How to find complete log likelihood for mixture of PPCA
$begingroup$
In Appendix C of a paper by Michael E. Tipping and Christopher M. Bishop about mixture models for probabilistic PCA, the probability of a single data vector $mathbf{t}$ is expressed as a mixture of PCA models (equation 69):
$$
p(mathbf{t}) = sum_{i=1}^Mpi_i p(mathbf{t}|i)
$$
where $pi$ is the mixing proportion and $p(mathbf{t}|i)$ is a single probabilistic PCA model.
The model underlying the probabilistic PCA method is (equation 2)
$$
mathbf{t} = mathbf{Wx} + boldsymbolmu + boldsymbolepsilon.
$$
Where $mathbf{x}$ is a latent variable. By introducing a new set of variables $z_{ni}$ "labelling which model is responsible for generating each data point $mathbf{t}_n$", Bishop formulates the complete log likelihood as (equation 70):
$$
mathcal{L}_C = sum_{n=1}^Nsum_{i=1}^Mz_{ni}ln{pi_ip(mathbf{t}_n, mathbf{x}_{ni})}.
$$
I would like to understand how he derives this expression as he doesn't provide a solution himself. How is this expression for the complete log likelihood found?
probability-theory machine-learning
asked Jan 4 at 11:07
SandiSandi
262112
$endgroup$
add a comment |
$begingroup$
In Appendix C of a paper by Michael E. Tipping and Christopher M. Bishop about mixture models for probabilistic PCA, the probability of a single data vector $mathbf{t}$ is expressed as a mixture of PCA models (equation 69):
$$
p(mathbf{t}) = sum_{i=1}^Mpi_i p(mathbf{t}|i)
$$
where $pi$ is the mixing proportion and $p(mathbf{t}|i)$ is a single probabilistic PCA model.
The model underlying the probabilistic PCA method is (equation 2)
$$
mathbf{t} = mathbf{Wx} + boldsymbolmu + boldsymbolepsilon.
$$
Where $mathbf{x}$ is a latent variable. By introducing a new set of variables $z_{ni}$ "labelling which model is responsible for generating each data point $mathbf{t}_n$", Bishop formulates the complete log likelihood as (equation 70):
$$
mathcal{L}_C = sum_{n=1}^Nsum_{i=1}^Mz_{ni}ln{pi_ip(mathbf{t}_n, mathbf{x}_{ni})}.
$$
I would like to understand how he derives this expression as he doesn't provide a solution himself. How is this expression for the complete log likelihood found?
probability-theory machine-learning
asked Jan 4 at 11:07
SandiSandi
262112
$endgroup$
add a comment |
$begingroup$
In Appendix C of a paper by Michael E. Tipping and Christopher M. Bishop about mixture models for probabilistic PCA, the probability of a single data vector $mathbf{t}$ is expressed as a mixture of PCA models (equation 69):
$$
p(mathbf{t}) = sum_{i=1}^Mpi_i p(mathbf{t}|i)
$$
where $pi$ is the mixing proportion and $p(mathbf{t}|i)$ is a single probabilistic PCA model.
The model underlying the probabilistic PCA method is (equation 2)
$$
mathbf{t} = mathbf{Wx} + boldsymbolmu + boldsymbolepsilon.
$$
Where $mathbf{x}$ is a latent variable. By introducing a new set of variables $z_{ni}$ "labelling which model is responsible for generating each data point $mathbf{t}_n$", Bishop formulates the complete log likelihood as (equation 70):
$$
mathcal{L}_C = sum_{n=1}^Nsum_{i=1}^Mz_{ni}ln{pi_ip(mathbf{t}_n, mathbf{x}_{ni})}.
$$
I would like to understand how he derives this expression as he doesn't provide a solution himself. How is this expression for the complete log likelihood found?
probability-theory machine-learning
asked Jan 4 at 11:07
SandiSandi
262112
$endgroup$
In Appendix C of a paper by Michael E. Tipping and Christopher M. Bishop about mixture models for probabilistic PCA, the probability of a single data vector $mathbf{t}$ is expressed as a mixture of PCA models (equation 69):
$$
p(mathbf{t}) = sum_{i=1}^Mpi_i p(mathbf{t}|i)
$$
where $pi$ is the mixing proportion and $p(mathbf{t}|i)$ is a single probabilistic PCA model.
The model underlying the probabilistic PCA method is (equation 2)
$$
mathbf{t} = mathbf{Wx} + boldsymbolmu + boldsymbolepsilon.
$$
Where $mathbf{x}$ is a latent variable. By introducing a new set of variables $z_{ni}$ "labelling which model is responsible for generating each data point $mathbf{t}_n$", Bishop formulates the complete log likelihood as (equation 70):
$$
mathcal{L}_C = sum_{n=1}^Nsum_{i=1}^Mz_{ni}ln{pi_ip(mathbf{t}_n, mathbf{x}_{ni})}.
$$
I would like to understand how he derives this expression as he doesn't provide a solution himself. How is this expression for the complete log likelihood found?
probability-theory machine-learning
probability-theory machine-learning
asked Jan 4 at 11:07
SandiSandi
262112
asked Jan 4 at 11:07
SandiSandi
262112
asked Jan 4 at 11:07
SandiSandi
262112
asked Jan 4 at 11:07
SandiSandi
262112
asked Jan 4 at 11:07
SandiSandi
262112
262112
add a comment |
add a comment |
1 Answer
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Let's concentrate for the time being on the $n^{rm th}$ datapoint, $mathbf t_n$. Suppose this datapoint is generated from the $i_n^{rm th}$ model. Then $$ z_{ni} = begin{cases} 1 & {rm if } i = i_n \ 0 & {rm otherwise}end{cases}.$$Thus we have
$$sum_{i=1}^M z_{ni} ln left(pi_i p(mathbf t_n ,mathbf x_{ni}) right) = ln left( pi_{i_n} p(mathbf t_n , mathbf x_{n{i_n}})right).$$
The expression on the right-hand side is log-likelihood for the $n$th datapoint. To spell it out:
$pi_{i_n}$ is the probability that the $n^{rm th}$ datapoint is generated by the $i_n^{rm th}$ model.
$p(mathbf t_n, mathbf x_{ni_n})$ is the probability of encountering this particular latent vector $mathbf x_{ni_n}$ and this particular visible vector $mathbf t_n$ for the $n^{rm th}$ datapoint, given that this datapoint is generated from the the $i_n^{rm th}$ model. [In fact, $$p(mathbf t_n, mathbf x_{ni_n}) = mathcal N(mathbf x_{ni_n} | mathbf 0, mathbf I) times mathcal N(mathbf t_n - mathbf W mathbf x_{ni_n} - mathbf mu_n |mathbf 0, sigma_n^2 mathbf I),$$ assuming that $mathbf epsilon_{n} sim mathcal N(mathbf 0, sigma_n^2 mathbf I)$.]
Since the datapoints are generated independently, the log-likelihood for the entire dataset is a sum over the log-likelihoods for the individual datapoints, giving the desired result.
answered Jan 6 at 20:56
Kenny WongKenny Wong
19.1k21441
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add a comment |
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1 Answer
1
active
oldest
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1 Answer
1
active
oldest
votes
active
oldest
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active
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votes
$begingroup$
Let's concentrate for the time being on the $n^{rm th}$ datapoint, $mathbf t_n$. Suppose this datapoint is generated from the $i_n^{rm th}$ model. Then $$ z_{ni} = begin{cases} 1 & {rm if } i = i_n \ 0 & {rm otherwise}end{cases}.$$Thus we have
$$sum_{i=1}^M z_{ni} ln left(pi_i p(mathbf t_n ,mathbf x_{ni}) right) = ln left( pi_{i_n} p(mathbf t_n , mathbf x_{n{i_n}})right).$$
The expression on the right-hand side is log-likelihood for the $n$th datapoint. To spell it out:
$pi_{i_n}$ is the probability that the $n^{rm th}$ datapoint is generated by the $i_n^{rm th}$ model.
$p(mathbf t_n, mathbf x_{ni_n})$ is the probability of encountering this particular latent vector $mathbf x_{ni_n}$ and this particular visible vector $mathbf t_n$ for the $n^{rm th}$ datapoint, given that this datapoint is generated from the the $i_n^{rm th}$ model. [In fact, $$p(mathbf t_n, mathbf x_{ni_n}) = mathcal N(mathbf x_{ni_n} | mathbf 0, mathbf I) times mathcal N(mathbf t_n - mathbf W mathbf x_{ni_n} - mathbf mu_n |mathbf 0, sigma_n^2 mathbf I),$$ assuming that $mathbf epsilon_{n} sim mathcal N(mathbf 0, sigma_n^2 mathbf I)$.]
Since the datapoints are generated independently, the log-likelihood for the entire dataset is a sum over the log-likelihoods for the individual datapoints, giving the desired result.
answered Jan 6 at 20:56
Kenny WongKenny Wong
19.1k21441
$endgroup$
add a comment |
$begingroup$
Let's concentrate for the time being on the $n^{rm th}$ datapoint, $mathbf t_n$. Suppose this datapoint is generated from the $i_n^{rm th}$ model. Then $$ z_{ni} = begin{cases} 1 & {rm if } i = i_n \ 0 & {rm otherwise}end{cases}.$$Thus we have
$$sum_{i=1}^M z_{ni} ln left(pi_i p(mathbf t_n ,mathbf x_{ni}) right) = ln left( pi_{i_n} p(mathbf t_n , mathbf x_{n{i_n}})right).$$
The expression on the right-hand side is log-likelihood for the $n$th datapoint. To spell it out:
$pi_{i_n}$ is the probability that the $n^{rm th}$ datapoint is generated by the $i_n^{rm th}$ model.
$p(mathbf t_n, mathbf x_{ni_n})$ is the probability of encountering this particular latent vector $mathbf x_{ni_n}$ and this particular visible vector $mathbf t_n$ for the $n^{rm th}$ datapoint, given that this datapoint is generated from the the $i_n^{rm th}$ model. [In fact, $$p(mathbf t_n, mathbf x_{ni_n}) = mathcal N(mathbf x_{ni_n} | mathbf 0, mathbf I) times mathcal N(mathbf t_n - mathbf W mathbf x_{ni_n} - mathbf mu_n |mathbf 0, sigma_n^2 mathbf I),$$ assuming that $mathbf epsilon_{n} sim mathcal N(mathbf 0, sigma_n^2 mathbf I)$.]
Since the datapoints are generated independently, the log-likelihood for the entire dataset is a sum over the log-likelihoods for the individual datapoints, giving the desired result.
answered Jan 6 at 20:56
Kenny WongKenny Wong
19.1k21441
$endgroup$
add a comment |
$begingroup$
Let's concentrate for the time being on the $n^{rm th}$ datapoint, $mathbf t_n$. Suppose this datapoint is generated from the $i_n^{rm th}$ model. Then $$ z_{ni} = begin{cases} 1 & {rm if } i = i_n \ 0 & {rm otherwise}end{cases}.$$Thus we have
$$sum_{i=1}^M z_{ni} ln left(pi_i p(mathbf t_n ,mathbf x_{ni}) right) = ln left( pi_{i_n} p(mathbf t_n , mathbf x_{n{i_n}})right).$$
The expression on the right-hand side is log-likelihood for the $n$th datapoint. To spell it out:
$pi_{i_n}$ is the probability that the $n^{rm th}$ datapoint is generated by the $i_n^{rm th}$ model.
$p(mathbf t_n, mathbf x_{ni_n})$ is the probability of encountering this particular latent vector $mathbf x_{ni_n}$ and this particular visible vector $mathbf t_n$ for the $n^{rm th}$ datapoint, given that this datapoint is generated from the the $i_n^{rm th}$ model. [In fact, $$p(mathbf t_n, mathbf x_{ni_n}) = mathcal N(mathbf x_{ni_n} | mathbf 0, mathbf I) times mathcal N(mathbf t_n - mathbf W mathbf x_{ni_n} - mathbf mu_n |mathbf 0, sigma_n^2 mathbf I),$$ assuming that $mathbf epsilon_{n} sim mathcal N(mathbf 0, sigma_n^2 mathbf I)$.]
Since the datapoints are generated independently, the log-likelihood for the entire dataset is a sum over the log-likelihoods for the individual datapoints, giving the desired result.
answered Jan 6 at 20:56
Kenny WongKenny Wong
19.1k21441
$endgroup$
Let's concentrate for the time being on the $n^{rm th}$ datapoint, $mathbf t_n$. Suppose this datapoint is generated from the $i_n^{rm th}$ model. Then $$ z_{ni} = begin{cases} 1 & {rm if } i = i_n \ 0 & {rm otherwise}end{cases}.$$Thus we have
$$sum_{i=1}^M z_{ni} ln left(pi_i p(mathbf t_n ,mathbf x_{ni}) right) = ln left( pi_{i_n} p(mathbf t_n , mathbf x_{n{i_n}})right).$$
The expression on the right-hand side is log-likelihood for the $n$th datapoint. To spell it out:
$pi_{i_n}$ is the probability that the $n^{rm th}$ datapoint is generated by the $i_n^{rm th}$ model.
$p(mathbf t_n, mathbf x_{ni_n})$ is the probability of encountering this particular latent vector $mathbf x_{ni_n}$ and this particular visible vector $mathbf t_n$ for the $n^{rm th}$ datapoint, given that this datapoint is generated from the the $i_n^{rm th}$ model. [In fact, $$p(mathbf t_n, mathbf x_{ni_n}) = mathcal N(mathbf x_{ni_n} | mathbf 0, mathbf I) times mathcal N(mathbf t_n - mathbf W mathbf x_{ni_n} - mathbf mu_n |mathbf 0, sigma_n^2 mathbf I),$$ assuming that $mathbf epsilon_{n} sim mathcal N(mathbf 0, sigma_n^2 mathbf I)$.]
Since the datapoints are generated independently, the log-likelihood for the entire dataset is a sum over the log-likelihoods for the individual datapoints, giving the desired result.
answered Jan 6 at 20:56
Kenny WongKenny Wong
19.1k21441
answered Jan 6 at 20:56
Kenny WongKenny Wong
19.1k21441
answered Jan 6 at 20:56
Kenny WongKenny Wong
19.1k21441
answered Jan 6 at 20:56
Kenny WongKenny Wong
19.1k21441
19.1k21441
add a comment |
add a comment |
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