Points of Significance: Bayes' theorem

Journal name:
Nature Methods
Volume:
12,
Pages:
277–278
Year published:
DOI:
doi:10.1038/nmeth.3335
Published online

Incorporate new evidence to update prior information.

At a glance

Figures

  1. Marginal, joint and conditional probabilities for independent and dependent events.
    Figure 1: Marginal, joint and conditional probabilities for independent and dependent events.

    Probabilities are shown by plots3, where columns correspond to coins and stacked bars within a column to coin toss outcomes, and are given by the ratio of the blue area to the area of the red outline. The choice of one of two fair coins (C, C′) and outcome of a toss are independent events. For independent events, marginal and conditional probabilities are the same and joint probabilities are calculated using the product of probabilities. If one of the coins, Cb, is biased (yields heads (H) 75% of the time), the events are dependent, and joint probability is calculated using conditional probabilities.

  2. Graphical interpretation of Bayes' theorem and its application to iterative estimation of probabilities.
    Figure 2: Graphical interpretation of Bayes' theorem and its application to iterative estimation of probabilities.

    (a) Relationship between conditional probabilities given by Bayes' theorem relating the probability of a hypothesis that the coin is biased, P(Cb), to its probability once the data have been observed, P(Cb|H). (b) The probability of the identity of the chosen coin can be inferred from the toss outcome. Observing a head increases the chances that the coin is biased from P(Cb) = 0.5 to 0.6, and further to 0.69 if a second head is observed.

  3. Disease predictions based on presence of markers.
    Figure 3: Disease predictions based on presence of markers.

    (a) Independent conditional probabilities of observing each marker (A, B) given a disease (X, Y, Z) (e.g., P(A|Y) = 0.9). (b) Posterior probability of each disease given a single observation that confirms the presence of one of the markers (e.g., P(Y|A) = 0.66). (c) Evolution of disease probability predictions with multiple assays. For a given disease, each path traces (left to right) the value of the posterior that incorporates all the assay results up to that point, beginning at the prior probability for the disease (blue dot). The assay result is encoded by an empty (marker absent) or a solid (marker present) dot. The red path corresponds to presence of A and B. The highest possible posterior is shown in bold.

References

  1. Eddy, S.R. Nat. Biotechnol. 22, 11771178 (2004).
  2. Krzywinski, M. & Altman, N. Nat. Methods 10, 809810 (2013).
  3. Oldford, R.W. & Cherry, W.H. Picturing probability: the poverty of Venn diagrams, the richness of eikosograms. http://sas.uwaterloo.ca/~rwoldfor/papers/venn/eikosograms/paperpdf.pdf (University of Waterloo, 2006)

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Author information

Affiliations

  1. Jorge López Puga is a Professor of Research Methodology at Universidad Católica de Murcia (UCAM).

  2. Martin Krzywinski is a staff scientist at Canada's Michael Smith Genome Sciences Centre.

  3. Naomi Altman is a Professor of Statistics at The Pennsylvania State University.

Competing financial interests

The authors declare no competing financial interests.

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Supplementary information

Other

  1. Supplementary Table 1 (71 KB)

    Worksheets illustrate calculations that support statements in the column or provide interactive versions of its figures.

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