Mixing
How to find most likely family of probability density polynomials given prior data?
$begingroup$
If we imagine we have a bunch of polynomials $P_n(t)$, each describe probability densities that an event happening at some time $t$ and we know that exactly one of the $N$ events will happen during this time.
$$int_{0}^{1}sum_n P_n(t)dt = 1, hspace{1cm}tin[0,1]$$
And we have the following events happening at time points $t_k$ at polynomial $n_k$.
Which mathematical techniques can we use to find the most likely polynomials ${P_n}$ given we know timepoints and polynomial for each event that has occurred?
It seems I will need to try to explain better what I mean.
Once an event occurs at one of the $n$ polynomials, probability density rakes up to 100% at this place, ok?
$$P_{m_k}(t_k) = delta_k(t-t_k)$$
But which polynomial $m_k$ of the $n$ it happens for can vary for each new sample: $k$, in this sense it is like a mixture as @nathan asked about. Every new sample could be from any of the $n$ polynomials and together they make up to 100 percent. But for every new sample we restart to count from $t=0$.
Assume we get $N$ samples at $left{P_{m_k}(t_k)right}$ So we may want to minimize in some sense
$$sum_{k=1}^{N}|P_{l}(t) - delta(t-t_k)cdot delta(m_k-l)|$$
probability analysis polynomials probability-distributions optimization
asked Jan 18 at 14:40
mathreadlermathreadler
15k72263
$endgroup$
add a comment |
$begingroup$
If we imagine we have a bunch of polynomials $P_n(t)$, each describe probability densities that an event happening at some time $t$ and we know that exactly one of the $N$ events will happen during this time.
$$int_{0}^{1}sum_n P_n(t)dt = 1, hspace{1cm}tin[0,1]$$
And we have the following events happening at time points $t_k$ at polynomial $n_k$.
Which mathematical techniques can we use to find the most likely polynomials ${P_n}$ given we know timepoints and polynomial for each event that has occurred?
It seems I will need to try to explain better what I mean.
Once an event occurs at one of the $n$ polynomials, probability density rakes up to 100% at this place, ok?
$$P_{m_k}(t_k) = delta_k(t-t_k)$$
But which polynomial $m_k$ of the $n$ it happens for can vary for each new sample: $k$, in this sense it is like a mixture as @nathan asked about. Every new sample could be from any of the $n$ polynomials and together they make up to 100 percent. But for every new sample we restart to count from $t=0$.
Assume we get $N$ samples at $left{P_{m_k}(t_k)right}$ So we may want to minimize in some sense
$$sum_{k=1}^{N}|P_{l}(t) - delta(t-t_k)cdot delta(m_k-l)|$$
probability analysis polynomials probability-distributions optimization
asked Jan 18 at 14:40
mathreadlermathreadler
15k72263
$endgroup$
$begingroup$
So, $P_n$ isn't necessarily a true probability density function, it's one scaled by some $p_nin (0,1]$? In which case $sum_{n}P_n$ is a mixture distribution?
$endgroup$
– nathan.j.mcdougall
Jan 18 at 18:23
$begingroup$
@nathan.j.mcdougall yep it is supposed to be a mixture of $N$ polynomials. Sorry that I was not clear enough when formulating the question.
$endgroup$
– mathreadler
Jan 19 at 7:49
$begingroup$
I'm still struggling to understand the question after you've updated it. Are you saying now that we don't just know $t_k$, we also know the "gaps" between events of $t_{k}-t_{k-1}$? And I absolutely have lost you in your sum of norms expression, I'm afraid.
$endgroup$
– nathan.j.mcdougall
Jan 20 at 2:51
add a comment |
$begingroup$
If we imagine we have a bunch of polynomials $P_n(t)$, each describe probability densities that an event happening at some time $t$ and we know that exactly one of the $N$ events will happen during this time.
$$int_{0}^{1}sum_n P_n(t)dt = 1, hspace{1cm}tin[0,1]$$
And we have the following events happening at time points $t_k$ at polynomial $n_k$.
Which mathematical techniques can we use to find the most likely polynomials ${P_n}$ given we know timepoints and polynomial for each event that has occurred?
It seems I will need to try to explain better what I mean.
Once an event occurs at one of the $n$ polynomials, probability density rakes up to 100% at this place, ok?
$$P_{m_k}(t_k) = delta_k(t-t_k)$$
But which polynomial $m_k$ of the $n$ it happens for can vary for each new sample: $k$, in this sense it is like a mixture as @nathan asked about. Every new sample could be from any of the $n$ polynomials and together they make up to 100 percent. But for every new sample we restart to count from $t=0$.
Assume we get $N$ samples at $left{P_{m_k}(t_k)right}$ So we may want to minimize in some sense
$$sum_{k=1}^{N}|P_{l}(t) - delta(t-t_k)cdot delta(m_k-l)|$$
probability analysis polynomials probability-distributions optimization
asked Jan 18 at 14:40
mathreadlermathreadler
15k72263
$endgroup$
If we imagine we have a bunch of polynomials $P_n(t)$, each describe probability densities that an event happening at some time $t$ and we know that exactly one of the $N$ events will happen during this time.
$$int_{0}^{1}sum_n P_n(t)dt = 1, hspace{1cm}tin[0,1]$$
And we have the following events happening at time points $t_k$ at polynomial $n_k$.
Which mathematical techniques can we use to find the most likely polynomials ${P_n}$ given we know timepoints and polynomial for each event that has occurred?
It seems I will need to try to explain better what I mean.
Once an event occurs at one of the $n$ polynomials, probability density rakes up to 100% at this place, ok?
$$P_{m_k}(t_k) = delta_k(t-t_k)$$
But which polynomial $m_k$ of the $n$ it happens for can vary for each new sample: $k$, in this sense it is like a mixture as @nathan asked about. Every new sample could be from any of the $n$ polynomials and together they make up to 100 percent. But for every new sample we restart to count from $t=0$.
Assume we get $N$ samples at $left{P_{m_k}(t_k)right}$ So we may want to minimize in some sense
$$sum_{k=1}^{N}|P_{l}(t) - delta(t-t_k)cdot delta(m_k-l)|$$
probability analysis polynomials probability-distributions optimization
probability analysis polynomials probability-distributions optimization
asked Jan 18 at 14:40
mathreadlermathreadler
15k72263
asked Jan 18 at 14:40
mathreadlermathreadler
15k72263
edited Jan 19 at 7:44
mathreadler
asked Jan 18 at 14:40
mathreadlermathreadler
15k72263
asked Jan 18 at 14:40
mathreadlermathreadler
15k72263
asked Jan 18 at 14:40
mathreadlermathreadler
15k72263
15k72263
$begingroup$
So, $P_n$ isn't necessarily a true probability density function, it's one scaled by some $p_nin (0,1]$? In which case $sum_{n}P_n$ is a mixture distribution?
$endgroup$
– nathan.j.mcdougall
Jan 18 at 18:23
$begingroup$
@nathan.j.mcdougall yep it is supposed to be a mixture of $N$ polynomials. Sorry that I was not clear enough when formulating the question.
$endgroup$
– mathreadler
Jan 19 at 7:49
$begingroup$
I'm still struggling to understand the question after you've updated it. Are you saying now that we don't just know $t_k$, we also know the "gaps" between events of $t_{k}-t_{k-1}$? And I absolutely have lost you in your sum of norms expression, I'm afraid.
$endgroup$
– nathan.j.mcdougall
Jan 20 at 2:51
add a comment |
$begingroup$
So, $P_n$ isn't necessarily a true probability density function, it's one scaled by some $p_nin (0,1]$? In which case $sum_{n}P_n$ is a mixture distribution?
$endgroup$
– nathan.j.mcdougall
Jan 18 at 18:23
$begingroup$
@nathan.j.mcdougall yep it is supposed to be a mixture of $N$ polynomials. Sorry that I was not clear enough when formulating the question.
$endgroup$
– mathreadler
Jan 19 at 7:49
$begingroup$
I'm still struggling to understand the question after you've updated it. Are you saying now that we don't just know $t_k$, we also know the "gaps" between events of $t_{k}-t_{k-1}$? And I absolutely have lost you in your sum of norms expression, I'm afraid.
$endgroup$
– nathan.j.mcdougall
Jan 20 at 2:51
$begingroup$
So, $P_n$ isn't necessarily a true probability density function, it's one scaled by some $p_nin (0,1]$? In which case $sum_{n}P_n$ is a mixture distribution?
$endgroup$
– nathan.j.mcdougall
Jan 18 at 18:23
$begingroup$
So, $P_n$ isn't necessarily a true probability density function, it's one scaled by some $p_nin (0,1]$? In which case $sum_{n}P_n$ is a mixture distribution?
$endgroup$
– nathan.j.mcdougall
Jan 18 at 18:23
$begingroup$
@nathan.j.mcdougall yep it is supposed to be a mixture of $N$ polynomials. Sorry that I was not clear enough when formulating the question.
$endgroup$
– mathreadler
Jan 19 at 7:49
$begingroup$
@nathan.j.mcdougall yep it is supposed to be a mixture of $N$ polynomials. Sorry that I was not clear enough when formulating the question.
$endgroup$
– mathreadler
Jan 19 at 7:49
$begingroup$
I'm still struggling to understand the question after you've updated it. Are you saying now that we don't just know $t_k$, we also know the "gaps" between events of $t_{k}-t_{k-1}$? And I absolutely have lost you in your sum of norms expression, I'm afraid.
$endgroup$
– nathan.j.mcdougall
Jan 20 at 2:51
$begingroup$
I'm still struggling to understand the question after you've updated it. Are you saying now that we don't just know $t_k$, we also know the "gaps" between events of $t_{k}-t_{k-1}$? And I absolutely have lost you in your sum of norms expression, I'm afraid.
$endgroup$
– nathan.j.mcdougall
Jan 20 at 2:51
add a comment |
1 Answer
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votes
$begingroup$
The probability that the first event occurs at time $t$ using polynomial $P_n$ is given by the product $P_n(t)P_{n_1}(t_1-t)$ where $n_1$ is the polynomial used for the event immediately following, $t_1$ is the time of that event, and we assume that all events are independent.
The actual probability is then given by integrating that expression,
$$int_{max{1,t_1}-1}^{min{1,t_1}}P_n(t)P_{n_1}(t_1-t),mathrm{d}t$$
and so we need only check each of the $n$ possible integrals to see which is largest, and hence likeliest. That integral should be very easy to evaluate for polynomials, of course.
answered Jan 18 at 19:11
nathan.j.mcdougallnathan.j.mcdougall
1,519818
$endgroup$
$begingroup$
Hmm I should have been more careful at describing the problem, it seems.
$endgroup$
– mathreadler
Jan 19 at 6:57
add a comment |
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1 Answer
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$begingroup$
The probability that the first event occurs at time $t$ using polynomial $P_n$ is given by the product $P_n(t)P_{n_1}(t_1-t)$ where $n_1$ is the polynomial used for the event immediately following, $t_1$ is the time of that event, and we assume that all events are independent.
The actual probability is then given by integrating that expression,
$$int_{max{1,t_1}-1}^{min{1,t_1}}P_n(t)P_{n_1}(t_1-t),mathrm{d}t$$
and so we need only check each of the $n$ possible integrals to see which is largest, and hence likeliest. That integral should be very easy to evaluate for polynomials, of course.
answered Jan 18 at 19:11
nathan.j.mcdougallnathan.j.mcdougall
1,519818
$endgroup$
$begingroup$
Hmm I should have been more careful at describing the problem, it seems.
$endgroup$
– mathreadler
Jan 19 at 6:57
add a comment |
$begingroup$
The probability that the first event occurs at time $t$ using polynomial $P_n$ is given by the product $P_n(t)P_{n_1}(t_1-t)$ where $n_1$ is the polynomial used for the event immediately following, $t_1$ is the time of that event, and we assume that all events are independent.
The actual probability is then given by integrating that expression,
$$int_{max{1,t_1}-1}^{min{1,t_1}}P_n(t)P_{n_1}(t_1-t),mathrm{d}t$$
and so we need only check each of the $n$ possible integrals to see which is largest, and hence likeliest. That integral should be very easy to evaluate for polynomials, of course.
answered Jan 18 at 19:11
nathan.j.mcdougallnathan.j.mcdougall
1,519818
$endgroup$
$begingroup$
Hmm I should have been more careful at describing the problem, it seems.
$endgroup$
– mathreadler
Jan 19 at 6:57
add a comment |
$begingroup$
The probability that the first event occurs at time $t$ using polynomial $P_n$ is given by the product $P_n(t)P_{n_1}(t_1-t)$ where $n_1$ is the polynomial used for the event immediately following, $t_1$ is the time of that event, and we assume that all events are independent.
The actual probability is then given by integrating that expression,
$$int_{max{1,t_1}-1}^{min{1,t_1}}P_n(t)P_{n_1}(t_1-t),mathrm{d}t$$
and so we need only check each of the $n$ possible integrals to see which is largest, and hence likeliest. That integral should be very easy to evaluate for polynomials, of course.
answered Jan 18 at 19:11
nathan.j.mcdougallnathan.j.mcdougall
1,519818
$endgroup$
The probability that the first event occurs at time $t$ using polynomial $P_n$ is given by the product $P_n(t)P_{n_1}(t_1-t)$ where $n_1$ is the polynomial used for the event immediately following, $t_1$ is the time of that event, and we assume that all events are independent.
The actual probability is then given by integrating that expression,
$$int_{max{1,t_1}-1}^{min{1,t_1}}P_n(t)P_{n_1}(t_1-t),mathrm{d}t$$
and so we need only check each of the $n$ possible integrals to see which is largest, and hence likeliest. That integral should be very easy to evaluate for polynomials, of course.
answered Jan 18 at 19:11
nathan.j.mcdougallnathan.j.mcdougall
1,519818
answered Jan 18 at 19:11
nathan.j.mcdougallnathan.j.mcdougall
1,519818
answered Jan 18 at 19:11
nathan.j.mcdougallnathan.j.mcdougall
1,519818
answered Jan 18 at 19:11
nathan.j.mcdougallnathan.j.mcdougall
1,519818
1,519818
$begingroup$
Hmm I should have been more careful at describing the problem, it seems.
$endgroup$
– mathreadler
Jan 19 at 6:57
add a comment |
$begingroup$
Hmm I should have been more careful at describing the problem, it seems.
$endgroup$
– mathreadler
Jan 19 at 6:57
$begingroup$
Hmm I should have been more careful at describing the problem, it seems.
$endgroup$
– mathreadler
Jan 19 at 6:57
$begingroup$
Hmm I should have been more careful at describing the problem, it seems.
$endgroup$
– mathreadler
Jan 19 at 6:57
add a comment |
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$begingroup$
So, $P_n$ isn't necessarily a true probability density function, it's one scaled by some $p_nin (0,1]$? In which case $sum_{n}P_n$ is a mixture distribution?
$endgroup$
– nathan.j.mcdougall
Jan 18 at 18:23
$begingroup$
@nathan.j.mcdougall yep it is supposed to be a mixture of $N$ polynomials. Sorry that I was not clear enough when formulating the question.
$endgroup$
– mathreadler
Jan 19 at 7:49
$begingroup$
I'm still struggling to understand the question after you've updated it. Are you saying now that we don't just know $t_k$, we also know the "gaps" between events of $t_{k}-t_{k-1}$? And I absolutely have lost you in your sum of norms expression, I'm afraid.
$endgroup$
– nathan.j.mcdougall
Jan 20 at 2:51