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gung - Reinstate Monica
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Say I have a logistic model, $$ Y_i \sim Bernoulli(logistic(\beta_0 + \beta_1 X_i)), $$$$ Y_i \sim {\rm Bernoulli}({\rm logistic}(\beta_0 + \beta_1 X_i)), $$ where $logistic(x) = e^x/(1+e^x)$ as usual.

I have

  1. $n$ observations $Y_i \in \{0,1\}$ ($i = 1, \ldots, n$), and also
  2. $m$ independent guestimates $\hat\pi_i$ ($i = n+1,\ldots,m+n$) of the probabilities $\pi_i = logistic(\beta_0 + \beta_1 X_i)$$\pi_i = {\rm logistic}(\beta_0 + \beta_1 X_i)$. Let's be generous and say $\hat\pi_i \sim N(\pi_i, \sigma^2)$, where $\sigma^2$ is small (and known, for the sake of argument).

I want to use all this information to get better estimates $\hat\beta_0$ and $\hat\beta_1$ for the model parameters.

I can think of a couple of approaches but this is surely a standard problem. If yes, what is it called? Any useful references?

Thanks!

Edit: so the sample data might look like:

X = -1: observations Y= 0,0,1

X = -2: observations Y= 0,1, $\hat\pi = 0.3$

X = 3: observations $\hat\pi = 0.5$, $\hat\pi =0.53$

Say I have a logistic model, $$ Y_i \sim Bernoulli(logistic(\beta_0 + \beta_1 X_i)), $$ where $logistic(x) = e^x/(1+e^x)$ as usual.

I have

  1. $n$ observations $Y_i \in \{0,1\}$ ($i = 1, \ldots, n$), and also
  2. $m$ independent guestimates $\hat\pi_i$ ($i = n+1,\ldots,m+n$) of the probabilities $\pi_i = logistic(\beta_0 + \beta_1 X_i)$. Let's be generous and say $\hat\pi_i \sim N(\pi_i, \sigma^2)$, where $\sigma^2$ is small (and known, for the sake of argument).

I want to use all this information to get better estimates $\hat\beta_0$ and $\hat\beta_1$ for the model parameters.

I can think of a couple of approaches but this is surely a standard problem. If yes, what is it called? Any useful references?

Thanks!

Edit: so the sample data might look like:

X = -1: observations Y= 0,0,1

X = -2: observations Y= 0,1, $\hat\pi = 0.3$

X = 3: observations $\hat\pi = 0.5$, $\hat\pi =0.53$

Say I have a logistic model, $$ Y_i \sim {\rm Bernoulli}({\rm logistic}(\beta_0 + \beta_1 X_i)), $$ where $logistic(x) = e^x/(1+e^x)$ as usual.

I have

  1. $n$ observations $Y_i \in \{0,1\}$ ($i = 1, \ldots, n$), and also
  2. $m$ independent guestimates $\hat\pi_i$ ($i = n+1,\ldots,m+n$) of the probabilities $\pi_i = {\rm logistic}(\beta_0 + \beta_1 X_i)$. Let's be generous and say $\hat\pi_i \sim N(\pi_i, \sigma^2)$, where $\sigma^2$ is small (and known, for the sake of argument).

I want to use all this information to get better estimates $\hat\beta_0$ and $\hat\beta_1$ for the model parameters.

I can think of a couple of approaches but this is surely a standard problem. If yes, what is it called? Any useful references?

Thanks!

Edit: so the sample data might look like:

X = -1: observations Y= 0,0,1

X = -2: observations Y= 0,1, $\hat\pi = 0.3$

X = 3: observations $\hat\pi = 0.5$, $\hat\pi =0.53$

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P.Windridge
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Say I have a logistic model, $$ Y_i \sim Bernoulli(logistic(\beta_0 + \beta_1 X_i)), $$ where $logistic(x) = e^x/(1+e^x)$ as usual.

I have

  1. $n$ observations $Y_i \in \{0,1\}$ ($i = 1, \ldots, n$), and also
  2. $m$ independent guestimates $\hat\pi_i$ ($i = n+1,\ldots,m+n$) of the probabilities $\pi_i = logistic(\beta_0 + \beta_1 X_i)$. Let's be generous and say $\hat\pi_i \sim N(\pi_i, \sigma^2)$, where $\sigma^2$ is small (and known, for the sake of argument).

I want to use all this information to get better estimates $\hat\beta_0$ and $\hat\beta_1$ for the model parameters.

I can think of a couple of approaches but this is surely a standard problem. If yes, what is it called? Any useful references?

Thanks!

Edit: so the sample data might look like:

X = -1: observations Y= 0,0,1

X = -2: observations Y= 0,1, $\hat\pi = 0.3$

X = 3: observations $\hat\pi = 0.5$, $\hat\pi =0.53$

Say I have a logistic model, $$ Y_i \sim Bernoulli(logistic(\beta_0 + \beta_1 X_i)), $$ where $logistic(x) = e^x/(1+e^x)$ as usual.

I have

  1. $n$ observations $Y_i \in \{0,1\}$ ($i = 1, \ldots, n$), and also
  2. $m$ independent guestimates $\hat\pi_i$ ($i = n+1,\ldots,m+n$) of the probabilities $\pi_i = logistic(\beta_0 + \beta_1 X_i)$. Let's be generous and say $\hat\pi_i \sim N(\pi_i, \sigma^2)$, where $\sigma^2$ is small (and known, for the sake of argument).

I want to use all this information to get better estimates $\hat\beta_0$ and $\hat\beta_1$ for the model parameters.

I can think of a couple of approaches but this is surely a standard problem. If yes, what is it called? Any useful references?

Thanks!

Say I have a logistic model, $$ Y_i \sim Bernoulli(logistic(\beta_0 + \beta_1 X_i)), $$ where $logistic(x) = e^x/(1+e^x)$ as usual.

I have

  1. $n$ observations $Y_i \in \{0,1\}$ ($i = 1, \ldots, n$), and also
  2. $m$ independent guestimates $\hat\pi_i$ ($i = n+1,\ldots,m+n$) of the probabilities $\pi_i = logistic(\beta_0 + \beta_1 X_i)$. Let's be generous and say $\hat\pi_i \sim N(\pi_i, \sigma^2)$, where $\sigma^2$ is small (and known, for the sake of argument).

I want to use all this information to get better estimates $\hat\beta_0$ and $\hat\beta_1$ for the model parameters.

I can think of a couple of approaches but this is surely a standard problem. If yes, what is it called? Any useful references?

Thanks!

Edit: so the sample data might look like:

X = -1: observations Y= 0,0,1

X = -2: observations Y= 0,1, $\hat\pi = 0.3$

X = 3: observations $\hat\pi = 0.5$, $\hat\pi =0.53$

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P.Windridge
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Hybrid estimation for logistic model

Say I have a logistic model, $$ Y_i \sim Bernoulli(logistic(\beta_0 + \beta_1 X_i)), $$ where $logistic(x) = e^x/(1+e^x)$ as usual.

I have

  1. $n$ observations $Y_i \in \{0,1\}$ ($i = 1, \ldots, n$), and also
  2. $m$ independent guestimates $\hat\pi_i$ ($i = n+1,\ldots,m+n$) of the probabilities $\pi_i = logistic(\beta_0 + \beta_1 X_i)$. Let's be generous and say $\hat\pi_i \sim N(\pi_i, \sigma^2)$, where $\sigma^2$ is small (and known, for the sake of argument).

I want to use all this information to get better estimates $\hat\beta_0$ and $\hat\beta_1$ for the model parameters.

I can think of a couple of approaches but this is surely a standard problem. If yes, what is it called? Any useful references?

Thanks!