S-Expression Syntax¶

symlik represents mathematical expressions as s-expressions, a notation from Lisp. This page teaches you the syntax so you can write custom models.

Basic Structure¶

An s-expression is either:

A number: 3, 2.5, -1
A symbol: 'x', 'lambda', 'mu'
A list: [operator, arg1, arg2, ...]

For example, \(x + 1\) becomes:

['+', 'x', 1]

And \(2x^2 + 3x\) becomes:

['+', ['*', 2, ['^', 'x', 2]], ['*', 3, 'x']]

Arithmetic Operators¶

Expression	S-expression	Notes
\(x + y\)	`['+', 'x', 'y']`	Addition
\(x - y\)	`['-', 'x', 'y']`	Subtraction
\(-x\)	`['-', 'x']`	Negation
\(x \cdot y\)	`['*', 'x', 'y']`	Multiplication
\(x / y\)	`['/', 'x', 'y']`	Division
\(x^n\)	`['^', 'x', 'n']`	Power

Mathematical Functions¶

Function	S-expression
\(\ln(x)\)	`['log', 'x']`
\(e^x\)	`['exp', 'x']`
\(\sqrt{x}\)	`['sqrt', 'x']`
\(\sin(x)\)	`['sin', 'x']`
\(\cos(x)\)	`['cos', 'x']`
\(\log \Gamma(x)\)	`['lgamma', 'x']`

Working with Data¶

Statistical models need to reference data. symlik provides special operators:

Operator	Meaning	Example
`['len', 'x']`	Length of x	Sample size
`['@', 'x', 'i']`	x[i] (1-based)	i-th observation
`['total', 'x']`	Sum of x	\(\sum x_i\)

Note: Indexing is 1-based, so ['@', 'x', 1] is the first element.

Summation¶

The sum operator iterates over indices:

['sum', 'i', n, body]

This computes \(\sum_{i=1}^{n} \text{body}\) where i takes values 1 through n.

Example: Sum of squared deviations from mean \(\mu\):

['sum', 'i', ['len', 'x'],
 ['^', ['-', ['@', 'x', 'i'], 'mu'], 2]]

This represents \(\sum_{i=1}^{n} (x_i - \mu)^2\).

Building a Log-Likelihood¶

Let's build the exponential log-likelihood step by step.

The likelihood for one observation is \(L_i = \lambda e^{-\lambda x_i}\), so the log-likelihood contribution is:

\[ \ell_i = \log(\lambda) - \lambda x_i \]

In s-expression form:

['+', ['log', 'lambda'], ['*', -1, ['*', 'lambda', ['@', 'x', 'i']]]]

Summing over all observations:

log_lik = ['sum', 'i', ['len', 'x'],
           ['+', ['log', 'lambda'],
            ['*', -1, ['*', 'lambda', ['@', 'x', 'i']]]]]

Nesting and Readability¶

Complex expressions can get hard to read. Use Python formatting:

# Normal log-likelihood (more complex)
n = ['len', 'x']
sum_sq = ['sum', 'i', n,
          ['^', ['-', ['@', 'x', 'i'], 'mu'], 2]]

log_lik = ['+',
           ['*', -0.5, ['*', n, ['log', ['*', 2, 3.14159]]]],
           ['+',
            ['*', -0.5, ['*', n, ['log', 'sigma2']]],
            ['*', -0.5, ['/', sum_sq, 'sigma2']]]]

Evaluating Expressions¶

Test your expressions with evaluate:

from symlik import evaluate

expr = ['+', ['*', 2, 'x'], ['^', 'y', 2]]
result = evaluate(expr, {'x': 3, 'y': 4})
print(result)  # 2*3 + 4^2 = 22

Common Patterns¶

Dot product \(\sum x_i y_i\):

['sum', 'i', ['len', 'x'],
 ['*', ['@', 'x', 'i'], ['@', 'y', 'i']]]

Sample mean \(\bar{x} = \sum x_i / n\):

['/', ['total', 'x'], ['len', 'x']]

Log of product \(\log \prod f_i = \sum \log f_i\):

['sum', 'i', n, ['log', body]]

Next Steps¶

Now that you know the syntax, try building custom models.