In his book Thinking Fast and Slow, which summarizes his life and Tversky’s work, Kahneman introduces biases that stem from misalignment — the false belief that a combination of two events is more likely than one event alone. Conjunction bias is a common error of reasoning in which we believe that two events occurring together are more likely than one of those events to occur on its own. While representativeness bias occurs when we ignore low base rates, conjunction error occurs when we attribute a higher probability to an event with a higher specificity. [Sources: 9]

However, the coincidence of probabilities was not taken into account, and regarding the choice of the model compared to the mean, the author points out that it is difficult to determine exactly which strategy was used by the participants. The observed probability matching behavior suggests that the nervous system samples the hypothesized distribution of the model at each trial. [Sources: 6]

These robust correspondences and discrepancies between human judgment and probability theory challenge non-sampling models of probabilistic bias; Costello and Watts (2014, 2016, 2018) have shown how a sampling model captures both biases and patterns in human probabilistic judgments, demonstrating that these judgments, they say, are, after all, “remarkably rational” and that irrational judgments are the result of noise [Sources: 2]

To illustrate, with one sample, each event will have a probability of 1 (i.e., 1 out of 1) or 0 (0 out of 1). If the brain can sample indefinitely, then under certain circumstances the sampling rates will match the “true” probabilities with arbitrary precision. One of the biggest problems with observational studies is that the likelihood of being exposed or unirradiated to a group is not accidental. [Sources: 2, 3]

The more correct covariates we use, the more accurate our prediction of the likelihood of exposure. We use covariates to predict the probability of exposure (PS). We want to match exposed and unexposed subjects in terms of their likelihood of being exposed (their PS). Below 0.01, we can get a lot of variability within the estimate because it is difficult for us to find matches, and this leads us to discard these items (incomplete match). [Sources: 3]

We would like to see a significant reduction in bias due to inconsistent and consistent analyzes. Ultimately, PSM scores are as good as the characteristics used for comparison. Since all characteristics related to treatment participation and outcomes are observed in the dataset and are known to the investigator, propensity scores provide significant matches for assessing the impact of the intervention. [Sources: 3, 8]

Specifically, PSM will calculate the probability of a unit participating in the program based on the observed characteristics. PSM then compares the processed units with the unprocessed units based on the propensity score. This ensures that units with the same covariate value have a positive chance of healing, but will not be cured. [Sources: 8]

Evaluate the impact of the intervention on the fit sample and calculate the standard errors. Using these coincidences, the researcher can assess the impact of the intervention. The obtained match pairs can also be analyzed using standard statistical methods, for example, Thus, if positive examples are observed in 60% of cases in the training sample, and negative examples are observed in 40% of cases, then the observer using the probability matching strategy predicts (for unlabeled examples) class label “positive.” in 60% of cases and class label “negative” in 40% of cases. [Sources: 3, 5, 8]

But the combination, that is, heads in two-thirds of the cases and tails in one-third of the cases, will be corrected with a probability of (2/3 x 2/3) + (1/3 x 1/3) = 5/9. … While probabilistic matching was a modal response strategy found in the current study, we are not suggesting that probabilistic matching is used in all perception problems, or even in all spatial problems. [Sources: 2, 6]

Recent research has shown that observer behavior is consistent with the expected loss function in a visual discrimination problem [40], but the results are ambiguous with respect to the specific decision-making strategy (mean, choice, and comparison of probabilities) they might make. similar predictions. Moreover, the effects of bias can be explained in terms of setting the response criteria, rather than the goodness-of-fit criteria, as is the case with the Ratcliffe approximations. Different probability distributions, rewards, or changes in context did not affect the results. It is more “rational” because deviations from probability theory arise from its use before improving probability estimates based on a small number of samples. [Sources: 1, 2, 6, 7]

It is also known that Thompson sampling is equivalent to probability matching, a heuristic often called suboptimal, but in fact it can work quite well under the assumption of a non-stationary environment. It also ties in with the converging evidence that participants can use a combination of direct and random exploration in multi-armed bandits, and I’m not sure how this can be accounted for in DBM. I believe that the direct inclusion of information acquisition in the model is analogous to the direct exploration strategy, while the softmax parameter can track random (value-based) exploration. [Sources: 10]

It is clear that the maximization strategy outweighs the coincidence strategy. However, the maximization strategy is rarely found in the biological world. From bees to birds to humans, most animals correspond to probabilities (Erev & Barron, 2005; CR Probabilistic Matching (PM) is a widely observed phenomenon in which subjects compare the likelihood of choice with the likelihood of reward in a stochastic context. [Sources: 1]

Matchmaking strategy is to pick A 70% of the time and B 30% of the time. Combination with substitution reduces distortion by better matching between objects. This effect is especially evident in the no-tip and no-feedback group, where the first ten-reel game was dominated by pairs (bottom right panel). [Sources: 1, 3, 4]

Of those participants who were asked to rate which strategy gives the higher expected return before forecasting, 74% correctly identified the maximization strategy as the best. When comparing these three strategies, the behavior of the overwhelming majority of observers in performing this perceptual task was more consistent with the comparison of probabilities. The third strategy is to select a causal structure in proportion to its likelihood, thus trying to match the likelihood of a putative causal structure. Choosing a matching strategy, subjects violate the axioms of decision theory, and therefore their behavior cannot be rationalized. [Sources: 1, 4, 6]

Based on the theory of optimal foraging (Stephens & Krebs, 1986), IFD predicts that the distribution of individuals between food stalls will match the distribution of resources, a pattern often observed in animals and humans (Grand, 1997; Harper, 1982; Lamb & Ollason, 1993 ; Madden et al., 2002; Sokolowski et al., 1999). There are discrepancies between the model and observed behavior, but foraging groups tend to approach IFD. [Sources: 1]

Indeed, this is the conditional expansion probability given the set of covariates Pr (E + | covariates). Therefore, the likelihood of being exposed is the same as the likelihood of not being exposed. [Sources: 3]

— Slimane Zouggari

##### Sources #####

[0]: https://stats.stackexchange.com/questions/392493/propensity-score-matching-bias-adjustment

[1]: http://naturalrationality.blogspot.com/2007/11/probability-matching-brief-intro.html

[2]: https://journals.sagepub.com/doi/full/10.1177/0963721420954801

[3]: https://www.publichealth.columbia.edu/research/population-health-methods/propensity-score-analysis

[4]: https://link.springer.com/article/10.3758/s13421-012-0268-3

[5]: https://en.wikipedia.org/wiki/Probability_matching

[6]: https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1000871

[7]: https://pubmed.ncbi.nlm.nih.gov/2913571/

[8]: https://dimewiki.worldbank.org/Propensity_Score_Matching

[9]: https://fs.blog/bias-conjunction-fallacy/

[10]: https://proceedings.neurips.cc/paper/2018/file/f55cadb97eaff2ba1980e001b0bd9842-Reviews.html