From 0aa4118a3aa1abd789c449b8f92b0865d8a70ffa Mon Sep 17 00:00:00 2001 From: Riku-Laine <28960190+Riku-Laine@users.noreply.github.com> Date: Mon, 8 Jul 2019 09:56:47 +0300 Subject: [PATCH] Fig updates, hierarchihcal model to section 7.6 --- analysis_and_scripts/notes.tex | 102 ++++++++++++------ figures/sl_diagnostic_bad_decider_with_Z.png | Bin 0 -> 187340 bytes .../sl_diagnostic_bernoulli_batch_with_Z.png | Bin 197405 -> 218599 bytes ...iagnostic_bernoulli_independent_with_Z.png | Bin 187197 -> 213713 bytes ...nostic_bernoulli_independent_without_Z.png | Bin 192085 -> 218838 bytes .../sl_diagnostic_biased_decider_with_Z.png | Bin 0 -> 187049 bytes .../sl_diagnostic_random_decider_with_Z.png | Bin 0 -> 187934 bytes .../sl_diagnostic_threshold_batch_with_Z.png | Bin 186242 -> 214792 bytes ...iagnostic_threshold_independent_with_Z.png | Bin 180778 -> 204956 bytes figures/sl_posterior_betax.png | Bin 0 -> 10016 bytes figures/sl_posterior_betaz.png | Bin 0 -> 8945 bytes 11 files changed, 70 insertions(+), 32 deletions(-) create mode 100644 figures/sl_diagnostic_bad_decider_with_Z.png create mode 100644 figures/sl_diagnostic_biased_decider_with_Z.png create mode 100644 figures/sl_diagnostic_random_decider_with_Z.png create mode 100644 figures/sl_posterior_betax.png create mode 100644 figures/sl_posterior_betaz.png diff --git a/analysis_and_scripts/notes.tex b/analysis_and_scripts/notes.tex index 3244a6e..c52a797 100644 --- a/analysis_and_scripts/notes.tex +++ b/analysis_and_scripts/notes.tex @@ -35,6 +35,7 @@ \newcommand{\M}{\mathcal{M}} % "fancy M" \newcommand{\B}{\mathcal{B}} % "fancy B" \newcommand{\RR}{\mathcal{R}} % supistusalgon R +\newcommand{\invlogit}{\text{logit}^{-1}} % supistusalgon R \renewcommand{\descriptionlabel}[1]{\hspace{\labelsep}\textnormal{#1}} @@ -235,7 +236,7 @@ Given the above framework, the goal is to create an evaluation algorithm that ca \end{figure} -\section{Modular framework -- based on 19 June discussion} \label{sec:modular_framework} +\section{Modular framework -- 19 June discussion} \label{sec:modular_framework} \emph{Below is the framework as was written on the whiteboard, then RL presents his own remarks on how he understood this.} @@ -313,9 +314,9 @@ Both of the data generating algorithms are presented in this chapter. In the setting without unobservables Z, we first sample an acceptance rate $r$ for all $M=100$ judges uniformly from a half-open interval $[0.1; 0.9)$. Then we assign 500 unique subjects for each of the judges randomly (50000 in total) and simulate their features X as i.i.d standard Gaussian random variables with zero mean and unit (1) variance. Then, probability for negative outcome is calculated as \begin{equation} \label{eq:inv_logit} - P(Y=0|X=x) = \dfrac{1}{1+\exp(-x)}=logit^{-1}(x). + P(Y=0|X=x) = \dfrac{1}{1+\exp(-x)}=\invlogit(x). \end{equation} -Because $P(Y=1|X=x) = 1-P(Y=0|X=x) = 1-logit^{-1}(x)$ the outcome variable Y can be sampled from Bernoulli distribution with parameter $1-logit^{-1}(x)$. The data is then sorted for each judge by the probabilities $P(Y=0|X=x)$ in descending order. If the subject is in the top $(1-r) \cdot 100 \%$ of observations assigned to a judge, the decision variable T is set to zero and otherwise to one. +Because $P(Y=1|X=x) = 1-P(Y=0|X=x) = 1-\invlogit(x)$ the outcome variable Y can be sampled from Bernoulli distribution with parameter $1-\invlogit(x)$. The data is then sorted for each judge by the probabilities $P(Y=0|X=x)$ in descending order. If the subject is in the top $(1-r) \cdot 100 \%$ of observations assigned to a judge, the decision variable T is set to zero and otherwise to one. \begin{algorithm}[] % enter the algorithm environment \caption{Create data without unobservables} % give the algorithm a caption @@ -325,8 +326,8 @@ Because $P(Y=1|X=x) = 1-P(Y=0|X=x) = 1-logit^{-1}(x)$ the outcome variable Y can \ENSURE \STATE Sample acceptance rates for each M judges from $U(0.1; 0.9)$ and round to tenth decimal place. \STATE Sample features X for each $N_{total}$ observations from standard Gaussian. -\STATE Calculate $P(Y=0|X=x)=logit^{-1}(x)$ for each observation -\STATE Sample Y from Bernoulli distribution with parameter $1-logit^{-1}(x)$. +\STATE Calculate $P(Y=0|X=x)=\invlogit(x)$ for each observation +\STATE Sample Y from Bernoulli distribution with parameter $1-\invlogit(x)$. \STATE Sort the data by (1) the judges and (2) by probabilities $P(Y=0|X=x)$ in descending order. \STATE \hskip3.0em $\rhd$ Now the most dangerous subjects for each of the judges are at the top. \STATE If subject belongs to the top $(1-r) \cdot 100 \%$ of observations assigned to a judge, set $T=0$ else set $T=1$. @@ -340,11 +341,11 @@ Because $P(Y=1|X=x) = 1-P(Y=0|X=x) = 1-logit^{-1}(x)$ the outcome variable Y can In the setting with unobservables Z, we first sample an acceptance rate r for all $M=100$ judges uniformly from a half-open interval $[0.1; 0.9)$. Then we assign 500 unique subjects (50000 in total) for each of the judges randomly and simulate their features X, Z and W as i.i.d standard Gaussian random variables with zero mean and unit (1) variance. Then, probability for negative outcome is calculated as \begin{equation} - P(Y=0|X=x, Z=z, W=w)=logit^{-1}(\beta_Xx+\beta_Zz+\beta_Ww)~, + P(Y=0|X=x, Z=z, W=w)=\invlogit(\beta_Xx+\beta_Zz+\beta_Ww)~, \end{equation} where $\beta_X=\beta_Z =1$ and $\beta_W=0.2$. Next, value for result Y is set to 0 if $P(Y = 0| X, Z, W) \geq 0.5$ and 1 otherwise. The conditional probability for the negative decision (T=0) is defined as \begin{equation} - P(T=0|X=x, Z=z)=logit^{-1}(\beta_Xx+\beta_Zz)+\epsilon~, + P(T=0|X=x, Z=z)=\invlogit(\beta_Xx+\beta_Zz)+\epsilon~, \end{equation} where $\epsilon \sim N(0, 0.1)$. Next, the data is sorted for each judge by the probabilities $P(T=0|X, Z)$ in descending order. If the subject is in the top $(1-r) \cdot 100 \%$ of observations assigned to a judge, the decision variable T is set to zero and otherwise to one. @@ -640,7 +641,7 @@ Given our framework defined in section \ref{sec:framework}, the results presente \subsection{Modular framework -- Monte Carlo evaluator} \label{sec:modules_mc} -For these results, data was generated either with module in algorithm \ref{alg:dg:coinflip_with_z} (drawing Y from Bernoulli distribution with parameter $\pr(Y=0|X, Z, W)$ as previously) or with module in algorithm \ref{alg:dg:threshold_with_Z} (assign Y based on the value of $logit^{-1}(\beta_XX+\beta_ZZ)$). Decisions were determined using one of the two modules: module in algorithm \ref{alg:decider:quantile} (decision based on quantiles) or \ref{alg:decider:lakkaraju} ("human" decision-maker as in \cite{lakkaraju17}). Curves were computed with True evaluation (algorithm \ref{alg:eval:true_eval}), Labeled outcomes (\ref{alg:eval:labeled_outcomes}), Human evaluation (\ref{alg:eval:human_eval}), Contraction (\ref{alg:eval:contraction}) and Monte Carlo evaluators (\ref{alg:eval:mc}). Results are presented in figure \ref{fig:modules_mc}. The corresponding MAEs are presented in table \ref{tab:modules_mc}. +For these results, data was generated either with module in algorithm \ref{alg:dg:coinflip_with_z} (drawing Y from Bernoulli distribution with parameter $\pr(Y=0|X, Z, W)$ as previously) or with module in algorithm \ref{alg:dg:threshold_with_Z} (assign Y based on the value of $\invlogit(\beta_XX+\beta_ZZ)$). Decisions were determined using one of the two modules: module in algorithm \ref{alg:decider:quantile} (decision based on quantiles) or \ref{alg:decider:lakkaraju} ("human" decision-maker as in \cite{lakkaraju17}). Curves were computed with True evaluation (algorithm \ref{alg:eval:true_eval}), Labeled outcomes (\ref{alg:eval:labeled_outcomes}), Human evaluation (\ref{alg:eval:human_eval}), Contraction (\ref{alg:eval:contraction}) and Monte Carlo evaluators (\ref{alg:eval:mc}). Results are presented in figure \ref{fig:modules_mc}. The corresponding MAEs are presented in table \ref{tab:modules_mc}. From the result table we can see that the MAE is at the lowest when the data generating process corresponds closely to the Monte Carlo algorithm. @@ -704,10 +705,10 @@ We have three different kinds of data generating modules (DG modules). The DG mo \label{tab:dg_modules} \begin{tabular}{@{}llcc@{}} \toprule - & & \multicolumn{2}{c}{Feature generation} \\ \midrule + & & \multicolumn{2}{c}{Feature generation} \\ \cmidrule(l){3-4} & & \multicolumn{1}{l}{With unobservables} & \multicolumn{1}{l}{Without unobservables} \\ \multicolumn{1}{c}{\multirow{2}{*}{Outcome}} & Drawn from Bernoulli & \ref{alg:dg:coinflip_with_z} & \ref{alg:dg:coinflip_without_z} \\ -\multicolumn{1}{c}{} & Assigned by threshold & \ref{alg:dg:threshold_with_Z} & \\ \cmidrule(l){2-4} +\multicolumn{1}{c}{} & Assigned by threshold & \ref{alg:dg:threshold_with_Z} & \\ \cmidrule(l){2-4} \bottomrule \end{tabular} \end{table} @@ -719,7 +720,7 @@ We have three different kinds of data generating modules (DG modules). The DG mo \ENSURE \FORALL{observations} \STATE Draw $x$ from from a standard Gaussian. - \STATE Draw $y$ from Bernoulli$(1-logit^{-1}(x))$. + \STATE Draw $y$ from Bernoulli$(1-\invlogit(x))$. \ENDFOR \RETURN data \end{algorithmic} @@ -733,7 +734,7 @@ We have three different kinds of data generating modules (DG modules). The DG mo \ENSURE \FORALL{observations} \STATE Draw $x, z$ and $w$ from from standard Gaussians independently. - \IF{$logit^{-1}(\beta_Xx+\beta_Zz+\beta_Ww) \geq 0.5$} + \IF{$\invlogit(\beta_Xx+\beta_Zz+\beta_Ww) \geq 0.5$} \STATE {Set $y$ to 0.} \ELSE \STATE {Set $y$ to 1.} @@ -751,7 +752,7 @@ We have three different kinds of data generating modules (DG modules). The DG mo \ENSURE \FORALL{observations} \STATE Draw $x, z$ and $w$ from from standard Gaussians independently. - \STATE Draw $y$ from Bernoulli$(1-logit^{-1}(\beta_Xx+\beta_Zz+\beta_Ww))$. + \STATE Draw $y$ from Bernoulli$(1-\invlogit(\beta_Xx+\beta_Zz+\beta_Ww))$. \ENDFOR \RETURN data \end{algorithmic} @@ -773,7 +774,7 @@ Below is presented the decision-maker by Lakkaraju \cite{lakkaraju17}. The decis \ENSURE \STATE Sample acceptance rates for each M judges from Uniform$(0.1; 0.9)$ and round to tenth decimal place. \STATE Assign each observation to a judge at random. -\STATE Calculate $\pr(T=0|X, Z) = logit^{-1}(\beta_XX+\beta_ZZ) + \epsilon$ for each observation and attach to data. +\STATE Calculate $\pr(T=0|X, Z) = \invlogit(\beta_XX+\beta_ZZ) + \epsilon$ for each observation and attach to data. \STATE Sort the data by (1) the judges and (2) by the probabilities in descending order. \STATE If subject belongs to the top $(1-r) \cdot 100 \%$ of observations assigned to that judge, set $T=0$ else set $T=1$. \STATE Set $Y=$ NA if decision is negative ($T=0$). @@ -781,7 +782,7 @@ Below is presented the decision-maker by Lakkaraju \cite{lakkaraju17}. The decis \end{algorithmic} \end{algorithm} -One discussed way of making the decisions independent was to "flip a coin at some probability". An implementation of that idea is presented below in algorithm \ref{alg:decider:coinflip}. As $\pr(T=0|X, Z) = logit^{-1}(\beta_XX+\beta_ZZ)$ the parameter for the Bernoulli distribution is set to $1-logit^{-1}(\beta_XX+\beta_ZZ)$. In the practical implementation, as some algorithms need to know the leniency of the decision, acceptance rate is then calculated then from the decisions. +One discussed way of making the decisions independent was to "flip a coin at some probability". An implementation of that idea is presented below in algorithm \ref{alg:decider:coinflip}. As $\pr(T=0|X, Z) = \invlogit(\beta_XX+\beta_ZZ)$ the parameter for the Bernoulli distribution is set to $1-\invlogit(\beta_XX+\beta_ZZ)$. In the practical implementation, as some algorithms need to know the leniency of the decision, acceptance rate is then calculated then from the decisions. \begin{algorithm}[H] % enter the algorithm environment \caption{Decider module: decisions from Bernoulli} % give the algorithm a caption @@ -789,16 +790,16 @@ One discussed way of making the decisions independent was to "flip a coin at som \begin{algorithmic}[1] % enter the algorithmic environment \REQUIRE Data with features $X, Z$, knowledge that both of them affect the outcome Y and that they are independent / Parameters: $\beta_X=1, \beta_Z=1$. \ENSURE -\STATE Draw $t$ from Bernoulli$(1-logit^{-1}(\beta_Xx+\beta_Zz))$ for all observations. +\STATE Draw $t$ from Bernoulli$(1-\invlogit(\beta_Xx+\beta_Zz))$ for all observations. \STATE Compute the acceptance rate. -\STATE Set $Y=$ NA if decision is negative ($T=0$). \emph{Optional.} +\STATE Set $Y=$ NA if decision is negative ($T=0$). \RETURN data with decisions. \end{algorithmic} \end{algorithm} -A quantile-based decider module is presented in algorithm \ref{alg:decider:quantile}. The algorithm tries to emulate Lakkaraju's decision-maker while giving out independent decisions. The independence is achieved by comparing the values of $logit^{-1}(\beta_Xx+\beta_Zz)$ to the corresponding value of the inverse cumulative distribution function $F^{-1}_{logit^{-1}(\beta_XX+\beta_ZZ)}$ or $F^{-1}$ in short. The derivation of $F^{-1}$ is deferred to the next section. The decisions have a guarantee that the fraction of positive decisions will converge to $r$ based on the law of large numbers. +A quantile-based decider module is presented in algorithm \ref{alg:decider:quantile}. The algorithm tries to emulate Lakkaraju's decision-maker while giving out independent decisions. The independence is achieved by comparing the values of $\invlogit(\beta_Xx+\beta_Zz)$ to the corresponding value of the inverse cumulative distribution function $F^{-1}_{\invlogit(\beta_XX+\beta_ZZ)}$ or $F^{-1}$ in short. The derivation of $F^{-1}$ is deferred to the next section. %The decisions have a guarantee that the fraction of positive decisions will converge to $r$ based on the law of large numbers. -\textbf{Example} Consider a decision-maker with leniency 0.60 who gets a new subject $\{x, z\}$ with a predicted probability $logit^{-1}(\beta_Xx+\beta_Zz)\approx 0.7$ for a negative outcome with some coefficients $\beta$. Now, as the judge has leniency 0.6 their cut-point $F^{-1}(0.60)\approx0.65$. That is, the judge will not give a positive decision to anyone with failure probability greater than 0.65 so our example subject will receive a negative decision. +\textbf{Example} Consider a decision-maker with leniency 0.60 who gets a new subject $\{x, z\}$ with a predicted probability $\invlogit(\beta_Xx+\beta_Zz)\approx 0.7$ for a negative outcome with some coefficients $\beta$. Now, as the judge has leniency 0.6, their cut-point $F^{-1}(0.60)\approx0.65$. That is, the judge will not give a positive decision to anyone with failure probability greater than 0.65, so our example subject will receive a negative decision. \begin{algorithm}[H] % enter the algorithm environment \caption{Decider module: "quantile decisions"} % give the algorithm a caption @@ -808,9 +809,9 @@ A quantile-based decider module is presented in algorithm \ref{alg:decider:quant \ENSURE \STATE Sample acceptance rates for each M judges from Uniform$(0.1; 0.9)$ and round to tenth decimal place. \STATE Assign each observation to a judge at random. -\STATE Calculate $\pr(T=0|X, Z) = logit^{-1}(\beta_XX+\beta_ZZ)$ for all observations. -\STATE If $logit^{-1}(\beta_Xx+\beta_Zz) \geq F^{-1}(r)$ set $t=0$, otherwise set $t=1$. -\STATE Set $Y=$ NA if decision is negative ($T=0$). \emph{Optional.} +\STATE Calculate $\pr(T=0|X, Z) = \invlogit(\beta_XX+\beta_ZZ)$ for all observations. +\STATE If $\invlogit(\beta_Xx+\beta_Zz) \geq F^{-1}(r)$ set $t=0$, otherwise set $t=1$. +\STATE Set $Y=$ NA if decision is negative ($T=0$). \RETURN data with decisions. \end{algorithmic} \end{algorithm} @@ -846,7 +847,7 @@ Black box predictive model. Another input to our framework is a black box predic True evaluation module computes the "true failure rate" of a predictive model \emph{had it been deployed to make independent decisions}. For computing the true failure rate "had the model been deployed" we need all outcome labels which is why the true failure rate can only be computed on synthetic data. -In practice, the module first trains a model $\B$ and assigns each observation with a probability score $\s$ using it as described above. Then the observations are sorted in ascending order by the scores so that most risky subjects are last (subjects with the highest predicted probability for a negative outcome). Taking the first $r \cdot 100\%$ of observations, the true failure rate can be computed straight from the ground truth. +In practice, the module first trains a model $\B$ and assigns each observation with a probability score $\s$ using it as described above. Then the observations are sorted in ascending order by the scores so that most risky subjects are last (subjects with the highest predicted probability for a negative outcome). Taking the first $r \cdot 100\%$ of observations, the true failure rate can be computed from the ground truth directly. \begin{algorithm}[H] % enter the algorithm environment \caption{Evaluator module: True evaluation} % give the algorithm a caption @@ -899,8 +900,7 @@ This [failure rate estimation for human decision-makers] can be done by grouping \REQUIRE Data $\D$ with properties $\{j_i, t_i, y_i\}$, acceptance rate r \ENSURE \STATE Assign judges with similar acceptance rate to $\mathcal{J}$ -\STATE $\D_{released} = \{(j, t, y) \in \D~|~t=1 \wedge j \in \mathcal{J}\}$ -\STATE \hskip3.0em $\rhd$ Subjects judged \emph{and} released by judges with correct leniency. +\STATE Assign subjects judged and released by judges in $\mathcal{J}$ to $\D_{released}$. \RETURN $\frac{1}{|\mathcal{J}|}\sum_{i=1}^{\D_{released}}\delta\{y_i=0\}$ \end{algorithmic} \end{algorithm} @@ -946,21 +946,21 @@ in probabilities or just deterministically \end{equation} In equations \ref{eq:Tprob} and \ref{eq:Tdet}, $\pr(Y=0|x, z, DG)$ is the predicted probability of a negative outcome given x and z. The probability $\pr(Y=0|x, z, DG)$ is predicted by the judge and here we used an approximation that \begin{equation} -\pr(Y=0|x, z, DG) = logit^{-1}(\beta_Xx+\beta_Zz) +\pr(Y=0|x, z, DG) = \invlogit(\beta_Xx+\beta_Zz) \end{equation} which is an increasing function of $z$ when $x$ is given. Now we do not know the $\beta$ coefficients so here we used the information that they are one. (In the future, they should be inferred.) The inverse cumulative function $F^{-1}(r)$ in equations \ref{eq:Tprob} and \ref{eq:Tdet} is the inverse cumulative distribution of \emph{logit-normal distribution} with parameters $\mu=0$ and $\sigma^2=2$, i.e. $F^{-1}$ is the inverse cumulative distribution function of the sum of two standard Gaussians after logistic transformation. If $\beta_X \neq 1$ and/or $\beta_Z \neq 1$ then from the basic properties of variance $\sigma^2=Var(\beta_XX+\beta_ZZ)=\beta_X^2Var(X)+\beta_Z^2Var(Z)$. Finally the inverse cumulative function -\begin{equation} -F^{-1}(r) = logit^{-1}\left(\text{erf}^{-1}(2r-1)\sqrt{2\sigma^2}-\mu\right) +\begin{equation} \label{eq:cum_inv} +F^{-1}(r) = \invlogit\left(\text{erf}^{-1}(2r-1)\sqrt{2\sigma^2}-\mu\right) \end{equation} where the parameters are as discussed and erf is the error function. -With this knowledge, it can be stated that if we observed $T=0$ with some $x$ and $r$ it must have been that $logit^{-1}(\beta_Xx+\beta_Zz) \geq F^{-1}(r)$. Using basic algebra we obtain that +With this knowledge, it can be stated that if we observed $T=0$ with some $x$ and $r$ it must have been that $\invlogit(\beta_Xx+\beta_Zz) \geq F^{-1}(r)$. Using basic algebra we obtain that \begin{equation} \label{eq:bounds} -logit^{-1}(x + z) \geq F^{-1}(r) \Leftrightarrow x+z \geq logit(F^{-1}(r)) \Leftrightarrow z \geq logit(F^{-1}(r)) - x +\invlogit(x + z) \geq F^{-1}(r) \Leftrightarrow x+z \geq logit(F^{-1}(r)) \Leftrightarrow z \geq logit(F^{-1}(r)) - x \end{equation} -as the logit and its inverse are strictly increasing functions and hence preserve the order of magnitude for all pairs of values in their domains. From equations \ref{eq:posterior_Z}, \ref{eq:Tprob} and \ref{eq:bounds} we can conclude that $\pr(Z < logit^{-1}(F^{-1}(r)) - x | T=0, X=x, R=r) = 0$ and that elsewhere the distribution of Z follows a truncated Gaussian with a lower bound of $logit(F^{-1}(r)) - x$. The expectation of Z can be computed analytically. All this follows analogically for cases with $T=1$ with the changes of some inequalities. +as the logit and its inverse are strictly increasing functions and hence preserve the order of magnitude for all pairs of values in their domains. From equations \ref{eq:posterior_Z}, \ref{eq:Tprob} and \ref{eq:bounds} we can conclude that $\pr(Z < \invlogit(F^{-1}(r)) - x | T=0, X=x, R=r) = 0$ and that elsewhere the distribution of Z follows a truncated Gaussian with a lower bound of $logit(F^{-1}(r)) - x$. The expectation of Z can be computed analytically. All this follows analogically for cases with $T=1$ with the changes of some inequalities. In practise, in lines 1--3 and 10--13 of algorithm \ref{alg:eval:mc} we do as in the True evaluation evaluator algorithm with the distinction that some of the values of Y are imputed with the corresponding counterfactual probabilities. In line 4 we compute the bounds as motivated above. In the for-loop (lines 5--8) we merely compute the expectation of Z given the knowledge of the decision and that the distribution of Z follows a truncated Gaussian. The equation \begin{equation} @@ -980,7 +980,7 @@ computes the correct expectation automatically. Using the expectation, we then c \STATE Compute bounds $Q_r = logit(F^{-1}(r)) - x$ for all judges. \FORALL{observations in test set} \STATE Compute expectation $\hat{z} = (1-t) \cdot E(Z | Z > Q_r) + t \cdot E(Z | Z < Q_r)$. % - \STATE Compute $\pr(Y(1) = 0) = logit^{-1}(x + \hat{z})$. + \STATE Compute $\pr(Y(1) = 0) = \invlogit(x + \hat{z})$. \ENDFOR \STATE Impute missing observations using the estimates $\pr(Y(1) = 0)$. \STATE Sort the data by the probabilities $\s$ to ascending order. @@ -990,6 +990,44 @@ computes the correct expectation automatically. Using the expectation, we then c \end{algorithmic} \end{algorithm} +%Comments this approach: +%\begin{itemize} +%\item Propensity ($\pr(T=1| X, Z)$) is taken as given and in correct form. In reality it is not known (?) +%\item The equation for the inverse cdf \ref{eq:cum_inv} assumes the joint pdf of $\invlogit(\beta_XX+\beta_ZZ)$ known when in real data X might be multidimensional and non-normal etc. +%\item +%\end{itemize} + +In the future, we should utilize a fully Bayesian approach to be able to include priors for the different $\beta$ coefficients into the model. + +The following hierarchical model was used as an initial approach to the problem. Data was generated with unobservables and both outcome Y and decision T were drawn from Bernoulli distributions. The $\beta$ coefficients were systematically overestimated as shown in figure \ref{fig:posteriors}. + +\begin{align} \label{eq1} + 1-t~|~x,~z,~\beta_x,~\beta_z & \sim \text{Bernoulli}(\invlogit(\beta_xx + \beta_zz)) \\ \nonumber + Z &\sim N(0, 1) \\ \nonumber +% \alpha_j & \sim N(0, 100), j \in \{1, \ldots, N_{judges} \} \\ + \beta_x & \sim N(0, 10^2) \\ \nonumber + \beta_z & \sim N_+(0, 10^2) +\end{align} + + +\begin{figure}[] + \centering + \begin{subfigure}[b]{0.475\textwidth} + \includegraphics[width=\textwidth]{sl_posterior_betax} + \caption{Posterior of $\beta_x$.} + %\label{fig:random_predictions_without_Z} + \end{subfigure} + \quad %add desired spacing between images, e. g. ~, \quad, \qquad, \hfill etc. + %(or a blank line to force the subfigure onto a new line) + \begin{subfigure}[b]{0.475\textwidth} + \includegraphics[width=\textwidth]{sl_posterior_betaz} + \caption{Posterior of $\beta_z$.} + \label{fig:posteriors} + \end{subfigure} + \caption{Coefficient posteriors from model \ref{eq1}.} + %\label{fig:random_predictions} +\end{figure} + \newpage \subsection{Summary table} diff --git a/figures/sl_diagnostic_bad_decider_with_Z.png b/figures/sl_diagnostic_bad_decider_with_Z.png new file mode 100644 index 0000000000000000000000000000000000000000..f3bc10405dcc7380822f32435c56f262b182c921 GIT binary patch literal 187340 zcmeEuX*ibs+V7=^B0~~N#z+*RQZi?VM5a)NC`9IYCaFXyL(+g$C_;+JRE7$bkYt`I zGnt9~yPmby`|kbWJ@(iAVLMjGv+BOx_ci?g=lPq?>&7Wf)y<o>Zz2d{v)XYb9fF_? z!k@!58}KjA`_mNhA4*q6H9Z>q_|aSl#osqN9XE0%2-+>=ABu*KfEE1XVK-$%H(f_7 zH&1gHOTymV&B@Nu&CbSx+r!eu)yC05LS(;)*k107Zf;IeqN4x(x`?BTwP@bhbykAl zCe)M^^t>L7c6b}>HLb}_Hs@@<KPq&eJM3&ha&WMq#deGB0wU=_7XF_<i@$OCegCXT z+>Sd|mO`n5DR=Is#&S`xTo#|FI?&F-s<)NtLQ$FA%9p>NEv<f1zMq=)llJ=cz_+e& zD6!CW=<9N#YnBEhC56(zf1c#I*%qz)&)@W3u@Sug{%LTbBbfgEV{>FfQ0TvZUURIM z|Iho+L{f17=l%OW5{&<T`|UgDiT}L)c<p}TKW~3B@c%!a>HnFFqt7fSC->=sIQ|!M z{d)NCcu})$jh5o<TefVew0vb<x2<u9q{B-2h_QB0PmhJn-<1{H7o7K63*IWLsHk_o 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