The Bayes programs on this page are all exact in the Bayesian sense, but not in the frequentist sense. Two of them are conservative in the frequentist sense. Those two are “Probabilities are uniform” and “Probabilities are proportional to numerators.” For those two the printed Bayesian likelihood ratio is an upper bound on the frequentist p-value. The frequentist null hypothesis for each of those two tests is its title in quotation marks. (Those two null hypotheses are simple, and the null hypothesis expected value of the reciprocal of the printed likelihood ratio is unity, and we use Markov’s inequality.)

All the Bayes programs on this page, except one, make use of Dirichlet distributions and Laplace’s Law of Succession. The exception is the program for “One parameter” which uses formulas furnished by the user and integrates by Romberg’s method applied to the midpoint rule. Any mistakes in the programs are mine, not Dirichlet’s or Laplace’s or Romberg’s.

In the following paragraphs are some examples showing use of the programs on this page. The user is respectfully invited to try out these examples or to use any others. The only thing to remember is: follow the grammatical rules of JavaScript. (This is because the “eval” method of JavaScript is used in picking up the data from the text area.) In particular, remember that integers beginning with a zero digit are in base 8.

If the user sees an answer “less than 5e-324”, then there was an underflow. 5e-324 is the smallest non-zero number. To top.

Which browsers?

[
[1768,  807, 189, 47],
[ 946, 1387, 746, 53],
[ 115,  438, 288, 16]
]

Here is a table from Agresti, A., A Survey of Exact Inference for Contingency Tables, Statistical Science 1992, Vol. 7, No.1, 131-177. It is Table 1 on page 132.

[
[17066,14464,788,126,37],
[   48,   38,  5,  1, 1]
]

The present program can do sparse tables. Here is a table copied from P. Diaconis and B. Sturmfels, Algebraic algorithms for sampling from conditional distributions, Annals of Statistics, 1998, Vol. 26, No. 1, 363-397. This table is on their page 364.

  [ [1,0,0,0,1,2,0,0,1,0,1,0],
    [1,0,0,1,0,0,0,0,0,1,0,2],
    [1,0,0,0,2,1,0,0,0,0,0,1],
    [3,0,2,0,0,0,1,0,1,3,1,1],
    [2,1,1,1,1,1,1,1,1,1,1,0],
    [2,0,0,0,1,0,0,0,0,0,0,0],
    [2,0,2,1,0,0,0,0,1,1,1,2],
    [0,0,0,3,0,0,1,0,0,1,0,2],
    [0,0,0,1,1,0,0,0,0,0,1,0],
    [1,1,0,2,0,0,1,0,0,1,1,0],
    [0,1,1,1,2,0,0,2,0,1,1,0],
    [0,1,1,0,0,0,1,0,0,0,0,0] ]

Any mistakes I have made in using Wilks’ formulas are my own mistakes, certainly not his. To top.

Probabilities are uniform

[82,0]

The program is not limited to only two cells. Here is a fictitious table for throwing a cheap variety-store plastic die one hundred times and counting how many times each face was on top. The numbers of times for 1 2 3 4 5 and 6 times are shown. As before, likelihood ratio larger than 1 is in favor of uniformity of probabilities, and smaller ratio is in favor of non-uniformity.

[21,25,26,16,0,12]

[
[315,101,108,32],
[  9,  3,  3, 1]
]

[
[ 102, function(t){ return (1-t)*(1-t)*(1-t) } ],
[ 301, function(t){ return   3*(1-t)*(1-t)*t } ],
[ 317, function(t){ return       3*(1-t)*t*t } ],
[  96, function(t){ return             t*t*t } ]
]

If instead the table is

[
[   0, function(t){ return (1-t)*(1-t)*(1-t) } ],
[ 301, function(t){ return   3*(1-t)*(1-t)*t } ],
[ 317, function(t){ return       3*(1-t)*t*t } ],
[  96, function(t){ return             t*t*t } ]
]

[
[ 301, function(t){ return   3*(1-t)*(1-t)*t } ],
[ 317, function(t){ return       3*(1-t)*t*t } ],
[  96, function(t){ return             t*t*t } ]
]

[
[50,8],
[92,50]
]

[
[50,92],
[8,50]
]

[
[100,100],
[100,100]
]

independent( [[50,8],[92,50]] )
*
independent( [[50,92],[8,50]] )

Another use for programming is testing an array for symmetry around its center. Here is an artificial example:

[50,0,49,1]

[50,50]

uniform( [50,1] )
*
uniform( [0,49] )

Besides the “independent” and “uniform” functions this file contains two other functions. There is a function called “proportional” to do what the “Probabilities are proportional to numerators” button does, and there is a function called “oneParameter” to do what the “One parameter” button does. To top.

Comparisons with non-Bayes inferences

The number 1.645 is copied out of the “normal” table used in many introductory courses in statistics. It is used in building a 5% one-sided boundary for a non-sequential test of the fairness of a coin. Here is a JavaScript program to find the number of heads that will cause that test to “reject,” and to use that number of heads in a Bayes inference.

var boundary=1.645;
var tosses=1e4;
var yatesCorrection=.5;
var heads=Math.ceil( yatesCorrection + tosses*.5 + boundary*Math.sqrt(tosses*.5*.5) );
var tails=tosses-heads;
var x=[heads,tails];
// throw( x );
uniform( x )

The number 2.706 is copied out of the “chi-squared” table used in many introductory courses in statistics. It is used in building a 5% one-sided boundary for a non-sequential test of the independence of a 2 by 2 contingency table. Here is a JavaScript program to find the contingency table entries that will cause that test to “reject,” and to use those entries in a Bayes inference.

var boundary=2.706;
var n=2e4;
var expected=n/4;
var yatesCorrection=0.5;
var higherObserved=Math.ceil( expected+yatesCorrection+Math.sqrt( boundary*expected/4 ) );
var lowerObserved=2*expected-higherObserved;
var x=[[higherObserved,lowerObserved],[lowerObserved,higherObserved]];
// throw( x );
independent( x )

The user will notice that for each of the two programs the Bayes decision and non-Bayes decision are 5% decisions, but in opposite directions. These are numerical examples of the Jeffreys-Lindley paradox. To top.

Enclose in brackets

var k1=10;
var k2=10;
var x=[];
	for( var eye=0;eye<k1;eye++ )
	{
	x[eye]=[];
	for( var j=0;j<k2;j++ )x[eye][j]=(eye+j)%2;
	}
independent( x )+"   "+incoherent( x );

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