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For each observation time, use only the currently susceptible risk set $S_t$, with new-infection indicators $Y_{it}$. PutThis is the spatial Bernoulli infection model, under conditional independence of the indicators. Distances must be positive. Already infected trees should not be included as new susceptible trials. The minus sign in the power kernel is present in the original PDF, although it is absent in the converted TeX.

The parameter $\alpha$ must be nonnegative: for $\alpha <0$ and positive exposure, $1-e^{-\alpha A_{it}}<0$ is not a probability. Therefore the stated unrestricted normal prior distribution cannot literally define the model. A coherent interpretation is a normal prior distribution restricted to $\alpha >0$,Here $\Phi$ is the standard normal distribution function. Use the same proper prior distribution in both kernel models. The given Gaussian prior distributions for $\beta$ and $\gamma$ are mathematically valid on the real line for finite positive distances, though restricting them to positive values would impose decaying kernels and would require the corresponding normalized prior distributions.

The full conditional kernels areNeither is generally normal or another elementary conjugate family: the infection factors contain $1-e^{-\alpha A}$, and $A$ depends nonlinearly on $\beta$. Thus a simple conjugate Gibbs sampler is unavailable. This does not make Gibbs sampling logically impossible; exact rejection or slice conditional draws could be used if implemented. A straightforward alternative is Metropolis–Hastings algorithm updates within each coordinate.

For example, put $\eta =\log \alpha$ and use symmetric Gaussian random-walk proposals for $(\eta ,\beta )$. The target in these coordinates isThe factor $e^{\eta }$ is the Jacobian determinant and must be retained. Accept according to the ratio of this target, either for a block or one coordinate at a time. Posterior averages and quantiles of $e^{\eta }$ and $\beta$ estimate the requested marginal posterior distributions.

To compare kernels, introduce $M\in \{1,2\}$ with positive model prior distribution probabilities $w_1,w_2$. Under model two replace $A_{it}$ by ${\sum }_{j\in I_t}{e}^{-\gamma d_{ij}}$, defining $L_2(\alpha ,\gamma )$. The joint model/parameter target iswhere $\zeta =\beta$ or $\gamma$ and $p_M$ is its normalized proper prior distribution. Within each model use the preceding random-walk updates. For a concrete model switch, keep $\alpha$, propose the other model's kernel parameter independently from its prior distribution, and use equal forward and reverse switch-selection probabilities. A move from $(1,\alpha ,\beta )$ draws $\gamma \sim p_{\gamma }$ and has ratioAccept with the minimum of this ratio and one. The auxiliary-coordinate swap $(\alpha ,\beta ,\gamma )\mapsto (\alpha ,\gamma ,\beta )$ has absolute Jacobian determinant one, so no additional factor occurs; unequal switch-selection probabilities would require their reverse/forward ratio. This is a valid model-switching Markov chain Monte Carlo construction even though the two active models have equal parameter dimension. Prior distribution proposals may be inefficient, so fitted proposals can be substituted provided their probability densities remain in the acceptance ratio.

After checking mixing, estimate posterior model probabilities by the fractions of retained iterations in each model. The Bayes factor in favor of the first kernel isUse the posterior distributions within each model as well as these odds to assess the competing distance kernels. Proper prior distributions and their model-dependent normalizing constants are essential for this comparison.