Propensity score weighting: Inverse of the probability this does not seem logical I hope someone can help me understand propensity score weights. When weighting a regression (or other analyses) by propensity score one uses 1/propensity score as the weight which means 1/probability of observing the given datum. This is counter intuitive as it increases the contribution to the analysis of rarely observed datum and decreases the contribution commonly observed datum. One would think that one would want to increase the weight of commonly seen datum and decrease the weight of an uncommonly observed datum. Compare the propensity score weight to the way one uses variances as weights. One weights by 1/variance which means that data that we have little confidence in (i.e. we don't often see) are down weighted while data in which we have great confidence in (i.e. we see often) are upweighted.  
 A: IPW up-weights individuals that are causally relevant and down-weights individuals that are causally irrelevant. Causally relevant control units are those with values similar to the those of the treated units, and vice versa. Causally irrelevant control units are those that bear little resemblance to treated units. A control unit with a propensity score of 0.9 is an unusual control unit but looks very similar to treated units; therefore, it is up-weighted because it does a good job of representing the outcome of a treated unit had they instead received control. The reason for doing this is so that the weighted outcomes of the control units simulate what the outcomes of the treated units would have been had they received control, which is the causal quantity of interest. Weighting by the variance is used to reduce the influence of outliers or poorly measured units (i.e., observations we have less confidence in), but weighting by the propensity score is used to reduce the influence of causally irrelevant units so that we can compare treated and control units validly.
Note that there are extensions to this method that weight individuals differently. Overlap weights weight treated units by $(1-ps)$ and control units by $(ps)$ (note no inverse occurs) (Li, et al. 2017). This also serves to improve overlap but up-weights those with propensity scores close to .5 (i.e., those who could have received treatment or control with approximately equal probability). Chan et al's (2016) empirical balancing calibration weights solve a three-way balancing problem: weight the treated and control units to minimize the imbalance between the treated and the control units, between the treated and the overall sample, and between the control and the overall sample. Li and Green's (2013) matching weights weight individuals in such a way to as simulate 1:1 matching. Remember that the key is balance, and any weights that achieve balance are valid weights for reducing confounding.

Chan, K. C. G., Yam, S. C. P., & Zhang, Z. (2016). Globally efficient non-parametric inference of average treatment effects by empirical balancing calibration weighting. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 78(3), 673–700. https://doi.org/10.1111/rssb.12129
Li, F., Morgan, K. L., & Zaslavsky, A. M. (2016). Balancing Covariates via Propensity Score Weighting. Journal of the American Statistical Association, 0(ja), 0–0. https://doi.org/10.1080/01621459.2016.1260466
Li, L., & Greene, T. (2013). A Weighting Analogue to Pair Matching in Propensity Score Analysis. The International Journal of Biostatistics, 9(2). https://doi.org/10.1515/ijb-2012-0030
A: John, 
You are correct in that this process does provide greater weight to those individuals outside of the mass of the propensity score distribution. This is also given your scores are: 0.0 > P < 1.0. These individuals with extreme propensities may be special in regards to their covariate imbalance and their contribution is under-scored if not addressed. The use of inverse weights, I believe, was introduced by James Robins. Below is a single resources out of countless that may be of value. I will note that there is another approach to using IPSW, since small propensities can be unstable, which is the use of standardized IP-weighting.   
Causal Inference book by Miguel A. Hernan and James M. Robins :https://www.hsph.harvard.edu/miguel-hernan/causal-inference-book/
