Math 2.0

Statistics Definition

Poisson regression

Builds on Generalized linear modelPoisson distributionMaximum likelihood estimation

Definition

Let x1,…,xn∈ℝpx_1, \dots, x_n \in \R^p be known regressor vectors. The Poisson regression model for responses y1,…,yn∈{0,1,2,… }y_1, \dots, y_n \in \set{0, 1, 2, \dots} states three things. The yiy_i are independent, yiy_i has the Poisson distribution with mean μi>0\mu_i > 0, and log⁡μi=xiTβ\log \mu_i = x_i^T \beta for an unknown β∈ℝp\beta \in \R^p.

Remarks

It is the generalized linear model with Poisson responses and the log link, which is the canonical link. The log-likelihood is ℓ(β)=∑i(yixiTβ−exiTβ−log⁡yi!)\ell(\beta) = \sum_i \big( y_i x_i^T \beta - e^{x_i^T \beta} - \log y_i! \big), a concave function. The maximum likelihood estimate solves ∑i(yi−μi)xi=0\sum_i (y_i - \mu_i) x_i = 0.

Example: with only an intercept, xi=1x_i = 1, the likelihood equation gives ∑i(yi−eβ)=0\sum_i (y_i - e^{\beta}) = 0, so eβ^=yˉe^{\hat\beta} = \bar y, the mean count.

When yiy_i counts events over an exposure ti>0t_i > 0, one takes log⁡μi=log⁡ti+xiTβ\log \mu_i = \log t_i + x_i^T \beta; the known term log⁡ti\log t_i is an offset, and exiTβe^{x_i^T \beta} is the rate per unit of exposure.

Non-example: if the counts have variance larger than their mean, the hypothesis that yiy_i is Poisson fails (overdispersion). The point estimates remain consistent when the mean is correctly specified, but the usual standard errors are too small.

Reading path

To understand this page from the ground up, read these 76 pages in order. The foundations come first.

  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Generalized linear model concept
  4. Integers object
  5. Ordered field definition
  6. Poisson distribution concept
  7. Proposition concept
  8. Random experiment concept
  9. Rational numbers object
  10. Real numbers object
  11. Set definition
  12. Ring of integers statement
  13. Disjunction definition
  14. Implication definition
  15. Predicate definition
  16. Ordered pair definition
  17. Union definition
  18. Sample space definition
  19. Existential quantifier definition
  20. Universal quantifier definition
  21. The rationals form a field statement
  22. Cartesian product definition
  23. Subset definition
  24. Uniqueness quantifier definition
  25. The rationals form an ordered field statement
  26. Binary relation definition
  27. Universal set definition
  28. Power set definition
  29. Order on the reals definition
  30. Complement definition
  31. Function definition
  32. Interval definition
  33. Sigma-algebra definition
  34. Injective function definition
  35. Preimage of a set definition
  36. Countable set definition
  37. Event definition
  38. Borel sigma-algebra definition
  39. Probability axioms concept
  40. Probability measure definition
  41. Probability space definition
  42. Probability distribution definition
  43. Conditional probability definition
  44. Random variable definition
  45. Independence of two events definition
  46. Discrete random variable definition
  47. Convergence in probability definition
  48. Cumulative distribution function definition
  49. Law of a random variable definition
  50. Random variable with a density definition
  51. Mutual independence of events definition
  52. Independence of two random variables definition
  53. Probability mass function definition
  54. Joint probability distribution definition
  55. Expected value definition
  56. Identically distributed random variables definition
  57. Probability density function definition
  58. Independent sigma-algebras definition
  59. Variance definition
  60. Marginal distribution definition
  61. Linearity of expectation statement
  62. Independence of random variables definition
  63. Covariance definition
  64. Independence means the joint law is the product of the marginals statement
  65. Variance of a sum statement
  66. Expectation of a product of independent random variables statement
  67. Independent and identically distributed random variables definition
  68. Independent random variables are uncorrelated statement
  69. Statistic definition
  70. Likelihood function definition
  71. Variance of a sum of independent random variables statement
  72. Estimator definition
  73. Consistent estimator definition
  74. Sample mean definition
  75. Maximum likelihood estimation concept
  76. Poisson regression definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Poisson_regression
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DefinesPoisson regression
UsesGeneralized linear model, Poisson distribution, Maximum likelihood estimation

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