Inference scratch

This is just a place to play around with different inference algorithms. Gitbook markdown is very application-specific so can't copy this algorithm text into other apps to play around with!

Current inference algorithm

• For $m=1 \dots M$, where $M$is the number of parallel MCMC chains (also known as slots)

  • Generate initial state

    • Generate an initial set of parameters $\Theta_{m,0}$, and copy this to both the global ($\Theta^G_{m,0}$) and chimeric ($\Theta^C_{m,0}$) parameter chain (sequence ;

    • Generate an initial epidemic trajectory $Z(\Theta_{m,0})$

    • Calculate and record the initial likelihood for each subpopulation, $$\mathcal{L_i}(D_i|Z_i(\Theta_{m,0}))$ ;

  • For $k= 1 ... K$ where $K$ is the length of the MCMC chain, add to the sequence of parameter values :

    • Generate a proposed set of parameters $\Theta^*$from the current chimeric parameters using the proposal distribution $$g(\Theta^*|\Theta^C_{m,k-1})$ ;

    • Generate an epidemic trajectory with these proposed parameters, $Z(\Theta^*)$

    • Calculate the likelihood of the data given the proposed parameters for each subpopulation, $\mathcal{L}_i(D_i|Z_i(\Theta^*))$

    • Calculate the overall likelihood with the proposed parameters, $\mathcal{L}(D|Z(\Theta^*))$

    • Make "global" decision about proposed parameters

      • Generate a uniform random number $u^G \sim \mathcal{U}[0,1]$

      • Calculate the overall likelihood with the current global parameters, $\mathcal{L}(D|Z(\Theta^G_{m,k-1}))$

      • Calculate the acceptance ratio $\alpha^G=\min \left(1, \frac{\mathcal{L}(D|Z(\Theta^*)) p(\Theta^*) }{\mathcal{L}(D|Z(\Theta^G_{m,k-1})) p(\Theta^G_{m,k-1}) } \right)$​

      • If $\alpha^G > u^G$: ACCEPT the proposed parameters to the global and chimeric parameter chains

        • Set

        • Set $\Theta_{m,k}^C=\Theta^*$

        • Update the recorded subpopulation-specific likelihood values (chimeric and global) with the likelihoods calculated using the proposed parameter ;

      • Else: REJECT the proposed parameters for the global chain and make subpopulation-specific decisions for the chimeric chain

        • Set $\Theta^G_{m,k} = \Theta^G_{m,k-1}$

        • Make "chimeric" decision:

          • For $i = 1 \dots N$

            • Generate a uniform random number $u_i^C \sim \mathcal{U}[0,1]$

            • Calculate the acceptance ratio $\alpha_i^C=\frac{\mathcal{L}_i(D_i|Z_i(\Theta^*)) p(\Theta^*) }{\mathcal{L}i(D_i|Z_i(\Theta^C_{m,k-1})) p(\Theta_{m,k-1}) }$

            • If $\alpha_i^C > u_i^C$: ACCEPT the proposed parameters to the chimeric parameter chain for this location

              • Set $\Theta_{m,k,i}^C = \Theta^*_{i}$

              • Update the recorded chimeric likelihood value for subpopulation $i$ to that calculated with the proposed parameter​

            • Else: REJECT the proposed parameters for the chimeric parameter chain for this location

              • Set $\Theta_{m,k,i}^C=\Theta_{m,k-1,i}$​

            • `End if ;

          • End for $N$subpopulations

        • End making chimeric decisions

      • End if

    • End making global decision

  • End for $K$ iterations of each MCMC chain

• End for $M$ parallel MCMC chains

• Collect the final global parameter values for each parallel chain $\theta_m = \{\Theta^G_{m,K}\}_m$

Making chimeric decision first