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📚 Statistical Inference for Everyone
Chapters ▾
1 Introduction to Probability2 Applications of Probability3 Random Sequences and Visualization4 Introduction to Model Comparison5 Applications of Model Comparison6 Introduction to Parameter Estimation7 Priors, Likelihoods, and Posteriors8 Common Statistical Significance Tests9 Applications of Parameter Estimation and Inference10 Multi-parameter Models11 Introduction to MCMC12 Concluding Thoughts13 Appendix A: Computational Analysis14 Appendix B: Notation and Standards15 Appendix C: Common Distributions and Their Properties16 Appendix D: Tables

6.5 MAP and Areas

Now we revisit the questions posed in Section 6.2 on page 121 about the bent coin, this time using the distribution found above, reproduced here in Figure 6.6.

Smooth posterior density for θ, the bent coin's chance of heads, given 3 heads and 9 tails. The curve rises from zero, peaks at θ=0.25 at a density of about 3.3, and is negligible beyond θ=0.7.
Figure 6.6. Posterior probability distribution for the θ values of the bent coin - the probability that the coin will land heads. The distribution is shown for data 3 heads and 9 tails, with a maximum at θ=0.25.

One answer to the first question can be accomplished by looking at the maximum of the posterior distribution, shown in Figure 6.6. By eye, it seems to have a maximum at θ=0.25. In fact one can demonstrate that this distribution has a maximum at

θ m a x = number of successes total number of attempts ,

where in our example, a success is head, and an attempt is a flip. We take up this question of the best estimate of θ, given the posterior probability for θ, in more detail in Section 6.7.

The answer to the second question can be done by looking at the area under the curve from θ=0, the “all heads” coin, to θ=0.5, the “fair” coin, as shown in Figure 6.7. This area represents the probability, given the data, that the coin is skewed towards heads or, in other words, how confident are we that this is an unfair coin. Given the value of P(θ<0.5)=0.954 we can say that this is “very likely” an unfair coin (see Table 1.1 on page 51).

The posterior density for θ given 3 heads and 9 tails with the region from θ=0 to θ=0.5 shaded; the shaded area is 0.954, the probability that the coin is biased toward tails.
Figure 6.7. Posterior probability distribution for the θ values of the bent coin - the probability that the coin will land heads. The distribution is shown for data 3 heads and 9 tails. The area under the curve from θ=0 (the “all heads” coin) to θ=0.5 (the “fair” coin) is 0.954.

Adapted from Statistical Inference for Everyone, by Brian Blais (Bryant University), licensed under CC BY-SA 4.0 (dual-licensed under the GNU FDL 1.2 or later; this adaptation uses the CC BY-SA grant). Changes were made; this adaptation is distributed under the same license. License: CC-BY-SA-4.0.