Data: Attributes ± Outcome Designation Array of of Apriori Probabilities

Data Input for Analysis of reference Data     The data is for a table with 2 columns     Each row contains data from a case from the reference data     Col 1 = Text string representing attributes of predictor     Col 2 = Text string representing Outcome Data Input for Interpretation     The data is for a table with single columns of attributes     Each row contains data from a case for interpretation Array of Apriori Probabilities     Probability for each outcome before attributes are known     Single row, number of columns = number of outcomes     Columns separated by spaces of tabs     Values representing relative probabilities

Program 1.

Reference Table of Counts

Input Table of Counts for Analysis of reference Table     This is an alternative input for analysis     The table is a count of the reference data     Col 1 contains attributes of predictor     The other columns are the outcomes     The first row contains the outcome names     Each following row represent attributes     Each cell is the count of the attribute (row) for that outcome (Col)

Program 2.

Probabilities for each attribute given the outcome P(a|o) Table of Probabilities

Input Table of Probability of attributes for each outcome P(a|o)     This is a table of probabilities P(a|o)     Col 1 contains attributes     The other columns are the outcomes     The first row contains the outcome names     Each following row shows each attribute, and its P(a|o)     Each cell is the Probability attribute for the outcome P(a|o)

Program 3.

Program 4.

Full explanations on Basic Bayes Probability, the terminology used, and the default example of this program are provided in the Introduction and Basic Bayes panels of Classification by Bayes Probability Explained Page . This panel presents brief summaries, sufficient only to help new users to negotiate data entry and program options.

Default Example

The data in the default example are artificially generated and does not represent reality. It purports to be from an exercise to use hair color (Dark and Light) and eye color (Blue, Brown, Others) to identify ethnicity (French, German, Italian). Once the model is established, it is used in a population where the ratios of French:German:Italians are 3:2:1 (normalized to 0.5:0.3333:0.1667).

Input

Predictor is what we use to predict outcome. The Basic Bayes model has only 1 predictor. When there are more than 1 variables used to predict, they must be combined into a single compound predictor. In our example, hair and eye colors are comined to form a compound predictor HairEyeColor

Attributes (a) are mutually exclusive alternatives in the predictor. In a compound predictor, the number of attributes is the multiple of attribute numbers from each original predictor. In our example, 2 hair and 3 eye color are combined into 2x3=6 attributes of DarkBlue, DarkBrown, DarkOthers, LightBlue, LightBrown and LightOthers.

Input Data are placed in the first text area labeled "Data" in Program 1, and can be in one or two formats

• For Program 1, for building and testing a model, both the attributes and the outcome names are required. These are in two columns separated by spaces of tabs. The first the attribute, the second the outcome. Each row consists of the combination of attributes and outcomes in a case.
• For all other programs in the program panel, data is optional, and when present is interpretted by the model. Only the first column of attributes is read and interpreted.

a priori Probability (π) is a believe in the probability of belonging to an outcome prior to executing the Baysian model. In data entry, any set of number representing relative proportions (sample size, ratios, percents) can be used, and the program will normalize them by dividing each by the total. The default example assigned the ratio of French:German:Italian to 3:2:1, normalized to 0.5:0.3333:0.1667.

Progam Output

Probabilities

During model development, the coefficient produced from the reference data is the Probability of attribute (a) given outcome (o), P(a|o). When using P(a|o) to predict, however, the following probabilities can be produced

• If a priori probabilities are not set (or same for all outcomes), the oucome probabilities are P(outcome|attribute) or P(o|a), also known as Maximum Likelihood. This describe the model, and the relationships between the attributes and outcomes.
• If s priori probabilities are set, the outcome probabilities are πP(o|a). This is the most commonly uese setting and output, and is referred to as Bayesian Probability or a posteriori Probability.

Missing Data: The program assumes all data used to develop the model to be valid, and included them as attributes or outcomes. When the developed coefficients are used to interpret attributes, any attribute string that does not match the list of reference attributes in the model is considered missing data, and the a priori probabilies are returned unchanged as results.

Javascript Program produces a function that will match a text string against the reference attributes, and returns the appropriate array of probabilities. The P(a|o) table must be available in the text area in Program 3, and the program will also read the a priori array in their textboxes near Program 1. The script produced can be copied and pasted into a html page and useed to produce probabilities. BasicBayesTemplate.html is a template html page demonstrating how the function produced by this page using default example data.

Programs

The programs on this page consists of a single program with 3 points of entry (Program 1-3), and a supplementary program (program 4) to create a Javascript interpreter. The Example buttons triggers the loading of the default example data for each entry, and runs the program using that data. Users can enter his/her own data and runs the program with the program buttons

Program 1 produces prediction model using raw data, and interprets the same data

1. Reads the 2 column data from the data text area
2. Counts all the attribute/outcome combinations to produce the count table
3. Presents the count table in the results and in the text area of Program 2
4. Passes the results to the algorithms of Program 2 to complete the program.

Program 2 produces the coefficients P(a|o) using a table of counts. On clicking the Example or Analyse Reference Table button, or if Program 1 is already running, the program will

1. Reads the count table from the text area of Program 2 or usees the table as passed from Program 1
2. Calculates the coefficients P(a|o), and deposits the coefficient table to the text area in Program 3
3. Passes the results to the algorithm of Program 3

Program 3 produces a posteriori probabilities using the table of coefficients P(a|o). On clicking the Example or Calculate Probabilities button, or if Program 1 or 2 is already running, the program will

1. Reads the count P(a|o) from the text area of Program 3 or the table passed from Program 1 or 2
2. Calculates and presents the a posterior probability table. This will be
   • the Maximum Likelihood P(o|a), if apriori probabilities are not set
   • the Bayesian Probability πP(o|a), if a priori probabilities are set
3. If there is no data in the Data text area of Program 1, then the program terminates. If data is available, the a posteriori probabilities for each row of attribute are calculated and presented as a table. The outcome with the highest probability is marked in bold, to indicate the preferred decision If there is more than one outcome with equal highest value, then only the first (alphabetically) is marked.

Program 4 is supplementary, and can be executed when the table of P(a|o) is available in the text area in Program 3. On clicking the Create Javascript Code for Interpretation button, the program will

1. Reads the table of P(o|a) from the text area of Program 3, the a priori probabilites from their text box of Program 1
2. Creates and presents the Javascript codes as a function, which accepts a text string as an attribute, and returns an array of probabilities

An example and template page can be examined to see how the Javascript function can be incorporated into a html file and used in other applications