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Discretize Chi-Square to K_J Support

Source: R/12_a2_kl.R
discretize_chisq.Rd

Converts a (possibly scaled) chi-square distribution into a discrete PMF on \(\{1, \dots, J\}\) using continuity-corrected binning: $$p(k) = P(k - 0.5 < X \le k + 0.5), \quad k = 1, \dots, J$$ followed by renormalization.

Usage

discretize_chisq(J, df, scale = 1)

Arguments

J

Integer; maximum value (support upper bound).

df

Numeric; degrees of freedom.

scale

Numeric; scale parameter (default 1). If scale != 1, assumes \(X = scale \cdot \chi^2_{df}\).

Value

Numeric vector of length J; PMF on \(\{1, \dots, J\}\).

Details

For a scaled chi-square distribution \(Y = scale \cdot X\) where \(X \sim \chi^2_{df}\):

  • \(E[Y] = scale \cdot df\)

  • \(Var[Y] = scale^2 \cdot 2 \cdot df\)

Matching target moments: Given target mean \(\mu_K\) and variance \(\sigma^2_K\):

  • \(scale = \sigma^2_K / (2 \mu_K)\)

  • \(df = 2 \mu_K^2 / \sigma^2_K\)

See also

DPprior_a2_kl

Examples

# Chi-square with target moments mu=5, var=8
mu_K <- 5
var_K <- 8
scale <- var_K / (2 * mu_K)
df <- 2 * mu_K^2 / var_K
pmf <- discretize_chisq(50, df = df, scale = scale)

# Verify moments
k_vals <- 1:50
sum(k_vals * pmf)  # ~5
#> [1] 5.01509