Log-odds based filtering and pairHMM alignment during local realignment

Question

[CY]

HC / Mutect2 uses PairHMM to do the alignment and output the score representing the likelihood of observing that read given that haplotype (P(G|D)). This P(G|D) will then be used in Bayes to call the variant.

My questions are:

1) Log-odds has been used to filter variants called from Mutect2. I have been having difficulty understanding the connect between log-odds ratio and the process described above. I feel this "ratio" has something to do with whatever resulted from the above process, right?

2) Does MAPQ in the bam file also has some connection with the output of PairHMM?

Really appreciate for any comments.