The bound is informative only once log M exceeds I + log 2.
Usage
Fano(M, mutual_info, base_e = TRUE)
Arguments
- M
Number of hypotheses, M >= 2.
- mutual_info
I(theta; X), non-negative.
- base_e
TRUE for nats, FALSE for bits.
Value
List with bound, raw_bound, log_M,
informative, M.
Details
Formula: P_err >= 1 - (I(theta; X) + log 2) / log M
References
Fano (1961), Transmission of Information, MIT Press, and
Cover & Thomas (2006), Elements of Information Theory, 2nd edition,
Theorem 2.10.1. The worklist filed this under "Ghosal Appendix K";
the copy of Ghosal & van der Vaart (2017) held in the corpus was
searched in full and the word "Fano" does NOT occur in it, so the
attribution could not be confirmed and the primary sources are cited.
Examples
Fano(M = 5L, mutual_info = 5L)
#> $bound
#> [1] 0
#>
#> $raw_bound
#> [1] -2.537351
#>
#> $log_M
#> [1] 1.609438
#>
#> $informative
#> [1] 0
#>
#> $M
#> [1] 5
#>
#> $method
#> [1] "Fano inequality lower bound"
#>
#> attr(,"class")
#> [1] "morie_rich_result" "list"
