothesis pair is:

$H_p$: all ref and quest come from the same source
$H_p$: quest comes from source 1, ref comes from source 2

See diwishart_inverse for the parametrization of the Inverted Wishart. See marginalLikelihood_internal for further documentation.

Normal-Inverse-Wishart model

Described in (Bozza et al. 2008) .

Observation level:

$$X_{ij} \sim N_p(\theta_i, W_i)$$ (i = source, j = items from source)

Group level:

$$\theta_i \sim N_p(\mu, B)$$
$$W_i \sim IW_p(\nu_w, U)$$

Hyperparameters:

$$B, U, \nu_w, \mu$$

Posterior samples of $\theta$, $W^{(-1)}$ can be generated with a Gibbs sampler.

Inverted Wishart parametrization (Press)

Uses (Press 2012) parametrization.

$$X \sim IW(\nu, S)$$ with $S$ is a $p \times p$ matrix, $\nu > 2p$ (the degrees of freedom).

Then: $$E[X] = \frac{S}{\nu - 2(p + 1)}$$

References

Bozza S, Taroni F, Marquis R, Schmittbuhl M (2008). “Probabilistic Evaluation of Handwriting Evidence: Likelihood Ratio for Authorship.” Journal of the Royal Statistical Society: Series C (Applied Statistics), 57(3), 329-341. ISSN 1467-9876, doi: 10.1111/j.1467-9876.2007.00616.x .

Press SJ (2012). Applied Multivariate Analysis: Using Bayesian and Frequentist Methods of Inference. Courier Corporation.

quest	the questioned dataset (a \(n_q \times p\) matrix)
ref	the reference dataset (a \(n_r \times p\) matrix)
n.iter	number of MCMC iterations excluding burn-in
B.inv	prior inverse of between-source covariance matrix
W.inv.1	prior inverse of within-source covariance matrix (questioned items)
W.inv.2	prior inverse of within-source covariance matrix (reference items)
nw	degrees of freedom
mu	prior mean (\(p \times 1\))
burn.in	number of MCMC burn-in iterations
verbose	if TRUE, be verbose
marginals	if TRUE, also return the marginal likelihoods in the LR formula (default: FALSE)