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diff.jl
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diff.jl
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##########################################################################################
#
# function 'derive' returning the expr of gradient
#
#
##########################################################################################
# TODO : add operators : hcat, vcat, ? : , map, mapreduce, if else
## macro to simplify derivation rules creation
macro dfunc(func::Expr, dv::Symbol, diff::Expr)
argsn = map(e-> isa(e, Symbol) ? e : e.args[1], func.args[2:end])
index = find(dv .== argsn)[1]
# change var names in signature and diff expr to x1, x2, x3, ..
smap = { argsn[i] => symbol("x$i") for i in 1:length(argsn) }
args2 = substSymbols(func.args[2:end], smap)
# diff function name
fn = symbol("d_$(func.args[1])_x$index")
fullf = Expr(:(=), Expr(:call, fn, args2...), Expr(:quote, substSymbols(diff, smap)) )
eval(fullf)
end
## common operators
# TODO : check if ds can be removed for clarity
@dfunc +(x::Real, y) x sum(ds)
@dfunc +(x::Array, y) x +ds
@dfunc +(x, y::Real) y sum(ds)
@dfunc +(x, y::Array) y +ds
@dfunc -x x -ds
@dfunc -(x::Real, y) x sum(ds)
@dfunc -(x::Array, y) x +ds
@dfunc -(x, y::Real) y -sum(ds)
@dfunc -(x, y::Array) y -ds
@dfunc sum(x) x +ds
@dfunc dot(x, y) x y .* ds
@dfunc dot(x, y) y x .* ds
@dfunc log(x) x ds ./ x
@dfunc exp(x) x exp(x) .* ds
@dfunc sin(x) x cos(x) .* ds
@dfunc cos(x) x -sin(x) .* ds
@dfunc abs(x) x sign(x) .* ds
@dfunc *(x::Real, y) x sum(ds .* y)
@dfunc *(x::Array, y) x ds * transpose(y)
@dfunc *(x, y::Real) y sum(ds .* x)
@dfunc *(x, y::Array) y transpose(x) * ds
@dfunc .*(x::Real, y) x sum(ds .* y)
@dfunc .*(x::Array, y) x ds .* y
@dfunc .*(x, y::Real) y sum(ds .* x)
@dfunc .*(x, y::Array) y ds .* x
@dfunc ^(x::Real, y::Real) x y * x ^ (y-1) * ds # Both args reals
@dfunc ^(x::Real, y::Real) y log(x) * x ^ y * ds # Both args reals
@dfunc .^(x::Real, y) x sum(y .* x .^ (y-1) .* ds)
@dfunc .^(x::Array, y) x y .* x .^ (y-1) .* ds
@dfunc .^(x, y::Real) y sum(log(x) .* x .^ y .* ds)
@dfunc .^(x, y::Array) y log(x) .* x .^ y .* ds
@dfunc /(x::Real, y) x sum(ds ./ y)
@dfunc /(x::Array, y::Real) x ds ./ y
@dfunc /(x, y::Real) y sum(- x ./ (y .* y) .* ds)
@dfunc /(x::Real, y::Array) y (- x ./ (y .* y)) .* ds
@dfunc ./(x::Real, y) x sum(ds ./ y)
@dfunc ./(x::Array, y) x ds ./ y
@dfunc ./(x, y::Real) y sum(- x ./ (y .* y) .* ds)
@dfunc ./(x, y::Array) y (- x ./ (y .* y)) .* ds
@dfunc max(x::Real, y) x sum((x .> y) .* ds)
@dfunc max(x::Array, y) x (x .> y) .* ds
@dfunc max(x, y::Real) y sum((x .< y) .* ds)
@dfunc max(x, y::Array) y (x .< y) .* ds
@dfunc min(x::Real, y) x sum((x .< y) .* ds)
@dfunc min(x::Array, y) x (x .< y) .* ds
@dfunc min(x, y::Real) y sum((x .> y) .* ds)
@dfunc min(x, y::Array) y (x .> y) .* ds
@dfunc transpose(x::Real) x +ds
@dfunc transpose(x::Array) x transpose(ds)
## Normal distribution
@dfunc logpdfNormal(mu::Real, sigma, x) mu sum((x - mu) ./ (sigma .* sigma)) * ds
@dfunc logpdfNormal(mu::Array, sigma, x) mu (x - mu) ./ (sigma .* sigma) * ds
@dfunc logpdfNormal(mu, sigma::Real, x) sigma sum(((x - mu).*(x - mu) ./ (sigma.*sigma) - 1.) ./ sigma) * ds
@dfunc logpdfNormal(mu, sigma::Array, x) sigma ((x - mu).*(x - mu) ./ (sigma.*sigma) - 1.) ./ sigma * ds
@dfunc logpdfNormal(mu, sigma, x::Real) x sum((mu - x) ./ (sigma .* sigma)) * ds
@dfunc logpdfNormal(mu, sigma, x::Array) x (mu - x) ./ (sigma .* sigma) * ds
## Uniform distribution
@dfunc logpdfUniform(a::Real, b, x) a sum((a .<= x .<= b) ./ (b - a)) * ds
@dfunc logpdfUniform(a::Array, b, x) a ((a .<= x .<= b) ./ (b - a)) * ds
@dfunc logpdfUniform(a, b::Real, x) b sum((a .<= x .<= b) ./ (a - b)) * ds
@dfunc logpdfUniform(a, b::Array, x) b ((a .<= x .<= b) ./ (a - b)) * ds
@dfunc logpdfUniform(a, b, x) x zero(x)
## Weibull distribution
@dfunc logpdfWeibull(sh::Real, sc, x) sh (r = x./sc ; sum(((1. - r.^sh) .* log(r) + 1./sh)) * ds)
@dfunc logpdfWeibull(sh::Array, sc, x) sh (r = x./sc ; ((1. - r.^sh) .* log(r) + 1./sh) * ds)
@dfunc logpdfWeibull(sh, sc::Real, x) sc sum(((x./sc).^sh - 1.) .* sh./sc) * ds
@dfunc logpdfWeibull(sh, sc::Array, x) sc ((x./sc).^sh - 1.) .* sh./sc * ds
@dfunc logpdfWeibull(sh, sc, x::Real) x sum(((1. - (x./sc).^sh) .* sh - 1.) ./ x) * ds
@dfunc logpdfWeibull(sh, sc, x::Array) x ((1. - (x./sc).^sh) .* sh - 1.) ./ x * ds
## Beta distribution
@dfunc logpdfBeta(a, b, x::Real) x sum((a-1) ./ x - (b-1) ./ (1-x)) * ds
@dfunc logpdfBeta(a, b, x::Array) x ((a-1) ./ x - (b-1) ./ (1-x)) * ds
@dfunc logpdfBeta(a::Real, b, x) a sum(digamma(a+b) - digamma(a) + log(x)) * ds
@dfunc logpdfBeta(a::Array, b, x) a (digamma(a+b) - digamma(a) + log(x)) * ds
@dfunc logpdfBeta(a, b::Real, x) b sum(digamma(a+b) - digamma(b) + log(1-x)) * ds
@dfunc logpdfBeta(a, b::Array, x) b (digamma(a+b) - digamma(b) + log(1-x)) * ds
## TDist distribution
@dfunc logpdfTDist(df, x::Real) x sum(-(df+1).*x ./ (df+x.*x)) .* ds
@dfunc logpdfTDist(df, x::Array) x (-(df+1).*x ./ (df+x.*x)) .* ds
@dfunc logpdfTDist(df::Real, x) df (tmp2 = (x.*x + df) ; sum( (x.*x-1)./tmp2 + log(df./tmp2) + digamma((df+1)/2) - digamma(df/2) ) / 2 .* ds )
@dfunc logpdfTDist(df::Array, x) df (tmp2 = (x.*x + df) ; ( (x.*x-1)./tmp2 + log(df./tmp2) + digamma((df+1)/2) - digamma(df/2) ) / 2 .* ds )
## Exponential distribution
@dfunc logpdfExponential(sc, x::Real) x sum(-1/sc) .* ds
@dfunc logpdfExponential(sc, x::Array) x (- ds ./ sc)
@dfunc logpdfExponential(sc::Real, x) sc sum((x-sc)./(sc.*sc)) .* ds
@dfunc logpdfExponential(sc::Array, x) sc (x-sc) ./ (sc.*sc) .* ds
## Gamma distribution
@dfunc logpdfGamma(sh, sc, x::Real) x sum(-( sc + x - sh.*sc)./(sc.*x)) .* ds
@dfunc logpdfGamma(sh, sc, x::Array) x (-( sc + x - sh.*sc)./(sc.*x)) .* ds
@dfunc logpdfGamma(sh::Real, sc, x) sh sum(log(x) - log(sc) - digamma(sh)) .* ds
@dfunc logpdfGamma(sh::Array, sc, x) sh (log(x) - log(sc) - digamma(sh)) .* ds
@dfunc logpdfGamma(sh, sc::Real, x) sc sum((x - sc.*sh) ./ (sc.*sc)) .* ds
@dfunc logpdfGamma(sh, sc::Array, x) sc ((x - sc.*sh) ./ (sc.*sc)) .* ds
## Cauchy distribution
@dfunc logpdfCauchy(mu, sc, x::Real) x sum(2(mu-x) ./ (sc.*sc + (x-mu).*(x-mu))) .* ds
@dfunc logpdfCauchy(mu, sc, x::Array) x (2(mu-x) ./ (sc.*sc + (x-mu).*(x-mu))) .* ds
@dfunc logpdfCauchy(mu::Real, sc, x) mu sum(2(x-mu) ./ (sc.*sc + (x-mu).*(x-mu))) .* ds
@dfunc logpdfCauchy(mu::Array, sc, x) mu (2(x-mu) ./ (sc.*sc + (x-mu).*(x-mu))) .* ds
@dfunc logpdfCauchy(mu, sc::Real, x) sc sum(((x-mu).*(x-mu) - sc.*sc) ./ (sc.*(sc.*sc + (x-mu).*(x-mu)))) .* ds
@dfunc logpdfCauchy(mu, sc::Array, x) sc (((x-mu).*(x-mu) - sc.*sc) ./ (sc.*(sc.*sc + (x-mu).*(x-mu)))) .* ds
## Log-normal distribution
@dfunc logpdflogNormal(lmu, lsc, x::Real) x ( tmp2=lsc.*lsc ; sum( (lmu - tmp2 - log(x)) ./ (tmp2.*x) ) .* ds )
@dfunc logpdflogNormal(lmu, lsc, x::Array) x ( tmp2=lsc.*lsc ; ( (lmu - tmp2 - log(x)) ./ (tmp2.*x) ) .* ds )
@dfunc logpdflogNormal(lmu::Real, lsc, x) lmu sum((log(x) - lmu) ./ (lsc .* lsc)) .* ds
@dfunc logpdflogNormal(lmu::Array, lsc, x) lmu ((log(x) - lmu) ./ (lsc .* lsc)) .* ds
@dfunc logpdflogNormal(lmu, lsc::Real, x) lsc ( tmp2=lsc.*lsc ; sum( (lmu.*lmu - tmp2 - log(x).*(2lmu-log(x))) ./ (lsc.*tmp2) ) .* ds )
@dfunc logpdflogNormal(lmu, lsc::Array, x) lsc ( tmp2=lsc.*lsc ; ( (lmu.*lmu - tmp2 - log(x).*(2lmu-log(x))) ./ (lsc.*tmp2) ) .* ds )
# TODO : find a way to implement multi variate distribs that goes along well with vectorization (Dirichlet, Categorical)
# TODO : other continuous distribs ? : Pareto, Rayleigh, Logistic, Levy, Laplace, Dirichlet, FDist
# TODO : other discrete distribs ? : NegativeBinomial, DiscreteUniform, HyperGeometric, Geometric, Categorical
## Bernoulli distribution (Note : no derivation on x parameter as it is an integer)
@dfunc logpdfBernoulli(p::Real, x) p sum(1. ./ (p - (1. - x))) * ds
@dfunc logpdfBernoulli(p::Array, x) p (1. ./ (p - (1. - x))) * ds
## Binomial distribution (Note : no derivation on x and n parameters as they are integers)
@dfunc logpdfBinomial(n, p::Real, x) p sum(x ./ p - (n-x) ./ (1 - p)) * ds
@dfunc logpdfBinomial(n, p::Array, x) p (x ./ p - (n-x) ./ (1 - p)) * ds
## Poisson distribution (Note : no derivation on x parameter as it is an integer)
@dfunc logpdfPoisson(lambda::Real, x) lambda sum(x ./ lambda - 1) * ds
@dfunc logpdfPoisson(lambda::Array, x) lambda (x ./ lambda - 1) * ds
# fake distribution to test gradient code
@dfunc logpdfTestDiff(x) x +ds
## returns sample value for the given Symobl or Expr (for refs)
hint(v::Symbol) = vhint[v]
hint(v) = v # should be a value if not a Symbol or an Expression
function hint(v::Expr)
assert(v.head == :ref, "[hint] unexpected variable $v")
v.args[1] = :( vhint[$(Expr(:quote, v.args[1]))] )
eval(v)
end
## Returns gradient expression of opex
function derive(opex::Expr, index::Integer, dsym::Union(Expr,Symbol)) # opex=:(z^x);index=2;dsym=:y
vs = opex.args[1+index]
ds = symbol("$DERIV_PREFIX$dsym")
args = opex.args[2:end]
val = map(hint, args) # get sample values of args to find correct gradient statement
fn = symbol("d_$(opex.args[1])_x$index")
try
dexp = eval(Expr(:call, fn, val...))
smap = { symbol("x$i") => args[i] for i in 1:length(args)}
smap[:ds] = ds
dexp = substSymbols(dexp, smap)
# unfold for easier optimization later
m = MCMCModel()
m.source = :(dummy = $dexp )
unfold!(m)
m.exprs[end] = m.exprs[end].args[2] # remove last assignment
m.exprs[end] = :( $(symbol("$DERIV_PREFIX$vs")) = $(symbol("$DERIV_PREFIX$vs")) + $(m.exprs[end]) )
return m.exprs
catch e
error("[derive] Doesn't know how to derive $opex by argument $vs")
end
end